Department of Mathematics and Statistics, Binghamton University seminars:arit

Fall 2014

2017-01-25T16:01:08-04:00

Fall 2014

September 12

Speaker: Alexander Borisov (Binghamton University)
Title: Dynamical properties of reductions of integer polynomial maps.

Abstract: Integer polynomial maps and their reductions module primes are among the simplest mathematical objects. Yet very little is known about their dynamical properties. I will describe several interesting examples and pose some questions and conjectures. Most of the talk will be accessible to most graduate sudents.

September 19

Speaker: Alexander Borisov (Binghamton University)
Title: Integer ratios of factorials, cyclic quotient singularities, Nyman-Beurling approach to the Riemann Hypothesis, and algebraic hypergeometric series.

Abstract: The topics listed in the title are in fact very closely related to each other. I will explain the connections between them and outline some natural open questions in this area.

October 10

Speaker: Robert Benedetto (Amherst College)
Title: Bounding Postcritically Finite Maps over Number Fields.

Abstract: A rational function f(z) is said to be postcritically finite (PCF) if all of its critical points have finite forward orbit under repeated application of f. Examples include monomials x^n, Chebyshev polynomials, and Lattes maps (which arise from elliptic curves), but there are many others. In this talk, we will discuss joint work with Ingram, Jones, and Levy, proving that for any number field K and degree d>1, there are only finitely many non-Lattes PCF maps of degree d in K(z). The proof rests on a deep result of McMullen combined with some p-adic analysis.

October 17

Speaker: Evan Dummit (University of Rochester)
Title: Counting Number Fields by Discriminant.

Abstract: The problem of analyzing the number of number field extensions L/K with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Ellenberg-Venkatesh, Kable-Yukie, and (especially) Bhargava. I will give an overview of the history of this problem and what results are known (or conjectured), and then discuss my work on a series of generalizations, using similar techniques to Ellenberg-Venkatesh, for giving an upper bound on the number of extensions L/K with fixed degree, bounded relative discriminant, and specified Galois closure.

October 31

Speaker: Patrick Milano (Binghamton University)
Title: Elliptic Curves and Mordell's Theorem

Abstract: The first part of this talk will be an introduction to elliptic curves. In the second part of the talk I'll outline a proof of Mordell's Theorem, which says that the group of rational points on an elliptic curve over Q is finitely generated.

November 4 (Cross-listing with the Combinatorics Seminar, 1:15-2:15 in OW 100-E)

Speaker: Emanuele Delucchi (Fribourg, Switzerland)
Title: Toric Arrangements – Towards Setting Up a Combinatorial Theory

Abstract: Recent work of De Concini, Procesi, and Vergne on vector partition functions gave a new impulse to the study of toric arrangements from algebraic, topological, and combinatorial points of view. In this context, many new discrete structures have appeared in the literature, each describing some aspect of the theory (i.e., either the arithmetic-algebraic one or the topological one) and trying to mirror the combinatorial framework which revolves around arrangements of hyperplanes. I will give a quick overview of the state of the art and, taking inspiration from some recent results of topological flavor, I will try to suggest a possible approach towards unifying these different objects.

November 7

Speaker: Micah Loverro (Binghamton University)
Title: Reducing an elliptic curve modulo p and the Sato-Tate conjecture.

Abstract: Click here.

November 14

Speaker: Farbod Shokrieh (Cornell University)
Title: Non-archimedean abelian varieties, uniformization, and faithful tropicalization.

Abstract: The skeleton of a Berkovich analytic space is a subspace onto which the whole space deformation retracts. For an abelian variety, the skeleton is a real torus with an “integral structure”. I will discuss “faithful tropicalization” of abelian varieties in terms of non-archimedean and tropical theta functions. The solution relies on interesting combinatorial facts about lattices, matroids, and Voronoi decompositions. This talk is based on joint projects with Tyler Foster, Joe Rabinoff, and Alejandro Soto.

December 5

Speaker: Daniel Vallieres (Binghamton University)
Title: Abelian Artin L-functions at s=0.

Abstract: We'll define some new arithmetical objects, called evaluators, which have not been studied before. These evaluators are intimately linked with the special value s=0 of S-truncated L-functions attached to abelian extensions of number fields. We'll see that a certain rationality property of these evaluators is equivalent to Stark's conjecture over Q as formulated by Tate. If time permits, we will also start to look at some of their arithmetical properties which is more delicate.

December 12

Speaker: Amod Agashe (Florida State University)
Title: A generalization of Kronecker's first limit formula.

Abstract: The classical Kronecker's first limit formula gives the constant term in the Laurent expansion of a certain two variable Eisenstein series, which in turn gives the constant term in the Laurent expansion of the zeta function of a quadratic imaginary field. We will recall this formula and sketch how it can be generalized to more general Eisenstein series and zeta functions of arbitrary number fields.

Fall 2015

2017-01-25T15:56:38-04:00

Fall 2015

September 4

Speaker: FirstName LastName (Some University)
Title: Organizational Meeting

Abstract: We will discuss schedule and speakers for this semester

September 11

Speaker: Jaiung Jun (Binghamton University)
Title: Algebraic geometry over semi-structures and hyper-structures of characteristic one (Part 1)

Abstract: Algebraic geometry over the “field with one element” (F_1-geometry) is the recent field of mathematics. In this talk, we introduce motivations and the meaning of working in characteristic one as well as recent developments. If time permits, I will also introduce a rather exotic algebraic structure called hyperfield.

September 18

Speaker: Jaiung Jun (Binghamton University)
Title: Algebraic geometry over semi-structures and hyper-structures of characteristic one (Part 2)

Abstract: This is the second part of the talk given at Sep 11. In this talk, we introduce a notion of algebraic geometry over semifield and hyperfield in connection to tropical geometry.

September 25

Speaker: Jaiung Jun (Binghamton University)
Title: Algebraic geometry over semi-structures and hyper-structures of characteristic one (Part 3)

Abstract: We introduce basic notions of tropical geometry and semiring schemes. Then we generalize $\hat{\textrm{C}}$ech cohomology theory and invertible sheaves to semiring schemes. In particular, when $X=\mathbb{P}^n_M$, a projective space over a totally ordered idempotent semifield $M$, we show that $\hat{\textrm{H}}^n(X,\mathcal{O}_X)$ is in agreement with the classical computation for all $n$. Finally, we classify all invertible sheaves on $X=\mathbb{P}^n_M$ by computing $\textrm{Pic}(X)$ explicitly.

October 2

Speaker: Kalina Mincheva (Johns Hopkins University)
Title: Nullstellensatz for tropical polynomials

Abstract: We improve on a result of A. Bertram and R. Easton which can be regarded as a Nullstellensatz for tropical polynomials. In order to do that we give a new definition of prime congruences in additively idempotent semiring using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences we show that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. We give a complete description of prime congruences in the polynomial and Laurent polynomial semirings over the tropical semifield $\mathbb{R}_{max}$, the semifield $\mathbb{Z}_{max}$ and the Boolean semifield $\mathbb{B}$. The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. We show that the radical of every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension 1.

October 9

Speaker: Alexander Borisov (Binghamton University)
Title: Labeled trees, divisorial valuations at infinity, and two-dimensional Jacobian Conjecture

Abstract: Starting from a projective plane and performing several blowups of points at infinity, one can get various smooth compactifications of the affine plane. Each irreducible component of the complement of the affine plane gives a valuation on the field of rational functions in two variables. These valuations are called divisorial valuations, and they play an important role in many geometric approaches to the two-dimensional Jacobian Conjecture. We will introduce two discrete invariants of these valuations, and discuss their significance and limitations for the Jacobian Conjecture. Despite the algebro-geometric motivation, our methods are essentially combinatorial, based on the (dual) intersection graph of curves at infinity.

October 16

Speaker: Wushi Goldring (Washington University, St. Louis)
Title: Group-theoretical Hasse invariants

Abstract: I will explain some aspects of my joint work with J.-S. Koskivirta that I will not have time to discuss in my colloquium talk. Specifically, I will talk about our construction of what we call “group-theoretical Hasse invariants”. I will recall the classical Hasse invariant of elliptic curves in characteristic p>0 and explain how our construction offers both a generalization in several directions and a group-theoretic reinterpretation of the classical Hasse invariant. Time permitting, I will sketch how we apply these group-theoretical Hasse invariants to answer questions both about the geometry and about the cohomology of certain Shimura varieties modulo a prime p.

November 2

Speaker: Alexander Borisov (Binghamton University)
Title: Labeled trees, divisorial valuations at infinity, and two-dimensional Jacobian Conjecture (Part 2)

Abstract: This is a continuation of the October 9 talk. I will remind of the two invariants of divisorial valuations introduced then, and explain their importance for the study of hypothetical counterexamples to the Jacobian Conjecture. Besides purely combinatorial arguments, I will introduce some algebraic geometry techniques, like adjunction inequalities and inversion of adjunction.

November 9

Speaker: Alexander Borisov (Binghamton University)
Title: The abc-polynomials

Abstract: If a=b+c is a coprime triple of natural numbers, one can define a polynomial $f(x)=\frac{bx^a-ax^b+c}{(x-1)^2}$. The motivation behind this definition is a naive approach to the Masser-Oesterle abc conjecture: attempt to follow the proof in the geometric case by using quantum deformation of integers instead of differentiation of polynomials. I introduced these polynomials in a 1998 paper, and proved some results about them (in particular that most of them are irreducible). Not much happened, until these polynomials unexpectedly reappeared around 2004 in a graduate course problem by Joe Harris, that turned out to be unexpectedly hard. After several months of attempts by Jason Starr, Izzet Coskun and others, it was ultimately solved by Noam Elkies by a short and beautiful argument. A particular case of this problem appeared on 2014 Putnam exam, most probably by Elkies's suggestion, and, unsurprisingly, turned out to be unsolvable “in real time”. While the significance of these polynomials is unclear, I hope to convince you that they are interesting. In particular, I will present the Elkies's beautiful proof.

November 30

Speaker: Dan Collins (Cornell)
Title: Anticyclotomic p-adic L-functions and Ichino's formula I

Abstract: A p-adic L-function is an analytic function of a p-adic variable, which is characterized by interpolating special values of a classical (complex analytic) L-functions. I will give an introduction to where p-adic L-functions come from and why they are studied. I will then discuss about a certain type of “anticyclotomic p-adic L-function” of Bertolini-Darmon-Prasanna associated to a pair consisting of a classical modular form and a Hecke character of an imaginary quadratic field. Finally, I'll talk about recent work of Skinner that uses this p-adic L-function to obtain new results towards the Birch and Swinnerton-Dyer conjecture for elliptic curves of rank 1.

December 7

Speaker: Dan Collins (Cornell)
Title: Anticyclotomic p-adic L-functions and Ichino's formula II

Abstract: In this talk I will discuss how one actually constructs p-adic L-functions: the definition as “a p-adic analytic function with certain specified values” provides no clue as to whether such an analytic function actually exists! I'll discuss constructions of the “anticyclotomic p-adic L-function” of my previous talk - the classical Rankin-Selberg unfolding argument comes close to giving a direct construction, but it fails in the most interesting cases. In my thesis I use a different automorphic identity (from a triple-product formula of Ichino) to carry out a different construction that always works; I will talk about my result and some arithmetic consequences to it.

December 8, 11:00 am Special talk - Ph.D. Defense

Speaker: Ding Ding (Binghamton)
Title: Canonical Barsotti–Tate Groups of Finite Level

Abstract: Let $k$ be an algebraically closed field of characteristic $p>0$. Let c, d be positive integers and $h=c+d$. Let $H$ be a p-divisible group of codimension $c$ and dimension $d$ over $k$ . For a positive integer $m$ let $H[p^m]$ be the kernel of multiplication by $p^m.$ It is a finite commutative group scheme over $k$ of p power order, called a Barsotti–Tate group of level $m.$ We study a particular type of p-divisible groups $H_\pi$ called canonical Barsotti–Tate groups, where $\pi$ is a permutation on the set ${1,2,\ldots,h}.$ We obtain new formulas of combinatorial nature for the dimension of $Aut(H_\pi[p^m])$ and for the number of connected components of $End(H_\pi[p^m]).$

Fall 2016

2019-09-09T07:14:40-04:00

Fall 2016

August 29

Speaker: N/A
Title: Organizational Meeting

Abstract: We will discuss schedule and speakers for this semester

September 12

Speaker: Alexander Borisov (Binghamton)
Title: Adjunction on Surfaces and Keller Maps

Abstract: We will first review the basic theory of adjunction, canonical class, intersection pairing, and resolution of singularities of algebraic surfaces. Then we will discuss a certain adjunction inequality and inversion of adjunction results that are related to our approach to the two-dimensional Jacobian Conjecture.

September 19

Speaker: Tom Price (Toronto)
Title: The Regev Stephens-Davidowitz Inequality

Abstract: This talk will be about a recent inequality of Oded Regev and Noah Stephens-Davidowitz regarding the sum of the Gaussian function over lattice cosets. We'll discuss applications to arithmetic and to diffusion on tori and abelian Cayley graphs. We'll also discuss relevant open questions and the potential for generalization.

September 26

Speaker: Jaiung Jun (Binghamton)
Title: Introduction to Weil conjectures.

Abstract: This talk is an expository talk to introduce Weil conjectures to graduate students. We explain Weil conjectures and give a sketch of proofs for the curve case. If time permits, we explain how this naturally motivates the recent work of Connes and Consani.

October 10

Speaker: Xiao Xiao (Utica College)
Title: Minimal $F$-crystals.

Abstract: Let $k$ be an algebraically closed field of characteristic $p>0$. A $p$-divisible group $D$ over $k$ is said to be minimal if the isomorphism type of $D$ is determined by the kernel of the endomorphism $p: D \to D.$ Oort has given a complete classification of minimal $p$-divisible groups in 2005. In this talk, we will generalize this to $F$-crystals and give a classification theorem of minimal $F$-crystals. As an application of minimal $F$-crystals, we will give an upper bound of the isomorphism numbers of isosimple $F$-crystals, whose isogeny type are determined by simple $F$-isocrystals, in terms of their ranks, Hodge slopes and Newton slopes. In many cases, this new upper bound is sharper than some of the known results.

October 20
(CROSS LISTING WITH THE COLLOQUIUM –Dean's Speaker Series in Geometry/Topology; SPECIAL DAY THURSDAY and TIME 4:30pm):
Speaker: Ted Chinburg (University of Pennsylvania)
Title: Capacity theory and cryptography

Abstract: This talk is about an unexpected connection between cryptography and the theory of electrostatics. RSA cryptography is based on the presumed difficulty of factoring a given large integer N. In the 1990's, Coppersmith showed how one could quickly determine whether there is a factor of N which is within N^{1/4} of a given number. Capacity theory originated in studying how charged particles distribute themselves on an object. I will discuss how an arithmetic form of capacity theory can be used to show that one cannot increase the exponent 1/4 in Coppersmith's method. This is joint work with Brett Hemenway, Nadia Heninger and Zach Scherr.

October 21
(DEAN'S SPEAKER SERIES IN GEOMETRY/TOPOLOGY)
Speaker: Ted Chinburg (University of Pennsylvania)
Title: Constructing elements of Brauer groups and Tate-Shafarevitch groups from knots

Abstract: This talk has to do with knots invariants which are elements of the Brauer groups and of the Tate-Shafarevitch groups of curves over number fields. Constructing these invariants involves a close analysis of the canonical Azumaya algebra which is defined over an open dense subset of Thurston's canonical curve in the representation variety of the knot group. This is joint work with Alan Reid and Matt Stover.

October 24

Speaker: Brian Hwang (Cornell)
Title: An application of automorphic forms to Galois theory

Abstract: A classical problem in Galois theory is a strong variant of the Inverse Galois Problem: “What finite groups arise as the Galois group of a finite Galois extension of the rational numbers, if you impose the additional condition that the extension can only ramify at finite set of primes?” This question is wide open in almost every nonabelian case, and one reason is our lack of knowledge about how to find number fields with prescribed ramification at fixed primes. While such fields are often constructed to answer arithmetic questions, there is currently no known way to systematically construct such extensions in full generality.
However, there are some inspiring programs that are gaining ground on this front. One method, initiated by Chenevier, is to construct such number fields using Galois representations and their associated automorphic representations via the Langlands correspondence. We will explain the method, show how some recent advances in these subfields allow us to gain some additional control over the number fields constructed, and indicate how this brings us closer to our goal. As a application, we will show how one can use this knowledge to study the arithmetic of curves over number fields.

October 31

Speaker: Patrick Milano (Binghamton)
Title: TBA

Abstract: TBA

November 7
(Joint with Algebra Seminar)
Speaker: Matthew Moore (McMaster University)
Title: Dualizable algebras omitting types 1 and 5 have a cube term

Abstract: An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if V(A) is congruence distributive and A is dualizable, then A has an NU term. An important generalization of the NU term for congruence distributive varieties is the cube term for congruence modular (CM) varieties, and it has been thought that a similar characterization of dualizability for algebras in a CM variety would also hold. We prove that if A omits tame congruence types 1 and 5 (all locally finite CM varieties omit these types) and is dualizable, then A has a cube term.

November 14

Speaker: Micah Loverro (Binghamton)
Title: Lie algebras of linear algebraic groups and their representations.

Abstract: I will discuss the Lie algebra g of a linear algebraic group G, and the problem of classifying representations of g.

November 21

Speaker: Noah Giansiracusa (Swarthmore College)
Title: Tropicalizing schemes

Abstract: I'll discuss joint work with my brother, Jeff Giansiracusa, in which we extend tropicalization to a scheme-theoretic setting by writing down explicit equations cutting out tropical varieties. Tropical geometry has been rapidly gaining momentum and achieving exciting results in a variety of areas; our hope is that by expanding the scope to allow non-reduced structure and basing tropical methods on algebraic foundations that the range of applications with increase, though the program is still in its early steps. Connections to matroids and to Berkovich analytification will be mentioned.

November 28

Speaker: Alexander Borisov (Binghamton)
Title: Rolle's Theorem, Belyi's Theorem, and more

Abstract: Suppose a polynomial with real coefficients has all real roots, i.e. it is split over the reals. Then Rolle's theorem implies that its derivative is also split over the reals. We will prove the following theorem. A polynomial with rational coefficients divides a derivative of a polynomial split over the rationals if and only if all of its irrational roots are real and simple. The proof is related to the famous Belyi's theorem and to the basic but mysterious similarity between the logarithmic differentiation of products of powers and the Lagrange interpolation formula.

