seminars:arit

## The Arithmetic Seminar

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.

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, Sayak Sengupta, 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).

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#### Spring 2018

• January 22
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester
• January 29
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 This refines and reobtains prior works due to Zink, de Jong–Oort, Vasiu, and Yang and provides two new proofs of an unpublished result of Deligne.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: TBA
• 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: Brian Hwang (Cornell)
Title: TBA
Abstract: TBA
• April 23
Speaker: Micah Loverro (Binghamton)
Title: TBA
Abstract: TBA
• April 30
Speaker: Changwei Zhou (Binghamton)
Title: TBA
Abstract: TBA
• May 7
Speaker: Patrick Milano (Binghamton)
Title: TBA
Abstract: TBA