TOPICS: Arithmetic in the broadest sense that includes Number Theory (Elementary Arithmetic, Algebraic, Analytic, Combinatorial, etc.), Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Geometry of Numbers, Tropical Geometry, Arithmetic Dynamics, etc.

PLACE and TIME: This semester the seminar meets on Mondays at 3:30 p.m. in WH 100E, with possible special lectures at other days. Before the talks, there will be refreshments in WH-102.

ORGANIZERS: Alexander Borisov, Marcin Mazur, Adrian Vasiu, Jaiung Jun, Patrick Milano, and Micah Loverro.

To receive announcements of seminar talks by email, please join the seminar's mailing list.

The number theory group at Binghamton University presently consists of three faculty members (Alexander Borisov, Marcin Mazur, and Adrian Vasiu), one post-doc (Jaiung Jun) and several Ph.D. students (John Brown, Patrick Carney, Micah Loverro, Patrick Milano, Changwei Zhou).

Past Ph.D. students in number theory related topics that graduated from Binghamton University: Ilir Snopce (Dec. 2009), Xiao Xiao (May 2011), Jinghao Li (May 2015), Ding Ding (Dec. 2015).

Previous talks

• Spring 2017
• Fall 2016
• Spring 2016
• Fall 2015
• Spring 2015
• Fall 2014

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• 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: TBA
  Abstract: TBA
  

• November 13
  Speaker: Tom Price (Toronto)
  Title: TBA
  Abstract: TBA
  

• November 20
  Speaker: Patrick Carney (Binghamton)
  Title: TBA
  Abstract: TBA
  

• November 27
  Speaker: Sayak Sengupta (Binghamton)
  Title: TBA
  Abstract: TBA
  

• December 4
  Speaker: Philipp Jell (Georgia Tech)
  Title: TBA
  Abstract: TBA
  

• December 6
  Speaker: Patrick Carney (Binghamton)
  Title: TBA
  Abstract: TBA