December 5

Speaker: Sayak Sengupta (Binghamton)
Title: Dual Spaces

Abstract: The talk primarily covers different properties of linear functionals on vector spaces supported by appropriate examples and theorems. We will discuss about these functionals over arbitrary fields, starting with the dual space of a vector space and then moving on to the double dual up to arriving at the conclusion that the double dual is same as the original space in the finite dimensional case.

Fall 2017

2018-01-18T08:48:42-04:00

Fall 2017

August 22 (Tuesday, 10:00 am – 12:00 pm)
Speaker: Micah Loverro (Binghamton)
Title: Relating G-modules and Lie(G)-modules
Abstract: Given a fixed representation V of G_K over a field K, where K is the field of fractions of a Noetherian normal domain R, and the group scheme G over R is reductive, we investigate relations between Lie(G)-modules and G-modules inside V. If M inside V is a G-module, then M is always a Lie(G)-module. We have conditions in some cases which imply that if M is a Lie(G)-module, then it is also a G-module. In particular, we show that we can reduce the problem to the case where R is a complete discrete valuation ring with residue field algebraically closed.

August 22 (Tuesday, 2:00 am – 4:00 pm)
Speaker: John Brown (Binghamton)
Title: Classifying finite hypergeometric groups, height one balanced integral factorial ratio sequences, and some step functions
Abstract: In this talk we will discuss some connections between hypergeometric series, factorial ratio sequences, and non-negative bounded integer-valued step functions. We will start with a finiteness criterion for hypergeometric groups by Beukers and Heckman, then show how this leads to the classification by Bober of integral balanced factorial ratio sequences of height one, and thus a proof that a conjectured classification of a certain class of step functions by Vasyunin is complete.

August 28
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester

September 11
Speaker: Jaiung Jun (Binghamton)
Title: Geometry over hyperfields
Abstract: In this talk, we illustrate how hyperfields can be used to show that certain topological spaces (underlying topological spaces of schemes, Berkovich analytification of schemes, and real schemes) are homeomorphic to sets of rational points of schemes over hyperfields.

September 18
Speaker: Martin Ulirsch (Michigan)
Title: Realizability of tropical canonical divisors
Abstract: We solve the realizability problem for tropical canonical divisors: Given a pair $(\Gamma, D)$ consisting of a stable tropical curve $\Gamma$ and a divisor $D$ in the canonical linear system on $\Gamma$, we develop a purely combinatorial condition to decide whether there is a smooth curve realizing $\Gamma$ together with a canonical divisor that specializes to $D$. In this talk I am going to introduce the basic notions needed to understand this problem and outline a comprehensive solution based on recent work of Bainbridge-Chen-Gendron-Grushevsky-M\”oller on compactifcations of strata of abelian differentials. Along the way, I will also develop a moduli-theoretic framework to understand the specialization of divisors to tropical curves as a natural tropicalization map in the sense of Abramovich-Caporaso-Payne.
This talk is based on joint work with Bo Lin, as well as on an ongoing project with Martin M\”oller and Annette Werner.

September 25
Speaker: Jaiung Jun (Binghamton)
Title: Picard groups for tropical toric varieties.
Abstract: From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (with some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$. In other words, we show that the two groups $Pic(X)$ and $Pic(X_S)$ are isomorphic. We also construct the group $CaCl(X_S)$ of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme $X_S$ and prove that $CaCl(X_S)$ is isomorphic to $Pic(X_S)$.

October 2
Speaker: Patrick Milano (Binghamton)
Title: Ghost spaces and some applications to Arakelov theory
Abstract: Arakelov theory provides a method for completing arithmetic curves like Spec(Z) by adding formal points “at infinity.” There is an Arakelov divisor theory for such completed arithmetic curves that is analogous to the theory of divisors on projective algebraic curves. In order to describe the cohomology of an Arakelov divisor, Borisov introduced the notion of a ghost space. After some background and motivation, we will define ghost spaces and look at some of their applications.

October 9
Speaker: Christian Maire (Cornell, Besançon)
Title: Fixed points in p-adic analytic extensions of number fields and ramification (joint work with Farhid Hajir)
Abstract: In this talk, I will present two arithmetic applications of the presence of fixed points in p-adic analytic extensions of number fields: (i) for the mu of the p-class group; (ii) for some evidences of the tame version of the Fontaine-Mazur conjecture. As we will see, the nature of the ramification (tame versus wild) is essential. The lecture will be accessible for non-specialists.

October 23
Speaker: Max Kutler (Yale)
Title: Faithful tropicalization of hypertoric varieties
Abstract: A hypertoric variety is a “hyperk\”ahler analogue” of a toric variety. Each hypertoric variety comes equipped with an embedding into a toric variety, called the Lawrence toric variety, and hence has a natural tropicalization. We explicitly describe the polyhedral structure of this tropicalization. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map.

October 30
Speaker: Alina Vdovina (CUNY, Newcastle)
Title: Buildings, quaternions and fake quadrics
Abstract: We'll present construction of buildings as universal covers of certain complexes. A very interesting case is when the fundamental group of such a complex is arithmetic, since the construction can be carried forward to get new algebraic surfaces, namely fake quadrics. Fake projective planes are already classified following series of works of D. Mumford, G. Prasad, S.-K. Young, D.Cartwright, T.Steger, but the fake quadrics remain mysterious.

November 6
Speaker: Micah Loverro (Binghamton)
Title: G-modules and Lie(G)-modules with examples from SL_2
Abstract: Given a fixed representation $V$ of a simply-connected semisimple group $G_K$ over a field $K$, we seek to determine which Lie$(G)$-modules $M$ inside $V$ are also $G$-modules, where $G$ is a smooth affine group scheme of finite type over a Noetherian normal domain $R$ whose field of fractions is $K$. Previously we showed that we can assume $R$ is a complete discrete valuation ring with algebraically closed residue field. In this talk, we will go through the details of the case when $G$ is $SL_2$, and then show how the Frobenius map could be used in a more general setting to produce Lie$(G)$-modules which are not $G$-modules under certain conditions depending on the weights of the representation.

November 13
Speaker: Tom Price (Toronto)
Title (tentative): A global sections functor for Arakelov bundles
Abstract: We exhibit a class of real-valued functions on Abelian groups, which have some non-trivial properties generalizing the behaviour of indicator functions of subgroups, such as the Regev Stephens-Davidowitz inequality. We construct a category of groups equipped with these functions, use this to create an analogue of a global sections functor for Arakelov bundles, and demonstrate that this functor has some properties we should expect.

November 20
Speaker: Patrick Carney (Binghamton)
Title: Geometry and divisors on rational curves and surfaces
Abstract: In his 2014 paper A. Borisov constructed two invariants of divisorial valuations at infinity. We will discuss some algebraic geometry notions and constructions used in that paper, specifically the theory of divisors and linear equivalence on the projective line, the projective plane, and other compactifications of the affine plane. The blow-up of a point construction will also be presented in detail. This is the first of two talks, that deals with the prerequisites for the paper. It will be followed by the second talk that discusses the combinatorial methods and results of that paper.

November 27
Speaker: Sayak Sengupta (Binghamton)
Title: Valuations
Abstract: The main topic of the talk is how to recover the valuation function from a valuation ring of a field. The talk starts with the definition of a valuation ring and an idea of how to construct valuation ring from any given field followed by a short discussion about valuation functions and discrete valuation functions leading to the final part of the talk i.e to establish the main topic of the talk.

December 4
Speaker: Philipp Jell (Georgia Tech)
Title: Non-archimedean Arakelov theory and cohomology of differential forms on Berkovich spaces
Abstract: Arakelov theory studies varieties over number fields by combining analytic geometry over the complex numbers (representing the infinite places) with algebraic intersection theory on suitable models over the ring of intergers (representing the finite places). However, due to lack of resolution of singularities in mixed characteristic, such models are hard to come by. It has always been a goal to unify the approaches and replace intersection theory on models by analytic geometry over the finite places.
In 2012, Chambert-Loir and Ducros made a promising step in this direction, introducing real-valued differential forms and currents on Berkovich analytic spaces and proving among other things a Poincaré-Lelong formula and existence of Chern classes for line bundles.
In this talk, we will give a brief introduction to Arakelov theory and introduce the forms defined by Chambert-Loir and Ducros. We will then discuss the cohomology theory defined by these forms for varieties over non-archimedean fields. In particular we explain a Poincaré lemma result and results on duality for curves.

December 6 (Room: WH 329)
Speaker: Patrick Carney (Binghamton)
Title: Geometry and divisors on rational curves and surfaces, Part 2
Abstract: This is a continuation of the November 20 talk. We will discuss the structure of the divisor class group on arbitrary compactifications of the affine plane that are obtained from the projective plane by a sequence of blowups. We will discuss the intersection form on this group and define the two invariants of the divisorial valuations at infinity studied in that 2014 paper by Borisov. We will explain the properties of these invariants and the main results regarding them.

Fall 2018

2019-01-22T09:07:14-04:00

Fall 2018

August 27 (Monday)
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester

September 18 (Tuesday)
Speaker: Vaidehee Thatte (Binghamton)
Title: Ramification Theory for Degree $p$ extensions of Arbitrary Valuation Rings in Positive Characteristic, part 1
Abstract: In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields. We would like to study arbitrary valuation rings with possibly imperfect residue fields and possibly non-discrete valuations of rank $\geq 1$, since many interesting complications arise for such rings. In particular, defect may occur (i.e. we can have a non-trivial extension, such that there is no extension of the residue field or the value group) when the characteristic is positive. We will discuss some new results in the equal characteristic case, similar results are true in the mixed characteristic $(0,p)$ case.
We will begin with a few examples of Artin-Schreier extensions of valuation fields and explicitly compute some invariants of ramification theory in each case. Time permitting, we will also discuss a generalization of the classical Swan conductor.

September 25 (Tuesday)
Speaker: Alexander Borisov (Binghamton)
Title: Singular Fano Varietes
Abstract: One of the Fields Medals this year went to Caucher Birkar for his birational geometry work, that included in particular a proof of so-called Borisov-Alexeev-Borisov conjecture. This conjecture was proposed independently by me and Valery Alexeev around 1993. I will give a short introduction to the subject, to state the BAB conjecture (now Birkar's Theorem) and discuss some related results and open questions.

October 2 (Tuesday)
Speaker: Alexander Borisov
Title: A search for Keller maps
Abstract: I will update on my, so far unsuccessful, search for Keller maps, i.e. counterexamples to the Jacobian Conjecture. In particular, I will present a detailed framework for such counterexample and explain how one can use computers to check if it leads to a Keller map.

October 9 (Tuesday)
Speaker: Vaidehee Thatte (Binghamton)
Title: Ramification Theory for Degree $p$ extensions of Arbitrary Valuation Rings in Positive Characteristic, part 2
Abstract: We will continue discussing Artin-Schreier extensions of valuation fields in positive characteristic. We will present some results that relate the “higher ramification ideal” of the extension with the ideal generated by the inverses of Artin-Schreier generators via the norm map. We will also introduce a generalization and further refinement of Kato's refined Swan conductor for such extensions.

October 15 (Monday)
Speaker: Viji Thomas (Cleveland State)
Title: A Report on Schurs Exponent Conjecture and some closure properties of the nonabelian tensor product of groups and the second stable homotopy group of the Eilenberg Maclane Space
Abstract: The talk has three parts. In the first part, I will describe Schurs Exponent conjecture, and the progress made so far on this conjecture. Then we will briefly describe the work done by us towards this conjecture. After this we will describe some closure properties of the nonabelian tensor product of groups and finally time permitting, we will discuss the relationship of the nonabelian tensor product with the second stable homotopy group of the Eilenberg-Maclane space.

October 23 (Tuesday)
Speaker: Vaidehee Thatte (Binghamton)
Title: Ramification Theory for Degree $p$ extensions of Arbitrary Valuation Rings in Positive Characteristic, part 3
Abstract: Let $K$ be a valued field of characteristic $p > 0$ with henselian valuation ring $A$. Let $L$ be a non-trivial Artin-Schreier extension of $K$ with $B$ as the integral closure of $A$ in $L$. In the classical theory, $B$ is generated as an $A$-algebra by a single element. This is not true when there is defect. We will discuss a result that allows us to write $B$ as a “filtered union over $A$”, in such cases.
We will conclude the series with some remarks on how to obtain analogous results in the mixed characteristic $(0, p)$ case.

November 6 (Tuesday)
Speaker: Marie Langlois (Cornell)
Title: Building Variable Homogeneous Integer-valued Polynomials Using Projective Planes
Abstract: A polynomial $f$ over $\mathbb{Q}[x,y,z]$ is integer-valued if $f(x,y,z)\in \mathbb{Z}$, whenever $x,\ y$ and $z$ are integers. This talk will go over various examples of these and general techniques to find bases for the modules they create. Then, the focus will be on the case of $f$ being homogeneous and how to construct polynomials such that the denominators are divisible by the highest possible power of $p=2$. Projective H-planes will be introduced, which are a generalization of finite projective planes over rings, to construct a correspondence between lines that cover H-planes and homogeneous IVPs that are a product of linear factors. This correspondence will be illustrated starting with the degree 8 case where we produce a polynomial with the largest possible denominator which factors as a product of linear polynomials.

November 13 (Tuesday)
Speaker: Andrew Obus (Baruch College)
Title: Explicit resolution of weak wild quotient singularities on arithmetic surfaces
Abstract: Given a smooth projective curve X over a complete discretely valued field K, it follows from well-known work of Abhyankar/Lipman that there is a regular model of X defined over the valuation ring of K. A particularly interesting case is when X has potentially good reduction. In this case, there is a natural model of X with so-called quotient singularities. Resolution of tame quotient singularities is well understood, and we will give a complete picture of the resolution of the simplest case of wild quotient singularities (which we call ”weak wild arithmetic quotient singularities”). Our techniques involve heavy use of deformation theory and valuation theory, in contrast to the more global techniques that have been used by Lorenzini on related problems.

November 19 (Monday)
Speaker: Isabel Leal (Courant)
Title: Generalized Hasse-Herbrand functions
Abstract: The classical Hasse-Herbrand function is an important object in ramification theory, related to higher ramification groups. In this talk, I will discuss generalizations of the Hasse-Herbrand function and go over some of their properties. These generalized Hasse-Herbrand functions are defined for extensions L/K of complete discrete valuation fields where the residue field k of K is perfect of characteristic p>0 but the residue field l of L is possibly imperfect.

November 27 (Tuesday)
Speaker: Changwei Zhou (Binghamton)
Title: Survey of some recent results in arithemetic surfaces and new observation
Abstract: In this talk I will review analytic torsion and show a proof that it has an upper bound only depending on genus when we give the surface Arakelov metric. To my knowledge this is the first result of this type in the literature. The work is a direct corollary of Jorgenson and Kramer's work on the non-completeness of Arakelov metric on the moduli space. I will also discuss some relevant work by Wilms, Bost, Soule, Wentworth and Faltings.

December 4 (Tuesday)
Speaker: Sayak Sengupta (Binghamton)
Title: An overview of Jacobian Conjecture in positive characteristic
Abstract: This is the first in the 2-talk series. I am going to start with the statement of Jacobian Conjecture along with some related examples. After that I am going to talk about our problem of interest which is an analogue of the Conjecture, establishing its connection and some very interesting examples.

Fall 2019

2020-01-28T11:46:10-04:00

Fall 2019

August 27 (Tuesday)
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester

September 10 (Tuesday)
Speaker: Alexander Borisov (Binghamton University)
Title: Geometrically nilpotent subvarieties for polynomial maps over finite fields
Abstract: For every dominant self-map of an affine space over a finite field, periodic orbits are Zariski dense. In particular, it is not possible that all points over the algebraic closure of the field are sent to one fixed point by some iteration of this polynomial map. However there may exist a proper subvariety of the affine space such that all its points over the algebraic closure are sent to a fixed point by some iteration of the map, yet the variety itself is not. I will give several examples of this phenomenon and discuss some related questions.

September 17 (Tuesday)
Speaker: Alexander Borisov (Binghamton University)
Title: Lattices in Euclidean Spaces, Part 1
Abstract: This is a first part of a series of two talks on lattices in Euclidean Spaces and their invariants.

September 24 (Tuesday)
Speaker: Alexander Borisov (Binghamton University)
Title: Lattices in Euclidean Spaces, Part 2
Abstract: This is a continuation of the talk from September 17. We will follow the recent preprint of Bost to prove the Banaszczyk's remarkable “Transference Inequality”.

October 7 (Monday)
Speaker: Huy Dang (University of Virginia)
Title: The refined Swan conductor and deformation of Artin-Schreier covers
Abstract: An Artin-Schreier curve is a G:=Z/p-branched cover of the projective line over a field of characteristic p>0. A unique aspect of characteristic p is that there exist flat deformations of a wildly ramified cover so that the number of branch points changes but the genus does not. Using refined Swan conductor, we give the necessary and sufficient conditions for the existence of a deformation between given Artin-Schreier curves. As an application, we show that the moduli space of Artin-Schreier covers of fixed genus g is connected when g is sufficiently large.

October 15 (Tuesday)
Speaker: Inna Sysoeva (Pittsburgh)
Title: Irreducible representations of braid groups
Abstract: In this talk I'm going to discuss the classification of the irreducible representations of Artin braid group $B_n$ on $n$ strings. All irreducible representations of $B_n$ of dimension less or equal to $n-1$ were classified by Ed Formanek in 1996; the irreducible representations of $B_n$ dimension $n$ for $n\geq 9$ were classified by the speaker in 1999, and for $n\leq 8$ they were classified by Formanek, Lee, Vazirani and the speaker in 2003.
I will give the overview of the known results, and I will talk about the work in progress aimed to classify all the irreducible representations of $B_n$ of dimension less than or equal to $2n-9$ for $n\geq 10.$

October 22 (Tuesday)
Speaker: Fikreab Solomon Admasu (Binghamton University)
Title: Zeta functions of classical groups and class two nilpotent groups
Abstract: We will discuss zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups. In particular, we will look at a simpler but less extensive approach to the zeta functions computed by J. Igusa and later generalized by M. du Sautoy and A. Lubotzky.

October 29 (Tuesday) (Note: this is joint with Combinatorics Seminar)
Speaker: Jaiung Jun (SUNY New Paltz)
Title: The Hall algebra of the category of matroids
Abstract: To an abelian category A satisfying certain finiteness conditions, one can associate an algebra H_A (the Hall algebra of A) which encodes the structures of the space of extensions between objects in A. For a non-additive setting, Dyckerhoff and Kapranov introduced the notion of proto-exact categories, as a non-additive generalization of an exact category, which is shown to suffice for the construction of an associative Hall algebra. In this talk, I will discuss the category of matroids in this perspective.

November 5 (Tuesday)
Speaker: Fikreab Solomon Admasu (Binghamton University)
Title: Zeta functions of classical groups via Hecke series
Abstract: We continue the talk from two weeks ago and look at two ways of deriving the zeta functions of classical groups. Making use of computations of Hecke series by A. N. Andrianov, T. Hina and T. Sugano, we express zeta functions of symplectic groups and even orthogonal groups in terms of the cotype zeta function of the integer lattice. This may lead to an alternate proof of properties of the zeta functions of the above classical groups such as local functional equations and the existence of natural boundaries.

November 11 (Monday)
Speaker: Andrew Kobin (University of Virginia)
Title: A stacky compactification of the ring of Witt vectors
Abstract: The ring of Witt vectors is an essential tool for understanding relationships between the worlds of characteristic 0 and finite characteristic algebra. In this talk, I will recall how the ring (scheme) of Witt vectors allows one to lift field extensions and covers of curves from characteristic p to characteristic 0. The latter situation motivated Garuti to define a projective scheme which compactifies the ring (scheme) of Witt vectors “equivariantly” (with respect to Witt vector addition). After describing Garuti's construction and its application to the study of covers of curves, I will introduce a new compactification in the category of algebraic stacks that I am currently using to describe the local structure of stacky curves in characteristic p.

November 18/19
Speaker: Dikran Karagueuzian (Binghamton University)
Title: Coalescence of Polynomials over Finite Fields
Abstract: A polynomial over a finite field can be regarded as a map of the finite field to itself. The coalescence, or variance of the inverse image sizes, has been studied in connection with whether such maps are a good substitute for random maps. We are able to show that polynomial maps are not random in the sense that the coalescence must be an integer, in an asymptotic sense as the size of the finite field becomes large. This is based on joint work with Per Kurlberg.

November 25 (Monday)
Speaker: Caleb McWhorter (Syracuse University)
Title: Mordell-Weil Groups of Elliptic Curves
Abstract: The Mordell-Weil Theorem states that for a number field $K$, the group of $K$-rational points on an elliptic curve form a finitely generated abelian group, i.e. $E(K) \cong \mathbb{Z}^{r_K} \oplus E(K)_{\text{tors}}$, where $r_K$ is the rank and $E(K)_{\text{tors}}$ is the torsion subgroup. Despite the apparent simplicity of $E(K)$, there is little known about the possible Mordell-Weil groups, especially in understanding the possible $r_K$. This talk will discuss the progress in understanding each of these pieces. We will briefly discuss the heuristics of Park-Poonen-Voight-Wood and Lozano-Robledo regarding the possible ranks $r_K$, and then discuss the work of Bhargava-Shankar regarding the average rank of elliptic curves. Then we will discuss the progress in classifying the possible torsion subgroups of elliptic curves over global fields, where there is much more progress. Finally, we will discuss the specific determination of the possibilities for $E(K)_{\text{tors}}$ when $E$ is a rational elliptic curve and $K$ is a nonic Galois field.

Fall 2020

2021-02-15T13:03:03-04:00

Fall 2020

September 1
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

September 8 (Joint with Algebra Seminar, at 2:50 -4:00 pm, Zoom link: https://binghamton.zoom.us/j/95069915454 )
Speaker: Fikreab Solomon Admasu
Title: Composition laws from Gauss to Bhargava
Abstract: One of the fundamental objects of interest in number theory is the composition law of binary quadratic forms. In this talk, we will start with a review of the original complicated formulation due to Gauss and then discuss subsequent simplifications by Dirichlet, et al. Another approach involves identifying equivalence classes of binary quadratic forms with ideal classes in a quadratic ring and using the natural group structure of the ideal class group to formulate the composition law of binary quadratic forms. After about 200 years, M. Bhargava gave a new perspective on Gauss' composition law using so-called 2x2x2 cubes of integers and derived more composition laws. We will discuss the main theorems in 'Higher Composition Laws I' by Bhargava and mention the applications if time permits.

September 15 (Joint with Algebra Seminar, at 2:50 -4:00 pm, Zoom link: https://binghamton.zoom.us/j/95069915454 )
Speaker: Fikreab Solomon Admasu
Title: Composition laws from Gauss to Bhargava, part 2
Abstract: Continuation of the talk from September 8, with focus on the newer Bhargava approach.

September 22
Speaker: Sayak Sengupta
Title: Polynomial Maps
Abstract: We will cover a couple topics in this talk.
(a) Polynomial maps over finite fields &
(b) Geometrically nilpotent subvarieties.
We will start with exploring quasi-fixed points of an affine algebraic variety, how they are, what they look like etcetera and then we will identify the difference between nilpotent subvarieties and geometrically nilpotent subvarieties with the help of an example.
All the topics in this talk are based on Prof. Borisov's papers with the same names.

September 29
Speaker: Caroline Matson
Title: The commutant monoid of a higher-dimensional power series map
Abstract: In a 1994 paper, Lubin examined nonarchimedean dynamical systems, or families of power series that commute under composition. He defined a stable power series to be a power series f(x) = bx + (higher degree terms) such that the linear coefficient b is neither zero nor a root of unity and showed that if f(x) is stable then for any constant c there exists a unique power series g(x) such that f(g(x) = g(f(x)) and g(x) = cx + (higher degree terms). In this talk we will generalize this problem to multiple dimensions and will explore the notion of stability in this more complicated setting.

October 6
Speaker: Jaiung Jun (SUNY New Paltz)
Title: Hall algebras in a non-additive setting
Abstract: To an abelian category A satisfying some conditions, one may attach an associative algebra H_A, called the Hall algebra of A. Two interesting examples of such abelian categories are Coh(X / F_q), the category of coherent sheaves on a projective variety X over F_q, and Rep(Q,F_q), the category of representations of a quiver Q over F_q. In fact, Dyckerhoff and Kapranov introduced the notion of proto-exact categories, and generalized the construction of Hall algebras to the non-additive setting. This allows one to define and study the Hall algebras for Coh(X / F_q) and Rep(Q,F_q) in the setting of ``the field with one element''. I will explain some of my recent work in this direction. This is joint work with Matt Szczesny.

October 13
Speaker: Alexander Borisov (Binghamton)
Title: Projective geometry approach to the Jacobian Conjecture
Abstract: Jacobian Conjecture is one of the oldest unsolved problems in Algebraic Geometry, going back to a 1939 paper by Keller. It says that if a polynomial self-map of a plane is locally invertible (that is, unramified) then it is globally invertible. This conjecture is infamous for the large number of incorrect proofs that have been proposed over the years. In fact, it is quite possible that the conjecture is false, even in dimension two and especially in higher dimensions. I will explain this and some related conjectures and describe my approach to it in dimension two using methods of projective geometry. This is work in progress, since about 2005.

October 20
Speaker: Leo Herr (Utah)
Title: Log Intersection Theory and the Log Product Formula
Abstract: Kato: Log structures are “magic powder” that makes mildly singular spaces appear smooth. Log schemes literally lie between ordinary schemes and tropical geometry, and are related to Berkovich Spaces. Problems in Gromov-Witten Theory demand intersection theoretic machinery for slightly singular spaces. Log structures have solved similar problems in Hodge Theory, D-modules, Connections and Riemann-Hilbert Correspondences, Abelian Varieties (esp. Elliptic Curves), etc. How can they be used to define a reasonable intersection theory for curve counting on singular spaces? We'll give a product formula as proof-of-concept for a whole toolkit under development to tackle these types of problems.

October 27
Speaker: Juan Diego Rojas (Uniandes, Colombia)
Title: Global Field Totients
Abstract: We define the Euler's totient function for a global field and recover the essential analytic properties of the classical arithmetical function, namely the product formula and holomorphicity of the associated zeta function. As applications, (1) we recover the global analog of the mean value theorem of Erdős, Dressler and Bateman via the Wiener-Ikehara theorem and, (2) a function field analogue of Mertens’ mean value theorem. Joint work with Santiago Arango-Piñeros

November 10
Speaker: Harpreet Bedi (Alfred University)
Title: Perfectoid Algebraic Geometry
Abstract: Elementary Algebraic Geometry can be described as the study of zeros of polynomials with integer degrees, this idea can be naturally carried over to ‘polynomials’ with degree in Z[1/p], which leads to the perfectoid version of algebraic geometry. In this talk the similarities and differences between integer and rational degree are discussed. We will construct line bundles of rational degree and compute their cohomology. The formal schemes are then obtained via p-adic completion. Finally, we talk about zeros of rational degree polynomials and prove the Nullstellensatz.

November 17
Speaker: Piotr Achinger (IMPAN)
Title: Regular connections on log schemes and rigid analytic spaces
Abstract: I will describe an extension of Deligne's results on regular connections to log schemes over C. This is part of a project in progress whose goal is to obtain a Riemann–Hilbert type correspondence for smooth rigid-analytic spaces over C( (t) ).

December 1
Speaker: Julia Hartmann (UPenn)
Title: Local-global principles for linear algebraic groups over arithmetic function fields
Abstract: We consider linear algebraic groups over arithmetic function fields, i.e., over one variable function fields over complete discretely valued fields. Such function fields naturally admit several collections of overfields with respect to which one can study local-global principles. We will recall results about local-global principles for rational linear algebraic groups, and then focus on new results concerning certain classes of non-rational groups.

Fall 2021

2022-02-05T19:52:54-04:00

Fall 2021

August 31
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

September 14
Speaker: Sailun Zhan (Binghamton)
Title: Counting rational curves on K3 surfaces with finite group actions
Abstract: The Yau-Zaslow formula describes the number of rational curves in a linear system on a smooth projective K3 surface in terms of a modular form. In this talk, I will review the Yau-Zaslow formula with some examples and then discuss an equivariant version of the formula. When the K3 surface admits a finite group G-action, we can consider a linear system with the induced action. It turns out that the equivariant version of the formula will count G-rational curves and it will also provide interesting modular forms.

September 20 (Monday!), 4:15-5:15, by Zoom
Speaker: K. V. Shuddhodan (Purdue)
Title: The (non-uniform) Hrushovski-Lang-Weil estimates
Abstract: In 1996 using techniques from model theory and intersection theory, Hrushovski obtained a generalization of the Lang-Weil estimates. Subsequently, the estimate has found applications in group theory, algebraic dynamics, and algebraic geometry. We shall discuss an l-adic proof of the non-uniform version of these estimates and also the rationality of the associated generating function.

September 28
Speaker: Sailun Zhan (Binghamton)
Title: Counting rational curves on K3 surfaces with finite group actions (continued)
Abstract: The Yau-Zaslow formula describes the number of rational curves in a linear system on a smooth projective K3 surface in terms of a modular form. In this talk, I will review the Yau-Zaslow formula with some examples and then discuss an equivariant version of the formula. When the K3 surface admits a finite group G-action, we can consider a linear system with the induced action. It turns out that the equivariant version of the formula will count G-rational curves and it will also provide interesting modular forms.

October 5
Speaker: Alexander Borisov (Binghamton)
Title: Bi-Euclidean spaces and coherent sheaves on Arakelov curves
Abstract: It is well-known that lattices in Euclidean spaces are arithmetic analogs of locally free sheaves over the compactified spectrum of the ring of integers. The main obstacle to generalizing this analogy to coherent sheaves is to understand what to do at infinity. We propose a natural, and essentially elementary, construction, that has the potential to greatly enhance Arakelov Geometry in several ways. The main object at infinity is, roughly speaking, a pair of positive quadratic functions on a real vector space, one greater than the other. The morphisms are linear maps that are non-expanding with respect to both functions, and our objects are formal quotients of two Euclidean spaces. The resulting category is a natural target for the direct image map from the category of Hermitian sheaves on an Arakelov variety. This is work in progress, joint with Jaiung Jun.

October 19
Speaker: Sailun Zhan (Binghamton)
Title: Counting rational curves on K3 surfaces with finite group actions (part 3)
Abstract: The Yau-Zaslow formula describes the number of rational curves in a linear system on a smooth projective K3 surface in terms of a modular form. In this talk, I will review the Yau-Zaslow formula with some examples and then discuss an equivariant version of the formula. When the K3 surface admits a finite group G-action, we can consider a linear system with the induced action. It turns out that the equivariant version of the formula will count G-rational curves and it will also provide interesting modular forms.

October 26
Speaker: Tian An Wong
Title: Prehomogeneous vector spaces and the Arthur-Selberg trace formula
Abstract: The Arthur-Selberg trace formula is a central tool in the theory of automorphic forms, and can be viewed as a nonabelian Poisson summation formula. Prehomogeneous vector spaces on the other hand, are arithmetic objects from which certain zeta functions can be defined. In this talk, I will give a gentle introduction to these ideas, then discuss an application of the theory of prehomogeneous vector spaces to the development of the trace formula, following earlier work of W. Hoffmann and P. Chaudouard.

November 2
Speaker: Sunil Chetty
Title: Selmer groups and ranks of elliptic curves
Abstract: In the theory of elliptic curves, understanding the behavior of rank is a central problem. In light of the Birch-Swinnerton-Dyer and Tate-Shafarevich Conjectures, there are three avenues for understanding rank of a given elliptic curve E/K: by the structure of the Mordell-Weil group E(K), by the vanishing of the associated L-function L(E/K,s), or by the structure of the associated Selmer group Sel{E}{K}. We will discuss some of the big ideas for attacking the rank problem over number fields via the Selmer group approach, as well as methods of comparing parallel tools in the L-function approach.

November 9, 4:30-5:30, by Zoom
Speaker: Joe Kramer-Miller
Title: Ramification of geometric p-adic representations in positive characteristic
Abstract: A classical theorem of Sen describes a close relationship between the ramification filtration and the p-adic Lie filtration for p-adic representations in mixed characteristic. Unfortunately, Sen's theorem fails miserably in positive characteristic. The extensions are just too wild! There is some hope if we restrict to representations coming from geometry. Let X be a smooth variety and let D be a normal crossing divisor in X and consider a geometric p-adic lisse sheaf on X-D (e.g. the p-adic Tate module of a fibration of abelian varieties). We show that the Abbes-Saito conductors along D exhibit a remarkable regular growth with respect to the p-adic Lie filtration.

November 16
Speaker: Guillermo Mantilla Soler
Title: Applications of Higher composition laws to the classification of number fields
Abstract: In this talk we will describe what natural invariants we have studied with the aim of characterizing number fields, and how some of those are related to the higher composition laws discovered by Bhargava at the beginning of this century.

November 30
Speaker: Alexander Borisov (Binghamton)
Title: Bi-Euclidean spaces and coherent sheaves on Arakelov curves, Part 2
Abstract: This will be a continuation (with some repetition) of the talk from October 5. It is well-known that lattices in Euclidean spaces are arithmetic analogs of locally free sheaves over the compactified spectrum of the ring of integers. The main obstacle to generalizing this analogy to coherent sheaves is to understand what to do at infinity. We propose a natural, and essentially elementary, construction, that has the potential to greatly enhance Arakelov Geometry in several ways. The main object at infinity is, roughly speaking, a pair of positive quadratic functions on a real vector space, one greater than the other. The morphisms are linear maps that are non-expanding with respect to both functions, and our objects are formal quotients of two Euclidean spaces. The resulting category is a natural target for the direct image map from the category of Hermitian sheaves on an Arakelov variety. This is work in progress, joint with Jaiung Jun.

December 7
Speaker: Alex Best
Title: A guided tour of Chabauty methods
Abstract: Chabauty's method is a p-adic technique for determining the set of rational points on a projective curve that can be made computationally effective. In the 80 years since the inception of this method many variants have been introduced which apply more generally or give stronger results. We will give a guided tour of the method of Chabauty, its extensions and some of its many applications within arithmetic geometry. This will include some discussion of the original method of Chabauty, number field Chabauty, Chabauty–Kim and quadratic Chabauty.

Fall 2022

2023-01-15T14:11:42-04:00

Fall 2022

August 30
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

September 13
Speaker: Sailun Zhan (Binghamton)
Title: P-adic integration and motivic integration
Abstract: We will give an introduction to some integration techniques in number theory and algebraic geometry, which allow us to compare the number of points over finite fields and some geometric properties between algebraic varieties. If there is time left, we will also talk about an equivariant version.

September 20
Speaker: Alexander Borisov (Binghamton)
Title: Geometry of algebraic surfaces
Abstract: I will discuss some standard material on algebraic surfaces, including some material on surfaces with singularities

October 11
Speaker: Sarah Lamoureux (Binghamton)
Title: ADOs on the Completion of the Maximal Unramified Extension
Abstract: This past spring, I gave a talk about arithmetic differential operators (ADOs) f : R^d → R, where R is a compact discrete valuation ring. The notion of an ADO generalizes to maps (\hat{R}^{\textup{ur}})^d → \hat{R}^{\textup{ur}}, where \hat{R}^{\textup{ur}} is the completion of the maximal unramified extension of R. This talk explores properties of these maps and their relationship to ADOs from R^d to R.

October 18
Speaker: Alexander Borisov (Binghamton)
Title: Geometry of algebraic surfaces, Part 2
Abstract: This is the continuation of my September 20 talk. In particular, I plan to discuss intersection theory on singular surfaces.

October 25
Speaker: Jaiung Jun (SUNY New Paltz)
Title: From chip-firing games to vector bundles for schemes over natural numbers
Abstract: In tropical geometry, finite (metric) graphs play a role of algebraic curves. Baker and Norine proved that an analogue of Riemann-Roch theorem holds in this setting. To generalize this result to higher dimension, one is naturally led to study the scheme theory over idempotent semifields (or more generally schemes over natural numbers). I will introduce basic notions and properties for line bundles and vector bundles in this setting. I will also discuss some related concepts (finiteness, flatness, projectivity). This is joint work with James Borger.

November 1
Speaker: Sayak Sengupta (Binghamton)
Title: Locally nilpotent polynomials over Z (Part III)
Abstract: This is a continuation of two talks on the subject which were given in the Spring semester of 2022. So far we have defined locally nilpotent polynomials at r, seen several examples of locally nilpotent polynomials for different r's and also stated and proved a complete classification of locally nilpotent polynomials at 1 and -1. In order to prove this classification we only needed tools from elementary number theory. In this talk we will analyze the locally nilpotent polynomials at r when r\in Z without \pm 1. Here we will use a very deep result from algebraic number theory and even then we will see that only the linear polynomials could be studied and understood. I will start with a brief recollection of the major definitions and results, along with some notation and terminology and build our way up to the “general r” case.

November 29
Speaker: Alexander Borisov
Title: Singularities in birational algebraic geometry
Abstract: I will give a light overview of various classes of singularities that appear in birational algebraic geometry, with a special emphasis on surface singularities.

December 6 (by Zoom: Zoom link )
Speaker: Krishna Kishore (Indian Institute of Technology (IIT) Tirupati)
Title: Matrix Waring Problem
Abstract: We will explain the following statement: Let $q$ be a prime power. For every positive integer $k$ there is a constant $C_k$ depending only on $k$ such that for all $q > C_k$ and for all $n \geq 1$ every matrix in $M_n(F_q)$ is a sum of two kth powers. Here $F_q$ denotes the finite field with $q$ elements.

Fall 2023

2024-01-31T11:38:13-04:00

Fall 2023

August 29
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

September 5 4:15-6:15 pm Special Event: PhD Defense
Speaker: Sarah Lamoureux (Binghamton)
Title: Arithmetic Differential Operators on Compact DVRs
Abstract: Let R be a compact DVR and S=\hat{R}^{ur} the completion of its maximal unramified extension. We investigate the relationship between so-called `arithmetic differential operators' and analytic maps in three contexts: maps S^d\to S, maps R^d\to S, and maps with domain the R-points of a smooth affine scheme of finite type over R.

September 12
Speaker: Sailun Zhan (Binghamton)
Title: Waring problem for matrices over finite fields
Abstract: The Waring problem for matrices is to address whether matrices over a ring can be expressed as a sum of two kth powers of matrices. We prove that for all integers k>=1, for all q>=(k−1)^4 + 6k, and for all m>=1, every matrix in M_m(Fq) is a sum of two kth powers. We also study the case when the matrices are invertible, cyclic, or split semisimple, when k is coprime to p, or when m is sufficiently large. We give a criterion for the Waring problem in terms of stabilizers. This is a joint work with Krishna Kishore and Adrian Vasiu.

October 3
Speaker: Sayak Sengupta (Binghamton)
Title: Action of SL_2
Abstract: A standard group action in complex analysis is the action of $GL_2(\mathbb{C})$ on the Riemann sphere $\mathbb{C}\cup\{\infty\}$ by linear fractional transformations. It is known that this action is transitive; in fact, if $GL_2(\mathbb{C})$ is replaced by $SL_2(\mathbb{C})$, the transitivity of the action is still maintained. One can see, by following similar arguments, that if $\mathbb{C}$ is replaced by any field $K$, the action is also transitive. However, the action of $SL_2(\mathbb{R})$ on the Riemann sphere is not transitive. In this talk we will briefly review the actions of $SL_2$ described above, and then we will look at the special case of $SL_2(\mathcal{O}_K)$ acting on $K$, where $K$ is a number field, and $\mathcal{O}_K$, is its corresponding ring of integers.

October 10
Speaker: Sayak Sengupta (Binghamton)
Title: Modular forms and discrete matrix group actions
Abstract: In this talk we will continue our discussion from last week. In particular, we will introduce congruence subgroups and modular forms on a few congruence subgroups.

November 7
Speaker: Mithun Veettil (Binghamton)
Title: Wreath Product of Groups and Indicatrix Polynomial of a Group Action
Abstract: First, we will define the wreath product of finite groups. Then we will define a polynomial called ”indicatrix of a group” that captures the fixed points of the action of the group on some set. It turns out that the indicatrix behaves ”nicely” upon taking the wreath product. If time permits, we shall go through specific examples; we will compute the indicatrix of the symmetric group on k letters, S_k, acting naturally on {1,2,…,k}.

November 14
Speaker: Alexander Borisov (Binghamton)
Title: An update on the search for Keller maps
Abstract: This is a continuation of some of my previous talks, on my ongoing search for the counterexamples to the two-dimensional Jacobian Conjecture. It will be based on the last section of my 2020 paper on frameworks: http://people.math.binghamton.edu/borisov/documents/papers/Frameworks_EJC_final.pdf. I will also talk about the connection between chains of exceptional curves and cyclic quotient singularities, the Farey fraction encoding (from Patrick Carney's thesis) and a present a new framework, which is somewhat different from those in the above paper.

November 28 (by Zoom: Zoom link )
Speaker: Daodao Yang (CICMA)
Title: Large values of derivatives and logarithmic derivatives of zeta and L-functions
Abstract: An important topic in analytic number theory is the study of extreme values of zeta and L-functions. In this talk, I will report some of my recent work on large values of derivatives and logarithmic derivatives of zeta and L-functions. If time permits, I will also discuss GCD sums, log-type GCD sums, and Dirichlet character sums, which are related topics.

December 5
Speaker: Hari Asokan (Binghamton)
Title: GIT quotient of SL_g action on symmetric matrices
Abstract: The special linear group SL_g acts on the vector space of symmetric matrices, V_g by congruence action. This action extends to (V_g)^{r+1}. We will discuss the GIT quotient and the invariant ring of this action.

Fall 2024

2025-01-16T08:21:07-04:00

Fall 2024

August 27
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

September 3 (Algebra Seminar, cross-listed, 2:50-3:50 pm)
Speaker: Dikran Karagueuzian (Binghamton)
Title: Elliptic Curves for Dummies
Abstract: A theorem on the random variable of inverse image sizes for a polynomial over a finite field of order q computes the moments of the random variable of inverse images sizes up to an error term. This error term decreases with the inverse of the square root of q. Standard results in the theory of elliptic curves will be used to show that this error term cannot generally be improved. No familiarity with the extensive theory of elliptic curves will be assumed.

September 10
Speaker: Alexander Borisov (Binghamton)
Title: A Primer on Modular Forms
Abstract: I will give a very brief introduction to the theory of modular forms. The topics will include the action of the group of invertible 2×2 integer matrices on the upper half plane, the classical fundamental region, the modular functions and forms of even weight, the Eisenstein series. If time permits, I will also talk about Hecke operators, Petersson inner product, and forms of higher level. The main reference for the talk is the last chapter of the classical book by J.-P. Serre “A Course in Arithmetic”.

September 17 (Unusual time: 4:45-5:45 pm)
Speaker: Noah Stephens-Davidowitz (Cornell)
Title: A reverse Minkowski theorem
Abstract: Minkowski's celebrated first theorem is one of the foundational results in the study of the geometry of numbers, and it has innumerable applications from number theory to convex geometry to cryptography. It tells us that a lattice (i.e., a linear transform of Z^n) that is globally dense (i.e., has low determinant) must be locally dense (i.e., must have many short vectors). We will show a proof of a nearly tight converse to Minkowski's theorem, originally conjectured by Daniel Dadush—a lattice with many short points must have a sublattice with small determinant. This “reverse Minkowski theorem” has numerous applications in, e.g., complexity theory, additive combinatorics, cryptography, the study of Brownian motion on flat tori, algorithms for lattice problems, etc. Just recently, it was used by Reis and Rothvoss to give the first asymptotic improvement to integer programming in nearly forty years.
Based on joint work with Oded Regev.

September 24
Speaker: Alexander Borisov (Binghamton)
Title: Locally Integer Polynomial Functions on Infinite Subsets of Integers
Abstract: A locally integer polynomial function on a subset of $\mathbb Z$ is an integer-valued function whose restriction to any finite subset is given by a polynomial with integer coefficients. For infinite domains these functions have some properties reminiscent of the properties of complex analytic functions. For example, a LIP function that takes value 0 at infinitely many inputs must be zero, so a “LIP continuation” from a smaller infinite set to a larger one is unique, if exists. The talk will be partially based on the preprint https://arxiv.org/pdf/2401.17955 but will also include more recent results related to LIP continuation and the structure of the corresponding rings. I spoke about this topic in the beginning of last semester, but will not assume any knowledge about the subject. A number of open questions will be proposed.

October 15
Speaker: Hari Asokan (Binghamton)
Title: GIT quotient of SL_g action on symmetric matrices
Abstract: The special linear group SL_g acts on the vector space of symmetric matrices, V_g by congruence action. This action extends to (V_g)^{r+1}. We are interested in the GIT quotient and the invariant ring of this action. In this talk we will discuss a linear algebra problem arising from the above situation.

October 22
Speaker: Anton Mosunov (Cornell)
Title: On the Area of the Fundamental Region of a Binary Form
Abstract: Let F(x,y) be a binary form with integer coefficients, of degree n > 2 and nonzero discriminant D_F. Let A_F denote the area of the fundamental domain {(x,y) in R^2 : |F(x,y)|\leq 1}. Back in the 90s my academic brother, Michael Bean, proved that the quantity |D_F|^{1/n(n-1)}*A_F achieves its maximum over all forms specified above when F(x,y) = xy(x-y). Moreover, when n is fixed, he conjectured that the maximum must be attained by the form
F_n*(x,y) = \prod_{k=1}^n (sin(kπ/n)*x – cos(kπ/n)*y)
I will talk about the recent work on this conjecture, as well as about a similar problem which concerns bounding the quantity h_F^{2/n} *A_F from below. Here h_F is an appropriately chosen height of F.

October 29
Speaker: Jaiung Jun (SUNY New Paltz, IAS)
Title: Schemes over the natural numbers
Abstract: In this talk, I will first explain how a notion of positivity in algebraic geometry / number theory could be captured in terms of semirings by providing an example of the narrow class group of a number field as a reflexive Picard group. Then, I will introduce a notion of equivariant vector bundles over the natural numbers, and prove a version of Klyachko classification theorem of toric vector bundles in this setting.

November 5
Speaker: Mithun Veettil (Binghamton)
Title: Kirch and Golomb Topology on $\mathbb N$ and Locally Integer Polynomials
Abstract: Golomb topology on a domain is a generalization of arithmetic topology on $\mathbb N$. In this talk, we will show that Golomb topology is connected on $\mathbb N$ but not locally connected. But, by modifying the basis ”a little bit” we obtain a new topology on $\mathbb N$, known as Kirch topology, and we will prove that $\mathbb N$ is locally connected with this new topology. With these topologies, we will explore the structure of locally integer polynomials on $\mathbb N$.

November 12
Speaker: Chris Schroeder (Binghamton)
Title: The O'Nan Scott Theorem
Abstract: The O'Nan Scott Theorem classifies the maximal subgroups of the finite symmetric groups. It is arguably the most influential theorem in the theory of permutation groups, and it has far-reaching consequences in finite group theory in light of the classification of finite simple groups. In this talk, which is a continuation of the talk in the Algebra Seminar immediately beforehand, we will prove it.

December 3
Speaker: Marcin Mazur (Binghamton)
Title: Zsigmondy's Theorem
Abstract: In my earlier talk in the Algebra Seminar I showed how Zsigmondy's theorem can be used to get simple proofs of some fundamental results in algebra. In this expository lecture I will provide an elementary proof of Zsigmondy's theorem which will showcase some nice techniques from elementary number theory.

Fall 2025

2026-01-21T10:57:12-04:00

Fall 2025

August 26
Speaker: NA
Title: Organizational Meeting
Abstract:

September 9
Speaker: Huy Dang (Binghamton)
Title: The lifting problem for curves
Abstract: The lifting problem for curves asks: given a smooth, projective, connected curve 𝐶 over a field 𝑘 of characteristic 𝑝 > 0, which finite Galois coverings of 𝐶 lift to characteristic zero? In this talk, we provide an overview of the central questions and techniques used to study this problem. We will also discuss connections with other areas of research, including deformation theory and ramification theory.

September 16
Speaker: Huy Dang (Binghamton)
Title: Lifting abelian isogenies from characteristic $p$ to characteristic $0$
Abstract: In characteristic $0$, cyclic field extensions are classified by Kummer theory. In characteristic $p$, in addition to Kummer theory, one also needs Artin–Schreier–Witt theory to describe these extensions. Matsuda constructed a formal morphism that connects these two theories, providing a bridge between characteristic $p$ and characteristic $0$. In this talk, we present an algebraization of Matsuda’s construction to study the lifting of abelian isogenies from characteristic $p$ to characteristic $0$. As an application, we show that every lift of an abelian étale cover of a local scheme arises as the pullback of such a lift of an abelian isogeny. This is joint work with Khai-Hoan Nguyen-Dang.

September 30
Speaker: Enrique Trevino (Lake Forest College)
Title: On sets whose subsets have integer mean
Abstract: We say a finite set of positive integers is balanced if all of its subsets have integer mean. For a positive integer $N$, let $M(N)$ be the cardinality of the largest balanced set all of whose elements are less than or equal to $N$, and let $S(N)$ be the cardinality of the largest balanced set with elements less than or equal to $N$ that has maximal sum. For example, for $N = 3000$, the largest balanced set is $\{3000, 2580, 2160, 1740, 1320, 980, 480, 60\}$ so $M(3000) = 8$, while the largest set with maximal sum is $\{3000, 2940, 2880, 2820, 2760, 2700, 2640\}$, so $S(3000)= 7$. In this talk we will study the question of when $M(N) = S(N)$.

October 16 (Thursday, 2:45-3:45 pm, cross-listed from Geometry and Topology Seminar)
Speaker: John Abou-Rached (Binghamton)
Title: Integral models for non-Shimura curves and the Eichler-Shimura congruence relation
Abstract: We construct integral models for an infinite family of algebraic curves that includes noncongruence modular curves, as well as curves whose uniformizers are non-arithmetic Fuchsian groups. Most of these curves are not Shimura curves. We affirm a conjecture of Mukamel that the set of primes of good reduction of such curves have arithmetic significance and obtain an explicit description of this set. We conjecture that a version of Deligne-Rapoport's study of the reduction of modular curves holds in this context, and conjecture that a version of the Eichler-Shimura congruence relation holds in this setting, in resonance with Shimura curves.

October 21
Speaker: Alexander Borisov
Title: A structure sheaf for Kirch topology on $\mathbb N$
Abstract: Kirch topology on $\mathbb N$ goes back to a 1969 paper of Kirch. It can be defined by a basis of open sets that consists of all infinite arithmetic progressions $a+d\mathbb N_0$, such that $gcd(a,d)=1$ and $d$ is square-free. It is Hausdorff, connected, and locally connected. One can hope that in the classical imperfect analogy between arithmetic and geometry this can serve as an arithmetic analog of the usual topology on $\mathbb C$. However, the usual topology on $\mathbb C$ comes with a structure sheaf of complex-analytic functions. As far as I know, no analog for Kirch topology has been proposed before me. I believe that I have stumbled upon just such a thing, more by accident than by a conscious effort: locally LIP functions. These are functions from Kirch-open sets to $\mathbb Z$ such that for every point in the domain there is a Kirch-open neighborhood on which the function is “locally integer polynomial” (LIP): its interpolation polynomial on every finite set has integer coefficients. I will explain why this seems to be a natural object, what I know about it and what I hope to achieve. Some of the material of this talk will be based on my recently published paper: https://math.colgate.edu/~integers/z41/z41.pdf

November 4
Speaker: Bhargavi Parthasarathy (Syracuse University)
Title: Homomorphisms of maximal Cohen-Macaulay modules over the cone of an elliptic curve
Abstract: Consider the ring $R=k[[x,y,z]]/(f)$ where $f=x^3+y^3+z^3$ with an algebraically closed field $k$ and $char(k)\neq 3$. In a 2002 paper, Laza, Popescu and Pfister used Atiyah’s classification of vector bundles over elliptic curves to obtain a description of the maximal Cohen-Macaulay modules (MCM) over $R$. In particular, the matrix factorizations corresponding to rank one MCMs can be described using points in $V(f)$. If $M, \, N$ are rank one MCMs over $R$, then so is ${\rm Hom}_R(M,N)$. In this talk, I will discuss how the elliptic group law on $f$ can be used to obtain the point in $V(f)$ that describes the matrix factorization corresponding to ${\rm Hom}_R(M,N)$.

November 18 (Joint with the Combinatorics Seminar)
Speaker: Jaeho Shin (Seoul National University)
Title: Biconvex Polytopes and Tropical Linear Spaces
Abstract: Tropical geometry is geometry over exponents of algebraic expressions, using the “logarithmized” operations (min,+) or (max,+). In this setting, one can define tropical convexity and the related notion of biconvex polytopes, which are convex both classically and tropically. There is also a tropical analogue of linear spaces, called tropical linear spaces. Sturmfels conjectured that every biconvex polytope arises as a cell of a tropical linear space. In this talk, I will outline a proof of this conjecture.

December 2
Speaker: Bogdan Ion (University of Pittsburgh)
Title: Bernoulli operators, Dirichlet series, and analytic continuation
Abstract: Bernoulli operators are distributions with discrete support associated to Dirichlet series (or rather to the corresponding power series). The most basic case, when the power series has a pole singularity at $z=1$ is analyzed in detail. Its main property is that it naturally acts on the vector space of analytic functions in the plane (with possible isolated singularities) that fall in the image of the Laplace-Mellin transform (for the variable in some half-plane). The action of the Bernoulli operator on the function $t^s$, provides the analytic continuation of the associated Dirichlet series and also detailed information about the location of poles, their resides, and special values. Using examples of arithmetic origin, I will attempt to illustrate what is reasonable to expect when the power series has a non-pole singularity at $z=1$, pointing to an extension of this theory to tempered distributions associated to modular forms.

Spring 2015

2019-08-28T17:05:52-04:00

Spring 2015

February 9

Speaker: Alexander Borisov (Binghamton University)
Title: Riemann-Roch Theorem of Tate and van der Geer-Schoof, and Arakelov geometry

Abstract: Arakelov geometry provides a way to “compactify” arithmetic varieties, by using techniques from analysis “at infinity”. An older theory of Tate, refined by van der Geer and Schoof, provides similar, but deeper results in dimension one (i.e. algebraic number fields). I will give an introduction to both theories and make a case for the existence of a unified theory of “heat transfer on arithmetic varieties”.

February 16

Speaker: Alexander Borisov (Binghamton University)
Title: Riemann-Roch Theorem of Tate and van der Geer-Schoof, and Arakelov geometry, Part 2

Abstract: This is the continuation of the February 9 talk, with the emphasis on the details of the arithmetic cohomology theory.

March 2

Speaker: Alexander Borisov (Binghamton University)
Title: Stein factorizations of resolutions of Keller maps with the P^2 target

Abstract: Keller maps are polynomial self-maps of an affine complex space that have constant non-zero jacobian and are not invertible. The Jacobian Conjecture asserts that such maps do not exist, and most approaches it concentrate on studying various properties of these possible counterexamples. I will describe the results and proofs of my recent paper on the subject. A major part of the talk will be a crash course in the classical theory of curves on algebraic surfaces: divisor classes, intersection form, canonical class, adjunction formula, etc.

March 9

Speaker: John R. Doyle (University of Rochester)
Title: Preperiodic portraits for unicritical polynomials

Abstract: In 1964, I. N. Baker showed that if f(z) is (almost) any polynomial map defined over an algebraically closed field K of characteristic zero, then f(z) admits periodic points of every period N > 0. One could then ask the following dual question: Given a point P in K, a positive integer N, and an integer d at least 2, does there exist a polynomial f(z) of degree d for which P is periodic of period N? It turns out to be a trivial consequence of Baker's theorem that the answer to this question is always “yes.” The question becomes more interesting, however, if we restrict which polynomials of degree d we are allowed to consider. I will therefore discuss the situation where we restrict our attention to the family of polynomials of the form z^d + c, and I will completely answer the question in this case. Finally, I will also state a more general result for strictly preperiodic points.

March 16

Speaker: Lev Borisov (Rutgers University)
Title: Eisenstein series and equations of certain modular curves

Abstract: I will discuss properties of certain holomorphic Eisenstein series on the upper half plane and explain how they can be used to give explicit equations of the modular curves $\bar X_1(p)$ for prime $p$.

March 23

Speaker: Andrew Bridy (University of Rochester)
Title: The Artin-Mazur Zeta Function of a Rational Map in Positive Characteristic

Abstract: The Artin-Mazur zeta function of a dynamical system is a generating function that captures information about its periodic points. In characteristic zero, the zeta function of a rational map from P^1 to P^1 is known to be a rational function. In positive characteristic, the situation is much less clear. I show that the zeta function can be understood for a family of maps in positive characteristic that come from endomorphisms of algebraic groups. Somewhat surprisingly, it usually fails to be rational and can be shown to be transcendental.

March 30

Speaker: Alexander Borisov (Binghamton University)
Title: Stein factorizations of resolutions of Keller maps with the P^2 target (Rescheduled from March 2)

Abstract: Keller maps are polynomial self-maps of an affine complex space that have constant non-zero jacobian and are not invertible. The Jacobian Conjecture asserts that such maps do not exist, and most approaches it concentrate on studying various properties of these possible counterexamples. I will describe the results and proofs of my recent paper on the subject. A major part of the talk will be a crash course in the classical theory of curves on algebraic surfaces: divisor classes, intersection form, canonical class, adjunction formula, etc.

April 20

Speaker: Daniel Vallieres (Binghamton University)
Title: Lorentzian Weyl groups inside Vahlen groups.

Abstract: This is work in progress with Alex Feingold. In this talk, we will explain our attempt to describe Weyl groups of certain Kac-Moody algebras. We will explain the notion of a Vahlen group which seems to give a nice conceptual framework to attack this problem.

April 24, 4:30 pm (Special Meeting: Dissertation Defense)

Speaker: Jinghao Li (Binghamton University)
Title: Purity Results on F-crystals

Abstract: The first half of the talk will be an introduction to Rings of Witt vectors, F-crystals and a survey of purity results for stratifications of reduced locally Noetherian schemes in positive characteristic associated to F-crystals. In the second half, I will present our new purity result on F-crystals and prove that it implies all the previous purity results.

April 27

Speaker: Patrick Milano (Binghamton University)
Title: Part 1: Effectivity of Arakelov divisors

Abstract: This talk will be based on parts of a paper by van der Geer and Schoof. We will introduce the notion of an Arakelov divisor D on a number field F, and define the effectivity of D. We will then define a real number h^0(D), which may be viewed as the analogue of the dimension of the space of global sections of a divisor on a complete algebraic curve. We'll use this to get an arithmetic version of the Riemann-Roch theorem.

Title: Part 2: Convolution structures and ghost-spaces

Abstract: This second talk will be based on a paper by Borisov, which builds on the work of van der Geer and Schoof. We'll define ghost-spaces and their dimensions, and then use those objects to define an appropriate notion of H^0(D) and H^1(D) for an Arakelov divisor D.

May 4

Speaker: Nicolas Templier (Cornell University)
Title: Weyl’s law in the theory of automorphic forms

Abstract: We will introduce some of the conjectures and probabilistic models concerning the automorphic spectrum. In works with S.W.Shin, P.Sarnak, J.Matz we have generalized some of the classical results to higher rank groups. Consequences are refinements of the Katz-Sarnak heuristics and a result towards the Ramanujan conjecture on average for SL(n,R)/SO(n).

Spring 2016

2017-01-25T15:17:32-04:00

Spring 2016

January 29

Speaker: FirstName LastName (Some University)
Title: Organizational Meeting

Abstract: We will discuss schedule and speakers for this semester

February 8

Speaker: Adrian Vasiu (Binghamton University)
Title: Classification of Lie algebras with perfect Killing form

Abstract: We will review basic properties of Lie algebras over arbitrary commutative rings. Then we will present the classification of Lie algebra over such rings whose killing forms are perfect. This re-obtains and generalizes prior works of Curtis, Seligman, Mills, Block–Zassenhaus, and Brown in late sixties and in seventies who worked over fields. This work will appear in Algebra and Number Theory journal.

February 15

Speaker: Adrian Vasiu (Binghamton University)
Title: Classification of Lie algebras with perfect Killing form, Part II

Abstract: We will go through few details of the proof of the classification stated last time that involve universal enveloping algebras, Casimir elements, and cohomology. Then we will talk about the main motivation behind this classification coming from extensions of group schemes.

February 22

Speaker: Jaiung Jun (Binghamton University)
Title: Berkovich Analytification, Tropicalization, and hyperfields.

Abstract: I will review the basic notions of Berkovich analytification in connection to tropicalization. Then I will explain how some basic definitions of Berkovich analytification can be restated by using hyperfields. In particular, this view can be linked to my previous work on hyperstructures of affine algebraic group schemes.

February 29

Speaker: Farbod Shokrieh (Cornell University)
Title: Classes of compactified Jacobians in the Grothendieck ring

Abstract: Let $C$ be a nodal curve over an algebraically closed field $k$. Denote with $\textrm{Pic}^0(C)$ the generalized Jacobian of $C$, which is the classifying space for line bundles on $C$ having degree zero on each irreducible component. If the dual graph of $C$ is not a tree, then $\textrm{Pic}^0(C)$ is not compact. But (many) nice compactifications of $\textrm{Pic}^0(C)$ are known. I will describe how one can use the combinatorics of the dual graph to compute the class of these compactifications in the ``Grothendieck ring of $k$-varieties''. This is ongoing joint work with Alberto Bellardini. The talk should be accessible to graduate students.

March 14

Speaker: Jaiung Jun (Binghamton University)
Title: Matroid theory for Algebraic Geometers

Abstract: This is an expository talk based on a survey paper “Matroid Theory for Algebraic Geometers” (by Eric Katz). We will introduce the basic definitions of matroid theory in connection to tropical linear spaces and explain the idea that tropical linear spaces and valuated matroids are the same things. We also review the recent paper “Matroids over hyperfields” (by Matt Baker) to see how various classes of matroids can be unified under one framework.

March 21

Speaker: Changwei Zhou (Binghamton University)
Title: Cohomology of Lie algebras

Abstract: In this talk we review the basic definition of cohomology of Lie algebras from an analytical point of view by tracing back the analytical theory of Lie groups using de Rham's theorem. If we have extra time we shall discuss related topics like the Loday-Quillen-Taygen theorem in cyclic homology, the unitary trick and some sample computations of groups.
Loday&Quillen's paper directly motivated the computation of Hochschild homology groups of differential operators, and much of later work on Hochschild homology on pseudo-differential operators is built up on it. Hopefully we can connect some of the dots in the talk to see a united picture.
The sources are Samuel&Ellenberg's paper “Cohomology groups of Lie groups and Lie algebras”, Loday&Quillen's paper “Cyclic homology and the Lie algebra homology of matrices”, and Melrose&Nister's paper “Homology of pseudodifferential operators I. Manifolds with boundary”.

April 11

Speaker: Patrick Milano (Binghamton University)
Title: Growth in Groups

Abstract: Let $G$ be a group and let $A$ be a finite subset of $G$. Write $A^k=\{x_1x_2\dots x_k:x_i\in A\}$. We can ask how $|A^k|$ grows as $k$ grows. We will survey some results and techniques related to this question, focusing on the case when $G$ is a linear algebraic group. The material in this talk is taken from a course taught by Harald Helfgott at the 2016 Arizona Winter School.

April 18

Speaker: Thomas Price (Toronto)
Title: Numerical Cohomology

Abstract: This talk will be an overview of a preprint of the same title. A lattice (i.e. a discrete subgroup of a finite-dimensional inner product space) can be thought of as a vector bundle over the “completion” of Spec(Z). We can associate numbers to a lattice that act like dimensions of cohomology vector spaces. Unfortunately, these numbers can be arbitrary nonnegative real numbers, and therefore can't literally be interpreted as dimensions of vector spaces. To get around this, we can develop a numerical approach to cohomology, where vector spaces and linear maps are replaced by real numbers.

April 25

Speaker: Jaiung Jun (Binghamton University)
Title: Analytic geometry over $\mathbb{F}_1$ as relative algebraic geometry

Abstract: Several years ago, Berkovich introduced a notion of analytic geometry over $\mathbb{F}_1$ by directly generalizing his construction of an analytic space over a non-Archimedean field. On the other hand, recently, Ben-Bassat and Kremnizer took a functorial approach of Toen and Vaquie on algebraic geometry over a closed symmetric monoidal category and proved that the category of analytic spaces (in the sense of Berkovich) embeds fully faithfully into the category of relative schemes which they constructed. I will present some of these ideas. The aim of this talk is to provide backgrounds on the material and explain my research projects in this direction.

April 26
(CROSS LISTING WITH THE ALGEBRA SEMINAR; SPECIAL DAY TUESDAY and TIME 2:50pm)
Speaker: An Huang (Harvard University)
Title: Riemann-Hilbert problem for period integrals

Abstract: Period integrals of an algebraic variety are transcendental objects that describe, among other things, deformations of the variety. They were originally studied by Euler, Gauss and Riemann, who inspired modern Hodge theory through the theory of periods. Period integrals also play a central role in mirror symmetry in recent years. In this talk, we will discuss a number of problems on period integrals that are crucial to understanding mirror symmetry for Calabi-Yau manifolds. We will see how the theory of D-modules have led us to solutions and deep insights into some of these problems.

May 2
(CROSS LISTING WITH THE COLLOQUIUM –Dean's Speaker Series in Geometry/Topology)
Speaker: Melvyn Nathanson (CUNY)
Title: Every Finite Set of Integes is an Asymptotic Approximate Group

Abstract: A set $A$ is an $(r, l)$-approximate group in the additive abelian group $G$ if $A$ is a nonempty subset of $G$ and there exists a subset $X$ of $G$ such that $|X| ≤ l$ and $rA ⊆ X + A.$ The set $A$ is an asymptotic $(r, l)$-approximate group if the sumset $hA$ is an $(r, l)$-approximate group for all sufficiently large integers $h.$ It is proved that every finite set of integers is an asymptotic $(r, r + 1)$-approximate group for every integer $r ≥ 2.$

May 3
(CROSS LISTING WITH THE COLLOQUIUM –Dean's Speaker Series in Geometry/Topology; SPECIAL DAY TUESDAY and TIME 4:30pm):
Speaker: Melvyn Nathanson (CUNY)
Title: Every Finite Subset of an Abelian group is an Asymptotic Approximate Group

Abstract: If $A$ is a nonempty subset of an additive abelian group $G$, then the $h$-fold sumset is \[hA = \{x_1 + \cdots + x_h : x_i \in A_i \text{ for } i=1,2,\ldots, h\}.\]
We do not assume that $A$ contains the identity, nor that $A$ is symmetric, nor that $A$ is finite. The set $A$ is an $(r,\ell)$-approximate group in $G$ if there exists a subset $X$ of $G$ such that $|X| \leq \ell$ and $rA \subseteq XA$. The set $A$ is an asymptotic $(r,\ell)$-approximate group if the sumset $hA$ is an $(r,\ell)$-approximate group for all sufficiently large $h.$ It is proved that every polytope in a real vector space is an asymptotic $(r,\ell)$-approximate group, that every finite set of lattice points is an asymptotic $(r,\ell)$-approximate group, and that every finite subset of every abelian group is an asymptotic $(r,\ell)$-approximate group.

May 9
(DEAN'S SPEAKER SERIES IN GEOMETRY/TOPOLOGY)
Speaker: Alexandru Buium (University of New Mexico)
Title: Arithmetic analogue of Painleve VI

Abstract: The Painleve VI equations are a family of differential equations appearing in a number of contexts in mathematics and theoretical physics.On the other hand the theory of differential equations possesses an arithmetic analoguein which derivatives are replaced by Fermat quotients. The aim of the talk isto explain how one can set up an arithmetic analogue of the Painleve' VI equations.We prove that this arithmetic analogue has a “Hamiltonian structure” analogous to the classical one.The talk is based on joint work with Yuri I. Manin.

May 10
(CROSS LISTING WITH THE COLLOQUIUM –Dean's Speaker Series in Geometry/Topology; SPECIAL DAY TUESDAY and TIME 4:30pm):
Speaker: Alexandru Buium (University of New Mexico)
Title: The differential geometry of Spec Z

Abstract: The aim of this talk is to show how one can develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers Z will play the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold will be played by the prime numbers.
The role of partial derivatives of functions with respect to the coordinates will be played by the Fermat quotients of integers with respect to the primes. The role of metrics (respectively 2-forms) will be played by symmetric (respectively antisymmetric) matrices with coefficients in Z. The role of connections (respectively curvature) attached to metrics or 2-forms will be played by certain adelic (respectively global) objects attached to matrices as above. One of the main conclusions of our theory will be that Spec Z is ``intrinsically curved;” the study of this curvature will then be one of the main tasks of the theory.

Spring 2017

2017-08-22T08:05:30-04:00

Spring 2017

January 23

Speaker: N/A
Title: Organizational Meeting

Abstract: We will discuss schedule and speakers for this semester

January 31 (Tuesday, 4:15–5:15)

Speaker: Alexander Borisov (Binghamton)
Title: Rigidity problems for polygons and polyhedra

Abstract: Every triangle can be uniquely determined, up to isometry, by three “simple measurements” (sides or angles). For a generic quadrilateral one needs five simple measurements. However some quadrilaterals, including squares, can be described by just four simple measurements. I will present a number of results regarding this and related phenomena, both positive and negative, for polygons and some polyhedra, based on my 2010 Monthly paper, joint with Mark Dickinson and Stuart Hastings. If time permits, I will also discuss some related notions and results: Cauchy rigidity theorem, flexible polyhedra, Bellows Conjecture, and Dehn invariant. Most of the talk will be elementary.

February 6

Speaker: Jaiung Jun (Binghamton)
Title: Geometry of hyperfields in a view of Berkovich theory

Abstract: I will discuss possible research directions on geometry of hyperfields in connection to abstract tropical curves and Berkovich theory of analytic spaces. In particular, we will discuss how the tropical projective line can be considered as the abstract curve associated to the tropical function field (properly defined).

February 13

Speaker: Changwei Zhou (Binghamton)
Title: Overview of Arakelov intersection theory

Abstract: In today’s talk we give an overview of the basic set up of Arakelov intersection theory and discuss some introductory material on Faltings-Riemann-Roch theorem using metrized line bundles. The talk roughly follows Lang’s book and Faltings’ original paper Calculus on Arithmetic Surfaces, plus some examples.

February 20

Speaker: TBA
Title: TBA

Abstract: TBA

February 27

Speaker: Patrick Milano (Binghamton)
Title: The Riemann-Hurwitz formula

Abstract: Let X and Y be compact Riemann surfaces, and let f be a non-constant holomorphic map from X to Y. The Riemann-Hurwitz formula relates the genus of X, the genus of Y, the degree of f, and the amount of ramification of f. We will outline a proof of the formula. As an application, we will compute the genus of the Fermat curve X^n+Y^n=Z^n.

March 14 (Tuesday, 4:15–5:15, room: 309 WH)

Speaker: Martin Ulirsch (University of Michigan)
Title: The moduli stack of tropical curves

Abstract: The moduli space of tropical curves (and its variants) are some of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve (at least in the positive genus case). The classical work of Deligne-Knudsen-Mumford has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces.
In this talk I am going to give an introduction to these fascinating moduli spaces and report on ongoing work with Renzo Cavalieri, Melody Chan, and Jonathan Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this $2$-categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Moreover, I will propose two different ways of describing the process of tropicalization: one via logarithmic geometry in the sense of Fontaine-Kato-Illusie and the other via non-Archimedean analytic geometry in the sense of Berkovich.

March 21 (Tuesday, 4:15–5:15)

Speaker: Jaiung Jun (Binghamton)
Title: Introduction to hyperrings and hyperfields

Abstract: We introduce hyperrings and hyperfields. These are algebraic structures which generalize the classical commutative rings and fields. In this talk, we aim to introduce these rather exotic structures and illustrate several examples. We also discuss how hyperrings and hyperfields show up and fit into the classical theory, in particular, algebraic geometry and combinatorics. If time permits, we discuss how hyperfields can be used to reformulate some basic definitions of Berkovich's theory of analytic spaces.

March 27

Speaker: Noah Giansiracusa (Swarthmore College)
Title(tentative): Tropicalizing schemes

Abstract(tentative): I'll discuss joint work with my brother, Jeff Giansiracusa, in which we extend tropicalization to a scheme-theoretic setting by writing down explicit equations cutting out tropical varieties. Tropical geometry has been rapidly gaining momentum and achieving exciting results in a variety of areas; our hope is that by expanding the scope to allow non-reduced structure and basing tropical methods on algebraic foundations that the range of applications with increase, though the program is still in its early steps. Connections to matroids and to Berkovich analytification will be mentioned.

April 4 (Tuesday, 4:15–5:15)

Speaker: Alexander Borisov (Binghamton)
Title: Looking for Keller maps: a report on the hunt

Abstract: Keller maps are the counterexamples to the Jacobian conjecture. While we do not know if they exist, we have some ideas on how they might “look like” and where they might “live”. I will describe my ongoing work on finding them, in dimension two. There will be complicated maps, large trees, mysterious drawings, and 100+ degree polynomials.

April 24

Speaker: Neil Epstein (George Mason)
Title: Generic matroids - a bilevel matroid-like structure on sets with topological structure

Abstract: Matroids are not a traditional go-to structure within the field of commutative algebra. However, we show that given a finitely generated standard graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of ‘generic exchange', allowing one to pass between any two of them in at most $d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $k$-algebras. We unify all these results into a common axiomatic framework by introducing a structure we call a generic matroid, which is a common generalization of a topological space and a matroid.
We suspect that generic matroids fit into a much more general context, hence my interest in presenting this work at a place like Binghamton which has several experts in matroids and matroid-like structures.
This work is joint with Joseph Brennan.

April 25 (Tuesday, 4:15–5:15)

Speaker: Micah Loverro (Binghamton)
Title: Affine group schemes and representations

Abstract: I will introduce smooth affine reductive and semisimple group schemes and a problem about classifying representations corresponding to representations of the Lie algebra.

May 1

Speaker: Junguk Lee (Yonsei University, Korea)
Title: On the structure of certain valued fields

Abstract: For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if each pair of $n$-th residue rings is isomorphic for each $n\ge1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1,n_2\ge 1$, if $n_2$ is large enough, then any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuation rings. We can find a lower bound for $n_2$ depending only on $K_2$. Moreover, we get a functor from a category of certain principal Artinian local rings of length $n$ to a category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. The result improves Basarab's generalization of the AKE-principle for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields. This is joint work with Wan Lee.

May 9 (Tuesday, 4:15–5:15)

Speaker: John Brown (Binghamton)
Title: Classifying finite hypergeometric groups, height one balanced integral factorial ratio sequences, and some step functions

Abstract: (This is the second half of the candidacy exam talk. The first part is at the Algebra Seminar. The title and abstract are for both parts). In this talk we will discuss some connections between hypergeometric series, factorial ratio sequences, and non-negative bounded integer-valued step functions. We will start with a finiteness criterion for hypergeometric groups by Beukers and Heckman, then show how this leads to the classification by Bober of integral balanced factorial ratio sequences of height one, and thus a proof that a conjectured classification of a certain class of step functions by Vasyunin is complete.

Spring 2018

2018-08-27T16:30:45-04:00

Spring 2018

January 22
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester

January 29
Speaker: Adrian Vasiu (Binghamton)
Title: Purity of Crystalline Strata
Abstract: We report on joint work with Jinghao Li on the purity of crystalline strata in characteristic p, such as Artin–Schreier strata, p-ranks strata, break points strata, and Newton polygon strata. This refines and reobtains prior works due to Zink, de Jong–Oort, Vasiu, and Yang and provides two new proofs of an unpublished result of Deligne.

February 5
Speaker: Alexander Borisov (Binghamton)
Title: A hunt for the plane Keller maps
Abstract: Plane Keller maps are the counterexamples to the two-dimensional Jacobian Conjecture. My approach to finding them (or proving that they do not exist) has been to compactify the input and output affine planes and to resolve the map. Smooth compactifications of the affine plane can be described by the graphs of curves at infinity. Over the past several years I found many natural restrictions on these graphs, but no “silver bullet” that would prevent the existence of Keller maps, thus proving the two-dimensional Jacobian Conjecture. And, indeed, the corresponding combinatorial problem actually has a solution! I will present this solution, which is a pair of trees with a “map” between them, and outline the approaches to the next challenge: finding the beast (Keller map) that “lives” on these trees.

February 12
Speaker: Sergio Da Silva (Cornell)
Title: Frobenius splittings and the desingularization of hypersurfaces in positive characteristic
Abstract: In its simplest form, a resolution of singularities is a birational map from a smooth algebraic variety to a singular one. Stronger versions require extra conditions such as the map being an isomorphism over the smooth locus. Hironaka's famous result provides an answer in characteristic zero, with various algorithmic approaches being later introduced. Desingularization in positive characteristic however has remained a difficult problem, mostly because characteristic zero techniques fail in this setting.
I will give an overview of this desingularization algorithm and introduce Frobenius split varieties. Working in the affine hypersurface case, I will show why curves and surfaces that define Frobenius splittings can be desingularized without alteration to the algorithm. No prior experience with the resolution of singularities or Frobenius splittings is required.

February 26
Speaker: Sayak Sengupta (Binghamton)
Title: Dimension Theory Part I
Abstract: I would like to start with the graded rings and graded modules over graded rings, their properties, briefly go over the definition and uses of additive functions over Z; discuss Hilbert-Serre Theorem on Poincare Series on a graded module and ultimately lay the groundwork for the Dimension Theorem with some propositions.

March 12
Speaker: Yuto Yamamoto (Yale)
Title: Tropical K3 surfaces
Abstract: Let $\Delta$ be a smooth reflexive polytope in dimension 3 and $W$ be a tropical polynomial whose Newton polytope is its polar dual. By contracting a tropical K3 hypersurface defined by $W$, we can construct a $2$-sphere equipped with an integral affine structure with singularities. We write the complement of the singularity as $i \colon B_0 \hookrightarrow B$, and the local system of integral tangent vectors on $B_0$ as $T$. The cohomology $H^1(B, i_\ast T)$ corresponds to the deformations of tropical structures of $B$. We show that there exists a primitive embedding of the Picard groupd of the toric variety associated with the normal fan of $\Delta$ into $H^1(B, i_\ast T)$.

March 19
Speaker: Daniel Le (Toronto)
Title: The weight part of Serre's conjecture
Abstract: In the 70's, Serre conjectured that all odd irreducible continuous mod p Galois representations arise from modular forms. A decade later, he conjectured a recipe for the weight and level of the modular forms in terms of the Galois representations–a recipe which would play a key role in the proof of Fermat's Last Theorem. In Serre's original context, these conjectures are now known. We survey recent conjectures and results about the weight part of Serre's conjecture for more general automorphic forms. The main ingredient is a description of local Galois deformation rings using local models. This is joint work with B.V. Le Hung, B. Levin, and S. Morra.

March 26
Speaker: Sayak Sengupta (Binghamton)
Title: Dimension Theory Part II
Abstract: We are going to start where we left off recalling the Hilbert-Serre Theorem and one of its consequences. Then we are going to introduce the concept of ideal-filtration of a ring and prove 3 results which eventually are going to lead us to the main result of the talk/; The Dimension Theorem (Proof included).

April 9
Speaker: Jacob Matherne (UMass Amherst)
Title: Derived geometric Satake equivalence, Springer correspondence, and small representations
Abstract: A recurring theme in geometric representation theory is the ability to describe representations in terms of the topology of certain spaces. Two major theorems in this area are the geometric Satake equivalence and the Springer correspondence, which state:

1. For G a semisimple algebraic group, we can realize Rep(G) using intersection cohomology of the affine Grassmannian for the Langlands dual group.
2. For W a Weyl group, we can realize Rep(W) using intersection cohomology of the nilpotent cone.

In the late 90s, M. Reeder computed the Weyl group action on the zero weight space of the irreducible representations of G, thereby relating Rep(G) to Rep(W). More recently, P. Achar, A. Henderson, and S. Riche established a functorial relationship between the two phenomena above. In my talk, I will review this story and discuss a result which extends their functorial relationship to the setting of mixed, derived categories.

April 16
Speaker: Changwei Zhou (Binghamton)
Title: Gaussian measure and discrete Laplacian
Abstract: The Poisson summation formula for Gaussians played an important role for 1D Arakelov theory. The analog of Gaussian measure in 2D Arakelov theory has been absent. In this talk we shall discuss some attempts to construct Gaussian measures in 2D setting using discrete Laplacian. Specifically we want to discuss a “no free lunch theorem” for discrete Laplacian. If time allows, we may talk about the connection between discrete Laplacian, discrete holomorphic derivative as well as the discrete Gaussian free field.

April 23
Speaker: Micah Loverro (Binghamton)
Title: G-modules and Lie(G)-modules
Abstract: I'll continue from my previous talk, where we have a finite dimensional representation V of a semisimple simply-connected algebraic group G_K, and we want to know when a Lie(G)-module M inside V is also a G-module, where G=Spec(A) is a smooth affine group scheme over a Noetherian domain R whose field of fractions is K, and G_K = Spec(A \otimes_R K). We have a condition on the highest weight of the representation which guarantees that such an M is a G-module. This time I'll show how to construct counterexamples when the condition is not met.

April 30
Speaker: Brian Hwang (Cornell)
Title: Local Models of Shimura Varieties and Limit Linear Series on Algebraic Curves
Abstract: Roughly speaking, Shimura varieties are moduli spaces of abelian varieties (with some additional structure) that are closely linked to the structure of reductive groups. They have demonstrated themselves as a useful tool for studying number theoretic questions as well as possessing an interesting geometry in and of themselves. However, their mod p reductions are notoriously difficult to study. One fruitful way to study these mod p reductions is by using local models, which can be patched together to obtain the entire space. Recently, we discovered that local models of certain Shimura varieties also arise in a different subject: limit linear series on algebraic curves, which concerns degenerations of line bundles on algebraic curves. This simplifies certain aspects of local models and highlights some strange connections between the two subjects that we will explain. This is joint work with Binglin Li.

May 7
Speaker: Patrick Milano (Binghamton) – Thesis Defense
Title: Mixed Ghost Spaces
Abstract: An Arakelov divisor $D$ on a number field $K$ is a formal finite sum of prime ideals of $O_K$ and infinite primes of $K$. A ghost space is a kind of object introduced by Borisov in order to define $H^0(D)$ and $H^1(D)$ for Arakelov divisors.
A mixed ghost space is a generalization of the ghost spaces used to construct $H^0(D)$ and $H^1(D)$. I will develop the theory of mixed ghost spaces. The main focus will be on ghost spaces given by Gaussian functions on $\mathbb{R}^n$, which can be completely classified up to some natural equivalence relations. I will also use mixed ghost spaces to construct long exact sequences in Borisov's cohomology.

Spring 2019

2019-08-22T08:01:27-04:00

Spring 2019

January 29 (Tuesday)
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester

February 4 (Monday)
Speaker: Xiao Xiao (Utica College)
Title: Automorphism group schemes at finite level of $F$-cyclic $F$-crystals
Abstract: Let $M$ be an $F$-crystal over an algebraically closed field of positive characteristic. For every integer $m \geq 1$, let $\gamma_{M}(m)$ be the dimension of the automorphism group scheme $\mathrm{Aut}_m(M)$ of $M$ at finite level $m$. In 2012, Gabber and Vasiu proved that $0 \leq \gamma_{M}(1) < \gamma_{M}(2) < \cdots < \gamma_{M}(n_{M}) = \gamma_{M}(n_{M}+1) = \cdots$ where $n_{M}$ is the isomorphism number of $M$, and that $\gamma_{M}(m+1)- \gamma_{M}(m) \leq \gamma_{M}(m)- \gamma_{M}(m-1)$ for all $m \geq 1$ if $M$ is a Dieudonn\'e module over $k$. We generalize the same result to arbitrary $F$-crystals in 2014. Questions have been asked whether $\gamma_{M}(m+1)- \gamma_{M}(m) < \gamma_{M}(m)- \gamma_{M}(m-1)$ for all $1 \leq m \leq n_{M}$ for any $F$-crystal $M$. In this talk, we will discuss a combinatorial formula that calculates $\gamma_{M}(m)$ for a certain family of $F$-crystals called $F$-cyclic $F$-crystals. This formula allows to give a negative answer to the aforementioned question in general but a positive answer to some family of Dieudonn\'e modules.

February 19 (Tuesday)
Speaker: Alexander Borisov (Binghamton)
Title: An update on the Keller map search
Abstract: I will describe in detail several possible frameworks for Keller maps, that is solutions to the combinatorial problem related to a possible counterexample to the two-dimensional Jacobian Conjecture. The talk will be based on my recent preprint, https://arxiv.org/abs/1901.04073 with some even newer results.

February 25 (Monday)
Speaker: Liang Xiao (UConn Storrs)
Title: Bloch–Kato conjecture for some Rankin-Selberg motives.
Abstract: The Birch and Swinnerton-Dyer conjecture is known in the case of rank 0 and 1 thanks to the foundational work of Kolyvagin and Gross-Zagier. In this talk, I will report on a joint work in progress with Yifeng Liu, Yichao Tian, Wei Zhang, and Xinwen Zhu. We study the analogue and generalizations of Kolyvagin's result to the unitary Gan-Gross-Prasad paradigm. More precisely, our ultimate goal is to show that, under some technical conditions, if the central value of the Rankin-Selberg L-function of an automorphic representation of U(n)*U(n+1) is nonzero, then the associated Selmer group is trivial; Analogously, if the Selmer class of certain cycle for the U(n)*U(n+1)-Shimura variety is nontrivial, then the dimension of the corresponding Selmer group is one.

March 5 (Tuesday)
Speaker: Alexander Borisov (Binghamton)
Title: Classification of finitely generated abelian groups and Singular Value Decomposition Theorem
Abstract: The Classification Theorem for finitely generated abelian groups is one of the basic theorems in Abstract Algebra. The Singular Value Decomposition Theorem is a basic fact in Linear Algebra, especially popular in applied mathematics. We will show that these two theorems, and also the classification theorem for finitely generated modules over a discrete valuation ring, are very much related to each other. In fact, they are similar not just in statement but also in proof. The talk is aimed primarily at graduate students, no advanced background is assumed.

March 12 (Tuesday)
Speaker: Vaidehee Thatte (Binghamton)
Title: N/A
Abstract: Organizational meeting of the new “No Theory” seminar for number theory and related areas

March 26 (Tuesday)
Speaker: Andrew Lamoureux
Title: Arithmetic Differential Operators on Z_p
Abstract: We will examine a paper of A. Buium discussing an analogue of the derivative for the p-adic integers. The corresponding notion of “analytic” turns out to be equivalent to that of an arithmetic differential operator, essentially a power series where variables are replaced with iterations of the derivative. Moreover, the number of variables in such a series is related to the radius of the power series making the function analytic. Time allowing, I'll explain how I'm working on generalizing this result.

April 2 (Tuesday)
Speaker: Changwei Zhou (Binghamton)
Title: Effective upper bound on analytic torsion for Arakelov metric
Abstract: In this talk I will review analytic torsion and show a proof that it has an upper bound only depending on genus when we give the surface Arakelov metric. To my knowledge this is the first result of this type in the literature. The work is a direct corollary of Jorgenson and Kramer's work on the non-completeness of Arakelov metric on the moduli space and Selberg zeta functions. I will also discuss some relevant earlier work by Wilms, Bost, Soule, Wentworth and Faltings.

April 9 (Tuesday)
Speaker: Renee Bell (UPenn)
Title: Local-to-Global Extensions for Wildly Ramified Covers of Curves
Abstract: Given a Galois cover of curves $X \to Y$ with Galois group $G$ which is totally ramified at a point $x$ and unramified elsewhere, restriction to the punctured formal neighborhood of $x$ induces a Galois extension of Laurent series rings $k((u))/k((t))$. If we fix a base curve $Y$, we can ask when a Galois extension of Laurent series rings comes from a global cover of $Y$ in this way. Harbater proved that over a separably closed field, every Laurent series extension comes from a global cover for any base curve if $G$ is a $p$-group, and he gave a condition for the uniqueness of such an extension. Using a generalization of Artin–Schreier theory to non-abelian $p$-groups, we fully characterize the curves $Y$ for which this extension property holds and for which it is unique up to isomorphism, but over a more general ground field.

April 15 (Monday)
Speaker: Samantha Wyler (Binghamton)
Title: Perfect Numbers, Amicable Numbers, and More
Abstract: A Perfect number, a positive integer that is equal to the sum of its proper divisors. The discovery of such numbers is lost in prehistory. However, mathematics going as far back as the Pythagoreans (founded c. 525 BCE) studied perfect numbers for their “mystical” properties. We will go over a few examples, and facts about these numbers. We will also look at amicable numbers, two numbers are amicable to each other if the sum of the proper divisors of each is equal to the other number. We will conclude with looking at a new definition I made up called “pseudo amicable numbers” and go over some examples and consequences.

Special talk – Thesis Defense
April 23 (Tuesday)
Speaker: Changwei Zhou (Binghamton)
Title: Some results on Arakelov theory of arithmetic surfaces
Abstract: In 1997, Gillet and Soule proposed a conjecture suggesting there is an upper bound of regularized determinant of Laplacian (analytic torsion) for any metrized line bundle over a Riemann surface independent of the metric. Here we verify it for the Arakelov metric and trivial line bundle. Our result indicates there is an asympototic upper bound of order g for genus large enough. This is built upon work of Bost, Deligne, Jorgenson, Kramer, Wilms, Wentworth. We also discuss some preliminary constructions we tried for constructing cohomology space on arithmetic surfaces.

April 29 (Monday)
Speaker: Serin Hong (Michigan)
Title: Surjective bundle maps and bundle extensions over the Fargues-Fontaine curve
Abstract: Vector bundles on the Fargues-Fontaine curve play a pivotal role in recent development of p-adic Hodge theory and related fields, as they provide geometric interpretations of many constructions in these fields. The most striking example is the geometrization of the local Langlands correspondence due to Fargues where the correspondence is stated in terms of certain sheaves on the stack of vector bundles on the Fargues-Fontaine curve.
In this talk, we give two classification theorems regarding vector bundles on the Fargues-Fontaine curve: a classification of all pairs of vector bundles with a surjective bundle map between them and a classification of extensions of two given vector bundles satisfying certain conditions. We also explain several applications of our classification theorems, some of which are closely related to the geometrization of the local Langlands correspondence. This talk is based on my recent work plus a previous joint work with C. Birkbeck, T. Feng, D. Hansen, Q. Li, A. Wang and L. Ye.

April 30 (Tuesday)
Speaker: Evangelia Gazaki (Michigan)
Title: A structure theorem for zero-cycles on products of elliptic curves over p-adic fields.
Abstract: In the mid 90's Colliot-Thélène formulated a conjecture about zero-cycles on smooth projective varieties over p-adic fields. A weaker form of this conjecture was recently established, but the general conjecture is only known for very limited classes of varieties. In this talk I will present some recent joint work with Isabel Leal, where we prove this conjecture for products of elliptic curves, under some assumptions on their reduction type. Our methods often allow us to obtain very sharp results about the structure of the group of zero-cycles on such products and also give us some promising global-to-local information.

Spring 2020

2020-09-01T17:21:06-04:00

Spring 2020

All remaining Arithmetic Seminar talks are cancelled/postponed. Some of the talks may be rescheduled in the online format, if the speaker is willing and able to do it.

January 28
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester

February 4
Speaker: Fikreab Admasu (Binghamton University)
Title: Buildings with examples
Abstract: This talk will be an introduction to buildings with examples. Jacques Tits introduced the notion of buildings to provide a systematic geometric description of certain groups such as finite simple groups. They have found applications in several other areas. We will look at some examples such as affine buildings and finite projective planes.

February 11
Speaker: Fikreab Admasu (Binghamton University)
Title: Buildings and an application
Abstract: In this second talk, we will look at the construction of buildings such as spherical buildings. One application of buildings, due toJ. Igusa, M. du Sautoy and A. Lubotzky is that the local factors of the zeta function of a class of nilpotent groups is expressed in terms of the combinatorics of the building of the associated algebraic group.

February 18
Speaker: Alexander Borisov (Binghamton University)
Title: Lehmer's Conjecture and Dimitrov's proof of Schinzel-Zassenhaus Conjecture
Abstract: Mahler measure $M(f)$ of a univariate monic integer polynomial $f$ is the product of absolute values of its complex roots outside of the unit circle. A long-standing conjecture, known as Lehmer's Conjecture, asserts that there is a positive constant $C$, such that $M(f)$ either equals $1$ or is at least $1+C$. A somewhat weaker conjecture, due to Schinzel and Zassenhaus, says that for some positive $C$, independent of the degree $d$ of the polynomial $f$, the maximum absolute value of roots of $f$ either equals $1$ or is at least $1+C/d$. In a very recent breakthrough development the proof of this weaker conjecture was announced by Vesselin Dimitrov. I will explain some background and sketch Dimitrov's argument.

February 25
Speaker: Mahdi Asgari (Cornell University and OSU)
Title: Symmetric Algebras and L-functions
Abstract: I will explain the role that decompositions of certain symmetric algebras play in linking the local and global Langlands L-functions through a process called “unramified computation” and present some results along these lines, both old and new.

March 3
Speaker: Bin Guan (CUNY)
Title: Averages of central values of triple product L-functions
Abstract: Feigon and Whitehouse studied central values of triple L-functions averaged over newforms of weight 2 and prime level. They proved some exact formulas applying the results of Gross and Kudla which link central values of triple L-functions to classical “periods”. In this talk, I will show more results of this problem for more cases using Jacquet's relative trace formula, and some application of these average formulas to the non-vanishing problem.

March 10
Speaker: Jaiung Jun (SUNY New Paltz)
Title: Lattices, spectral spaces, and closure operations on idempotent semirings
Abstract: Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this talk, we introduce an analogous statement for idempotent semirings - a topological space is spectral if and only if it is the saturated prime spectrum of an idempotent semiring. We then introduce more examples of spectral spaces arising from idempotent semirings.

March 17 CANCELLED/POSTPONED
Speaker: Biao Wang (SUNY Buffalo)
Title: Some arithmetic functions and Chebotarev densities
Abstract: For M\”{o}bius function $\mu$, it is well-known that the prime number theorem is equivalent to $\sum_{n=1}^\infty\frac{\mu(n)}{n}=0$. In 1977, Alladi showed a formula on the restricted sum of $\frac{\mu(n)}{n}$ over the conjugacy class of smallest prime divisor of $n$. In 2017, Dawsey generalized Alladi's result to the setting of Chebotarev densities for finite Galois extensions of $\mathbb{Q}$. In this talk, we will introduce the analogues of their formulas with respect to the Liouville function and the Ramanujan sum, and propose a conjecture for more general arithmetic functions.

March 24 CANCELLED/POSTPONED
Speaker: Bogdan Ion (University of Pittsburgh)
Title: Bernoulli polynomials and Dirichlet series
Abstract: For a given sequence one can associate a power series and a Dirichlet series. We investigate the relationship between possible singularities that appear when we analytically continue both of these series. The most basic case, when the power series has a pole singularity at z=1 is analyzed in detail by employing some (infinite order) discrete derivative operator (associated to the power series) that we call Bernoulli operator. Its main property is that it naturally acts on the vector space of analytic functions in the plane (with possible isolated singularities) that fall in the image of the Laplace-Mellin transform (for the variable in some half-plane). The action of the Bernoulli operator on the function t^s, provides the analytic continuation of the associated Dirichlet series and also detailed information about the location of poles, their resides, and special values. Using examples of arithmetic origin, I will attempt to illustrate what is reasonable to expect when the power series has a non-pole singularity at z=1.

March 31 CANCELLED/POSTPONED
Speaker: Andrew Lamoureux (Binghamton University)
Title: TBA
Abstract: TBA

April 14 CANCELLED/POSTPONED
Speaker: Vaidehee Thatte (Binghamton University)
Title: TBA
Abstract: TBA

April 21 CANCELLED/POSTPONED
Speaker: Sayak Sengupta (Binghamton University)
Title: TBA
Abstract: TBA

April 28 CANCELLED/POSTPONED
Speaker: Patrick McGinty (Binghamton University)
Title: TBA
Abstract: TBA

May 5 CANCELLED/POSTPONED
Speaker: Helene Esnault (IAS and FU Berlin)
Title: TBA
Abstract: TBA

Spring 2021

2021-08-07T16:04:21-04:00

Spring 2021

February 16
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

February 23
Speaker: Fikreab Solomon Admasu (Binghamton)
Title: Bhargava's composition law for binary cubic forms
Abstract: The Delone-Fadeev correspondence shows that binary cubic forms with integer coefficients parametrize orders in cubic fields. With this result in mind, Bhargava constructs a binary cubic form from 2x3x3 boxes of integers and proves that there is a natural composition law for the boxes of integers. The group resulting from this law is then shown to be isomorphic to the class group of a corresponding cubic order. This is a cubic analogue of Gauss's theory of composition for binary quadratic forms and its relation to ideal classes of quadratic orders. The talk is based on Bhargava's “Higher composition laws II: On cubic analogues of Gauss composition.”

March 2
Speaker: Hyuk Jun Kweon (MIT)
Title: Bounds on the Torsion Subgroups of Néron-Severi Group Schemes
Abstract: Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$, and $\mathbf{Pic}\, ^0 X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron–Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}\, ^0 X)_{\mathrm{red}}$, and the N\'eron–Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$. We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_{et}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements.

March 9
Speaker: Thomas Morrill (Trine)
Title: Sign Changes in the Prime Number Theorem
Abstract: The prime number theorem is a lovely part of 19th century mathematics, leveraging the techniques of complex analysis in order to find results in the discrete setting, namely that the number of primes between $1$ and $x$ may be estimated by $x/log(x).$ However, to an analyst, the theorem gives a statement about the asymptotic growth of two real-valued functions. In fact, the recognizable $\pi(x) \sim x/log(x)$ is one of several equivalent statements of the theorem, depending on how one chooses to weight the count of primes from $1$ to $x$. We cover the fundamentals of this topic, and present our results on the sign of $\phi(x) - x$ as $x\to \infty$.

March 16
Speaker: Fikreab Solomon Admasu (Binghamton)
Title: Bhargava's composition law for binary cubic forms 2
Abstract: This is a continuation of the talk from February 23 with a focus on illustrative examples and some questions.

March 23
Speaker: Alexander Borisov (Binghamton)
Title: Coherent sheaves over the compactified Spec(Z)
Abstract: Lattices in finite-dimensional Euclidean spaces can be viewed as Arakelov analogs of locally free sheaves of finite rank over a compact curve. There are many indications that this category can be extended to a category of “coherent sheaves”. I will discuss some motivation, basic definitions, and and some work in progress, joint with Jaiung Jun.

March 30
Speaker: Andrew Lamoureux (Binghamton)
Title: Differential Algebra
Abstract: This talk will explore the basics of differential algebra, such as differential rings, differential ideals, and the ring of differential polynomials. I will then focus on differential field theory, emphasizing its parallels to ordinary field theory.

April 6
Speaker: Biao Wang (SUNY Buffalo)
Title: Alladi's formula and its analogues
Abstract: In 1977, Alladi introduced a duality between prime factors of integers, from which along with the prime number theorem in arithmetic progressions he showed the following beautiful formula on the M\"{o}bius function $\mu(n)$: if $(\ell,k)=1$, then $$ -\sum_{\begin{smallmatrix}n\ge 2\\ p_{\min}(n)\equiv \ell (\operatorname{mod}k) \end{smallmatrix}}\frac{\mu(n)}{n}=\frac1{\varphi(k)},$$ where $p_{\min}(n)$ is the smallest prime factor of $n$ and $\varphi$ is the Euler's totient function. Recently, this formula has been generalized from the Chebotarev density to the natural density of sets of primes in works by Dawsey, Sweeting and Woo, Kural, McDonald and Sah. In this talk I will discuss the generalizations of these results to other arithmetic functions and the analogues over function fields.

April 12 (Monday), 12-1 pm
Speaker: James Myer (CUNY)
Title: The alterations paradigm shift for the problem of resolution of singularities
Abstract: Heisuke Hironaka solved the problem of resolution of singularities for a variety over a field of characteristic zero in 1964, although the story goes that his proof was so complicated as to stump Alexander Grothendieck! Efforts have been made since then to simplify the proof and see if it can be made to work for positive characteristic, but to no avail.
Regardless of characteristic: The singularities of a curve are resolved in one fell swoop by the normalization, a powerful tool hailing from algebra made to work for us in geometry, by Jean-Pierre Serre’s Criterion for Normality, and we owe much to Oscar Zariski for teaching us about the normalization. Joseph Lipman dealt with surfaces in as much generality as one could hope for. Threefolds are rumored to have been handled by Vincent Cossart and Olivier Piltant, although their proof is quite long and there are some restrictions on the characteristic. Fourfolds and up are uncharted territory for the most part.
In his 1996 paper Smoothness, Semi-Stability, and Alterations, Johan de Jong introduces a paradigm shift for solving the problem of resolution of singularities by relaxing a resolution of singularities to what he calls an alteration. The distinction is that a resolution is a birational morphism (generically one-to-one), whereas an alteration is a generically finite morphism (generically finite-to-one). Every variety can be altered to a nonsingular variety regardless of the characteristic. In fact, de Jong’s technique paired up with Dan Abramovich’s geometric insight yields a proof that every variety over a field of characteristic zero admits a resolution of its singularities in a paper consisting of only twelve pages, see Smoothness, Semi-Stability, and Toroidal Geometry. de Jong’s technique relies on (at least) two ingenious ideas: The first is the statement that there is a simple blowup of any variety that admits a morphism to a projective space of one less dimension whose fibers are curves ~ intuitively, any variety can be modified to be a family of curves. The second is that the curves in the family of curves that results can be marked so as to become stable (in the sense of Pierre Deligne and David Mumford) so that it gives rise to morphism into the moduli space of stable curves, where we may take advantage of established facts about the moduli space of stable curves, including the fact that its compactification has curves with at worst nodal singularities. My talk has the goal of introducing Johan de Jong’s paradigm shift, and indicating, to the extent that I can, the idea of the proof that every variety may be altered to a nonsingular variety. An emphasis will be placed on the picture that demonstrates that there is a simple blowup of any variety that admits a morphism to a projective space of one less dimension whose fibers are curves.

April 27
Speaker: Andrew Lamoureux (Binghamton)
Title: Differential Algebra (continued)
Abstract: This is a continuation of my talk from March 30. In particular, I will discuss the basics of differential Galois theory and then p-derivations, which are arithmetic analogues of derivations.

May 4
Speaker: Francois Greer (IAS)
Title: Cycle-valued quasi-modular forms
Abstract: The theta correspondence of automorphic forms produces a rich supply of modular forms with “special cycle” coefficients in the cohomology of Shimura varieties. I will describe what happens when we take the closure of the special cycles in toroidal compactifications, focusing on the case of orthogonal type. This has concrete applications in enumerative algebraic geometry, and leads to a related program outside the Shimura variety context.

May 11
Speaker: Garen Chiloyan (UConn)
Title: A study of isogeny-torsion graphs
Abstract: An isogeny-torsion graph is a nice visualization of the $\mathbb{Q}$-isogeny class of an elliptic curve defined over $\mathbb{Q}$. A theorem of Kenku shows sharp bounds on the number of distinct isogenies that a rational elliptic curve can have (in particular, every isogeny graph has at most 8 vertices). In this talk, we classify what torsion subgroups over $\mathbb{Q}$ can occur at each vertex of a given isogeny-torsion graph of elliptic curves defined over the rationals. Then we will determine which isogeny-torsion graphs correspond to an infinite set of $\textit{j}$-invariants. This is joint work with \'Alvaro Lozano-Robledo.

Spring 2022

2022-08-25T15:33:15-04:00

Spring 2022

February 1
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

February 8
Speaker: Sailun Zhan (Binghamton)
Title: Monodromy of rational curves on K3 surfaces of low genus
Abstract: In many situations, the monodromy group of enumerative problems will be the full symmetric group. In this talk, we introduce a similar phenomenon on the rational curves in |O(1)| on a generic K3 surface of fixed genus over C as the K3 surface varies.

February 15
Speaker: Sayak Sengupta (Binghamton)
Title: Nilpotent and nilpotent modulo polynomials over $Z$
Abstract: We will define and describe two new kinds of polynomials over $Z$, named nilpotent and nilpotent modulo polynomials in one variable. These polynomials are truly fascinating and have some amazing behaviors. We will study these properties in this talk.

February 22
Speaker: Sayak Sengupta (Binghamton)
Title: Nilpotent and nilpotent modulo polynomials over $Z$ (part II)
Abstract: We will define and describe two new kinds of polynomials over$Z$, named nilpotent and nilpotent modulo polynomials in one variable. These polynomials are truly fascinating and have some amazing behaviors. We will study these properties in this talk.

March 1
Speaker: Alexander Borisov (Binghamton)
Title: Projective approach to the two-dimensional Jacobian conjecture: an introduction
Abstract: This is a first in a series of 2-3 talks aimed at explaining my approach to the two-dimensional Jacobian Conjecture, using methods of projective geometry. I am planning to discuss some foundational material, like divisors and divisor classes, intersection theory, canonical class, and adjunction formulas.

March 8
Speaker: Alexander Borisov (Binghamton)
Title: Projective approach to the two-dimensional Jacobian conjecture: an introduction, Part 2
Abstract: I will continue my talk from March 1, focusing on the theory of complex projective surfaces.

March 22
Speaker: Andrew Lamoureux (Binghamton)
Title: Arithmetic Differential Operators over Compact DVRs
Abstract: In 2011, Alexandru Buium, Claire C. Ralph, and Santiago Simanca proved that a map f: Z_p → Z_p is an 'arithmetic differential operator or order m' if and only if it is 'analytic of level m'. Both notions can be generalized first to maps f: R^d → R, where R is a compact DVR, and then to maps f: X(R) → Y(R), where X and Y are two smooth affine schemes of finite type over R. In this talk, we will see that these notions are still equivalent in this more general context and that every analytic map of manifolds f:X(R) → Y(R) is analytic of level m for some m.

March 29
Speaker: Stephen Pietromonaco (UBC)
Title: The Enumerative Geometry of Orbifold K3 Surfaces
Abstract: A few of the most celebrated theorems in enumerative geometry (both predicted by string theorists) surround curve-counting for K3 surfaces. The Yau-Zaslow formula computes the honest number of rational curves in a K3 surface, and was generalized to the Katz-Klemm-Vafa formula computing the (virtual) number of curves of any genus. In this talk, I will review this story and then describe a recent generalization to orbifold K3 surfaces. One interpretation of the new theory is as producing a virtual count of curves in the orbifold, where we track both the genus of the curve and the genus of the corresponding invariant curve upstairs. As one example, we generalize the counts of hyperelliptic curves in an Abelian surface carried out by Bryan-Oberdieck-Pandharipande-Yin. This is work in progress with Jim Bryan.

April 5
Speaker: Andrew Lamoureux (Binghamton)
Title: Arithmetic Differential Operators over Compact DVRs, part 2
Abstract: This is the continuation of the March 22 talk. In 2011, Alexandru Buium, Claire C. Ralph, and Santiago Simanca proved that a map f: Z_p → Z_p is an 'arithmetic differential operator or order m' if and only if it is 'analytic of level m'. Both notions can be generalized first to maps f: R^d → R, where R is a compact DVR, and then to maps f: X(R) → Y(R), where X and Y are two smooth affine schemes of finite type over R. In this talk, we will see that these notions are still equivalent in this more general context and that every analytic map of manifolds f:X(R) → Y(R) is analytic of level m for some m.

April 12
Speaker: Hari Asokan (Binghamton)
Title: Chow rings and Steiner's conic problem
Abstract: This is the first in the two talks aimed to discuss a classical problem in enumerative geometry (Steiner's conic problem). In this talk we will define Chow rings of varieties, compute them for some easy examples and explore some properties.

April 19
Speaker: Xuqiang Qin (UNC)
Title: Birational geometry of the Mukai system on a K3 surface
Abstract: The Mukai system on a K3 surface is a moduli space of torsion sheaves, admitting a Lagrangian fibration given by mapping each sheaf to its support. In this talk, we will focus on a class of Mukai systems which are birational to Hilbert scheme of points. Using the wall crossing technique from Bridgeland stability, we decompose the birational map into a sequence of flops. As a result, we give a full description of the birational geometry of such a Mukai system. This is based on joint work with Justin Sawon.

April 26
Speaker: Alexander Borisov (Binghamton)
Title: Projective approach to the two-dimensional Jacobian conjecture: an introduction. Part 3.
Abstract: This is the third in a series of talks aimed at explaining my approach to the two-dimensional Jacobian Conjecture, using methods of projective geometry.

May 3
Speaker: Hari Asokan (Binghamton)
Title: Chow rings and Steiner's conic problem, part 2
Abstract: This is the continuation of the talk from April 12.

May 10
Speaker: Aranya Lahiri (UCSD)
Title: Irreducibility of rigid analytic vectors in p-adic principal series representations
Abstract: For the $L$-rational points $G:=\mathbb{G}(L)$ of a p-adic reductive group, let $Ind^G_B(\chi)$ be the continuous p-adic principal series representations. Here $L$ is a finite extension of $\mathbb{Q}_p$, $B$ is the Borel subgroup corresponding to a maximal torus $T$ and $\chi$ is a character of $T$. We will consider the globally analytic vectors of the pro-p Iwahori group $I$ in the principal series representations. This is done by endowing the pro-p Iwahori with a $p$-valuation and subsequently giving it a structure of a rigid analytic group, thus generalizing the work of Lazard. The main result of this talk will be the topological irreducibility of these globally analytic vectors under certain assumptions on $\chi$. This is a generalization of works of Clozel and Ray in the case of $G:= GL_n(L)$. This is joint work with Claus Sorensen.

Spring 2023

2023-08-24T18:03:52-04:00

Spring 2023

January 24
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

January 31
Speaker: Alexander Borisov (Binghamton)
Title: The abc-polynomials
Abstract: To every triple of coprime natural numbers a=b+c, one can associate an interesting polynomial $f_{abc}(x)=\frac{bx^a-ax^b+c}{(x-1)^2}$. These abc-polynomials were introduced and studied in my old (1998) paper: http://people.math.binghamton.edu/borisov/documents/papers/abc.pdf In that paper I proved several results on the irreducibility of these polynomials, with the main theorem being that almost all of them, in the density sense, are irreducible. These results can certainly be strengthened and generalized in a number of ways, and the primary reason why this hasn't been done is that, probably, few people tried. I will give an overview of the motivation, results, and methods of this paper, in the hope that some of the graduate students in the audience get interested in pushing this topic further.

February 7
Speaker: Sailun Zhan (Binghamton)
Title: Punctual Hilbert schemes of points of affine spaces in the Grothendieck group of varieties
Abstract: We will give an introduction to punctual Hilbert schemes of n points of affine spaces A^{m+1}, which parametrizes (x_0,…,x_m)-primary ideals in k[x_0,…,x_m] of codimension n. Then we will give an explicit stratification of it with respect to m-dimensional partitions in the Grothendieck group of varieties. We will illustrate the idea by the cases when m=1 and 2.

February 14
Speaker: Sailun Zhan (Binghamton)
Title: Punctual Hilbert schemes of points of affine spaces in the Grothendieck group of varieties, part 2
Abstract: This is a continuation of the February 7 talk.

February 21
Speaker: Sayak Sengupta (Binghamton)
Title: Heights in Diophantine Geometry
Abstract: The height of a rational number is a measure of its arithmetic complexity. For example, although the numbers 5 and 500000/100001 are close, the second is arithmetically more complex, and has a much larger height. The height of a rational number is easy to define: it is simply the maximum of the absolute value of the numerator and the denominator when the number is expressed in lowest form. It is not immediately clear how one can extend this definition to more general algebraic numbers, such as the square root of 2. In this talk we will start with the definition of height of points in $\mathbb{Q}$ and build our way up to the general definition of heights of points in $\mathbb{P}^n_K$, $K$ being a number field.

February 28
Speaker: Alexander Borisov (Binghamton)
Title: Some applications of the Lagrange interpolation formula
Abstract: This elementary talk will highlight two applications of the Lagrange interpolation formula. The first one is my old result on the “split Rolle closure” of the rational numbers http://people.math.binghamton.edu/borisov/documents/papers/cb2.pdf and the second one is my solution of a conjecture by Dinesh Thakur, see Theorem 6 in https://arxiv.org/abs/2211.01076

March 7
Speaker: Alexander Borisov (Binghamton)
Title: Discrete invariants of compactifications of the affine plane
Abstract If a complete algebraic surface contains an affine plane, its discrete invariants are determined by the weighted graph (tree) of the curves at infinity. I will describe this structure and some associated invariants. The talk will be based on my 2014 paper http://people.math.binghamton.edu/borisov/documents/papers/divisorialvaluations.pdf I will also describe some recent improvements by my student Patrick Carney.

March 14
Speaker: Mithun Padinhare Veettil (Binghamton)
Title: Nilpotent and Geometrically Nilpotent Subvarieties.
Abstract: In this talk, we will define nilpotent and geometrically nilpotent subvarieties, an idea grew that out of density theorem proved by Prof. Borisov. Then we will discuss two explicit constructions that give rise to geometrically nilpotent but not nilpotent subvarieties (in fact, infinitely many of them!)

March 21
Speaker: Alexander Borisov (Binghamton)
Title: Discrete invariants of compactifications of the affine plane , part 2
Abstract: This is the continuation of the March 7 talk. I plan describe the determinant labels and Patrick Carney's improvements to the way they can be calculated

March 28
Speaker: Sayak Sengupta (Binghamton)
Title: Elliptic curves and heights in Elliptic curves
Abstract: Elliptic curve has been a central object of study in arithmetic geometry and number theory for a long time. In this talk we will explore the richness of elliptic curves as an abelian group with more emphasis on examples rather than proofs of theorems. We will define the two height concepts, viz. naive and canonical heights, on elliptic curve $E/\mathbb{Q}$ and build our way up to defining the well-known N\'eron-Tate pairing attached to $E$ which is a non-degenerate, symmetric, bilinear form on $E(\mathbb{Q})$, modulo the torsion points.

April 1-2
Special Event: Eleventh Upstate New York Number Theory Conference, at the University of Rochester
https://sites.google.com/view/upstate-ny-nt-conf-2023/home

April 11 (by Zoom: Zoom link )
Speaker: John Lee (UIC)
Title: Acyclic Reduction of Elliptic Curves for Primes in Arithmetic Progression
Abstract: Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $p$ a rational prime. $\tilde{E}_p$ denotes the reduction of $E$ modulo $p$. Recently, Akbal and G \”{u}lo\v{g}lu considered the question of cyclicity of $\tilde{E}_p(\mathbb{F}_p)$ under the restriction that lies in an arithmetic progression. In this talk, we study the issue of which arithmetic progressions $k$ mod $n$ have the property that, for all but finitely many primes $p \equiv k$ mod $n$, the group $\tilde{E}_p(\mathbb{F}_p)$ is NOT cyclic. Furthermore, we study the statistical congruence class bias of primes of cyclic reduction for generic elliptic curves. This is a joint work with Nathan Jones.

April 18 (by Zoom: Zoom link )
Speaker: Paolo Dolce (Ben-Gurion University)
Title: Introduction to Diophantine approximation and a generalization of Roth’s theorem
Abstract: Classically, Diophantine approximation deals with the problem of studying “good” approximations of a real number by rational numbers. I will explain the meaning of “good approximants” and the classical main results in this area of research. In particular, Klaus Roth was awarded with the Fields medal in 1955 for proving that the approximation exponent of a real algebraic number is 2. I will present a recent extension of Roth’s theorem in the framework of adelic curves. These mathematical objects, introduced by Chen and Moriwaki in 2020, stand as a generalization of global fields.

April 25
Speaker: Hari Asokan (Binghamton)
Title: Geometric invariant theory
Abstract: Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, developed by David Mumford in 1965. GIT theory is used to construct moduli spaces of geometric objects in algebraic geometry. The theory also has interactions with differential geometry and symplectic geometry.
In this talk we will discuss some basic concepts and definitions in geometric invariant theory. We will focus on the congruence action of special linear group on multiple quadratic forms. The geometric invariant theory of multiple quadratic forms is motivated by applications to the study of quadratic forms and, due to the recent work of Buium-Vasiu, of \delta-modular forms on moduli spaces of principally polarized abelian varieties of arbitrary dimension.

May 1 (Monday), 3:30-5:30 pm Special Event: PhD Defense
Speaker: Patrick Carney (Binghamton)
Title: Application of the Theory of Farey Fractions to the Combinatorics of Some Compactifications of the Affine Plane
Abstract: The combinatorial structure of the algebraic surfaces obtained from the complex projective plane by a sequence of blow-ups of points at infinity is classically described by a certain weighted graph whose vertices correspond to irreducible curves at infinity. In 2014, Alexander Borisov associated to each such curve a pair of integers that are invariant under subsequent blow-ups. We investigate these invariants and furnish a much more efficient method of calculating them based on the classical theory of Farey fractions.

Spring 2024

2024-08-17T14:59:18-04:00

Spring 2024

January 23
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester

February 13
Speaker: Alexander Borisov (Binghamton)
Title: Locally Integer Polynomial Functions
Abstract: I will discuss the ring of integer-valued functions on the integers with a peculiar property: when restricted to any finite subset, the interpolation polynomial has integer coefficients. The original motivation for this comes from Sayak Sengupta's work on iterations of integer polynomials, but this and related objects appear to be of independent interest.

February 20
Speaker: Sayak Sengupta (Binghamton)
Title: Nilpotent and Infinitely Nilpotent Integer Sequences
Abstract: We say that an integer sequence $\{r_n\}_{n\ge 0}$ has a generating polynomial $u(x)$ over $\mathbb{Z}$ if for every positive integer $n$ one has $u^{(n)}(r_0)=r_n$. In addition, if such a sequence satisfies the condition that $r_n=0$ for some positive integer $n$ (respectively, $r_n=0$ for infinitely many positive integers $n$), then we say that $\{r_n\}_{n\ge 0}$ is a nilpotent sequence (respectively, $\{r_n\}_{n\ge 0}$ is an infinitely nilpotent sequence). In this talk we will provide (and discuss) some important characteristics of nilpotent and infinitely nilpotent sequences.

March 12 by Zoom: Zoom link
Speaker: Haiyang Wang (UGA)
Title: Elliptic curves with potentially good supersingular reduction and coefficients of the classical modular polynomials
Abstract: Let $O_K$ be a Henselian discrete valuation domain with field of fractions $K$. Assume that $O_K$ has algebraically closed residue field $k$. Let $E/K$ be an elliptic curve with additive reduction. The semi-stable reduction theorem asserts that there exists a minimal extension $L/K$ such that the base change $E_L/L$ has semi-stable reduction. It is natural to wonder whether specific properties of the semi-stable reduction and of the extension $L/K$ impose restrictions on what types of Kodaira type the special fiber of $E/K$ may have.
In this talk we will discuss the restrictions imposed on the reduction type when the extension $L/K$ is wildly ramified of degree 2, and the curve $E/K$ has potentially good supersingular reduction. We will also talk about the possible reduction types of two isogenous elliptic curves with these properties and its relation to the congruence properties of the coefficients of the classical modular polynomials.

March 19
Speaker: Shane Chern (Dalhousie)
Title: The Seo-Yee conjecture: Nonmodular infinite products, seaweed algebras, and integer partitions
Abstract: In this talk, I will present my recent work on the Seo-Yee conjecture, which claims the nonnegativity of coefficients in the expansion of a q-series infinite product. The Seo-Yee conjecture arises from the study of seaweed algebras (a special type of Lie algebra), and is closely tied with the enumeration of the index statistic of integer partitions. Our proof of the Seo-Yee conjecture is built upon the asymptotic analysis for a generic family of nonmodular infinite products near each root of unity.

March 26
Speaker: Alexander Borisov (Binghamton)
Title: On irreducibility of higher derivatives of polynomials x^n+…+x+1
Abstract: In our 1999 joint paper with Filaseta, Lam, and Trifonov we proved, among other results, that for every fixed positive integer k the k-th derivatives of the polynomials in the title are irreducible over the rationals for a density one set of natural n. The proof relies on understanding the “location” of the roots of these derivatives in complex numbers and in p-adic complex numbers for primes dividing (n+1)n…(n+1-k). I will explain the main ideas of the proof while trying to avoid the rather formidable technical details.

April 9 4:00-6:00 pm Special Event: PhD Defense
Speaker: Sayak Sengupta (Binghamton)
Title: Iteration of Polynomials over Integers
Abstract: For a polynomial $u=u(x)$ over $\mathbb Z$ and $r\in\mathbb Z$, we consider the orbit of $u$ at $r$, denoted and defined by $\mathcal{O}_u(r):=\{u(r),u(u(r)),\ldots\}$. There are two main questions that we plan to answer: (1) what are the polynomials $u$ for which $0\in \mathcal{O}_u(r)$, and (2) what are the integer polynomials $u$ that satisfies the condition that for each prime number $p$ there is some iteration $m_p$ of $u$ such that $p|u^{(m_p)}(r)$? In this talk we will provide partial answer to (1), and a complete answer to (2).

April 16
Speaker: Alexander Borisov (Binghamton)
Title: On the Nyman-Beurling-Baez-Duarte criterion for the Riemann Hypothesis
Abstract: I will talk about an attractive criterion for the Riemann Hypothesis, originally due to Nyman and Beurling in early 1950s and strengthened by Baez-Duarte in early 2000s. The talk will be partially based on my 2005 paper https://people.math.binghamton.edu/borisov/documents/papers/quot-fact-rh.pdf and will also include some more recent unpublished considerations.

April 29 (Monday) 4:00-6:00 pm Special event: Admission to candidacy
Speaker: Mithun Veettil (Binghamton)
Talk 1 (4:00-4:55)
Title: Hilbert's Irreducibility Theorem
Abstract: Hilbert's irreducibility theorem deals with the following problem: Let $f(t,x)$ be an irreducible polynomial in $K[t,x]$. Then for which field $K$ is it true that there are infinitely many specializations $t\mapsto t_0\in K$ such that $f(t_0,x)$ is irreducible in $K[x]$? Surprisingly, it turns out that $\mathbb{Q}$ and function fields have this property.
Talk 2 (5:00-5:55)
Title: Golomb Topology on a Domain
Abstract: Golomb topology on a domain is a generalization of arithmetic topology on $\mathbb{Z}^+,$ appearing in Furstenberg's proof of the infinitude of primes. This paves way for the otherwise rare examples of countably infinite connected Hausdorff spaces. Following this, I shall conclude with a homeomorphism problem of Golomb topology on Dedekind domains. This talk is based on the 2019 paper by Pete Clark, Noah Lebowitz-Lockard, and Paul Pollack http://alpha.math.uga.edu/~pete/CLLP_November_30_2017.pdf

Spring 2025

2025-08-16T19:33:59-04:00

Spring 2025

January 21 Organizational Meeting
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January 28 No Talk (Slot taken by Interview Candidate)
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February 4
Speaker: Adrian Vasiu (SUNY Binghamton)
Title: Matrix invertible extensions over commutative rings A
Abstract: We report on some parts of the paper at the link https://arxiv.org/abs/2404.05780

February 11
Speaker: Hari Asokan (SUNY Binghamton)
Title: Elimination Theory and Resultants I
Abstract: Elimination Theory is used to study systems of polynomial equations in several variables. In this introductory talk we will discuss some classical results related to elimination theory and resultants of systems of polynomial equations.

February 18
Speaker: Hari Asokan (SUNY Binghamton)
Title: Elimination Theory and Resultants II
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February 25
Speaker: Adrian Vasiu (SUNY Binghamton)
Title: Matrix invertible extensions over commutative rings B
Abstract: We first present several criteria on a $2\times 2$ unimodular matrix to be simply extendable or extendable. Then we recall basic properties of stable ranges of rings and show their relevance to the study of different classes of rings such as EDR, Hermite, Bézout, $SE_2$, $E_2$, and $\pi_2$.

March 4
Speaker: Ruoxi Li (University of Pittsburgh)
Title: Motivic Classes of Varieties and Stacks with Applications to Higgs Bundles
Abstract: In this talk, we will first discuss the motivations for motivic classes coming from point counting over finite fields. Then we will give the definitions of the motivic classes of varieties, in particular we explain that an extra relation is needed in finite characteristic. We will introduce symmetric powers and motivic zeta functions that are universal versions of local zeta functions. For the second part of the talk, we will focus on the motivic classes of stacks. In particular, we will give the explicit formulas for the motivic classes of moduli of Higgs bundles.

March 18
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March 25
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April 1
Speaker: Effie Shani (SUNY Albany)
Title: Almost Trivial Units in Group Rings
Abstract: We study units in group rings $R[G]$ of finite abelian groups $G$ with coefficients in rings $R$ of algebraic integers in number fields. Let $\Sigma$ be the sum of all the elements of $G$ in the group ring $R[G]$. Units of the form $ug$ in $R[G]$ (or $ug+(\Sigma)$ in $R[G]/(\Sigma)$, respectively) for $u\in R^\times$ and $g\in G$ are called trivial units. Units in $R[G]$ that project to trivial units in $R[G]/(\Sigma)$ are called almost trivial units. Higman in 1939 classified all finite groups $G$ for which $\mathbb{Z}[G]$ contains only trivial units, and Herman and Li in 2005 generalized his results to coefficients in rings of algebraic integers. We characterize all finite abelian groups $G$ and rings of algebraic integers $R$ such that the only units of the reduced group ring $R[G]/(\Sigma)$ are trivial units (and so all units of the group ring $R[G]$ are almost trivial units). This is joint work with Anupam Srivastav, which extends the results of Brian Rich’s and my Ph.D. theses under Srivastav’s supervision.

April 8
Speaker: Alexander Betts (Cornell University)
Title: Galois sections and p-adic obstructions
Abstract: Grothendieck's anabelian programme is an attempt to study the arithmetic of curves over number fields through their etale fundamental groups. The centrepiece of this programme is the still-unproven Section Conjecture, which asserts that the rational points on a curve X should be in bijection with the set of splittings of a certain homotopy exact sequence on fundamental groups. In this talk, I will describe what is known about the Section Conjecture, and outline a few of my own results, which attempt to control splittings of the homotopy exact sequence using p-adic obstruction theory.

April 15
Speaker: Sarah Lamoureux (SUNY Binghamton)
Title: Cofinality of Ordered Modules over an Ordered Integral Domain
Abstract: A subset C of a poset P is cofinal if for all p in P, there is a c in C such that p \leq c. The cofinality of P is the smallest cardinality of a cofinal subset of P. In this talk, A is a (totally) ordered integral domain such that whenever 0 < a < b, there is a c > 0 such that b < ac. We present two structure theorems for a (totally) ordered A-module M based on its cofinality. The first is for the case when M is A-CPA (there exists an m in M such that A_{>0}m is a cofinal set); the second is for the case when M is not A-CPA. (Both theorems require either M or a particular submodule to be A-divisible.) We'll mention as well how these notions of “cofinality” and “A-CPA” relate to a larger project about valuation rings.

April 29
Speaker: Andreea Iorga (Cornell University)
Title: Realising certain semi-direct products as Galois groups
Abstract: In this talk, I will prove that, under a specific assumption, any semi-direct product of a p-group G with a group of order prime-to-p can appear as the Galois group of a tower of extensions M/L/K with the property that M is the maximal pro-p extension of L that is unramified everywhere, and Gal(M/L) = G. At the end, I will show that a nice consequence of this is that any local ring admitting a surjection to Z_5 or to Z_7 with finite kernel can be written as a universal everywhere unramified deformation ring.

May 6
Speaker: Mithun Padinhare Veettil (SUNY Binghamton)
Title: Some results on Locally Integer Polynomials
Abstract: Locally Integer Polynomial (LIP) functions are Z-valued functions on an infinite subset X of Z that are given by polynomials with integer coefficients in every finite subset of X. In this talk, we will explore some properties of rings of LIP functions.