The Arithmetic Seminar
TOPICS: Arithmetic in the broadest sense that includes Number Theory, Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Arithmetic Dynamics, etc.
PLACE and TIME: The seminar meets either on Tuesdays at 4:15 p.m. or on Fridays at 3:30 p.m. in LN 2205. Before the talks, there will be refreshments in the Anderson Reading Room at 4:00 p.m.(Tuesday) and 3:30 p.m. (Friday).
ORGANIZERS: Dikran Karagueuzian, Marcin Mazur, Adrian Vasiu, Joel Robert Dodge, Daniel Vallieres, Ding Ding, Michael Fink and Jinghao Li.
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The number theory group at Binghamton University presently consists of two faculty members (Marcin Mazur and Adrian Vasiu), four Ph.D. students (Ding Ding, Michael Fink, Jinghao Li and Daniel Williams) and two post-docs (Joel Dodge and Daniel Vallières).
Past Ph.D. students in number theory related topics that graduated from Binghamton University: Ilir Snopce (Dec. 2009), Xiao Xiao (May 2011).
Spring 2014: * talks
February 14 : Jinghao Li (Binghamton University)
Title: Purity Results on F-crystals.
Abstract: We will first review some basic facts about Newton polygons and F-crystals. Then we will present a survey of purity results for stratifications in positive characteristic associated to F-crystals, including some recent new purity results. In the first talk, we will focus on reviewing the definitions of Newton polygons and F-crystals and we will define purity notions.
February 21 : Jinghao Li (Binghamton University)
Title: Purity Results on F-crystals.
Abstract: This is a continuation of last talk. We will present some known purity results for Newton polygon stratifications in positive characteristic associated to F-crystals and introduce some recent new ones.
March 14 : David Zywina (Cornell University)
Title: Elliptic curves and the Inverse Galois Problem.
Abstract: By studying the Galois action on etale cohomology groups arising from families of elliptic curves, we will prove several new cases of the regular Inverse Galois Problem. In particular, we will explain why each of the simple groups PSp_4(Fp) occur as the Galois group of a regular extension of the function field Q(t). The key ingredients will be a big monodromy result along with some known cases of the Birch and Swinnerton-Dyer conjecture.
April 26 - April 27: Fourth Annual Upstate New York Number Theory Conference
Conference web page available here.
April 28 (SPECIAL DAY MONDAY AND TIME 4:40 IN LN 2205): Alexander Borisov (University of Pittsburgh)
Title: Two-dimensional Jacobian Conjecture: a birational geometry approach.
Abstract: The Jacobian Conjecture is one of the oldest unsolved problems in algebraic geometry, going back to the 1939 paper of O.-H. Keller. It states that if a polynomial map from a complex affine space to itself is locally 1-to-1 (i.e. has a non-zero Jacobian), then it is globally 1-to-1 (and, in fact, has a polynomial inverse). This conjecture, especially its dimension 2 case, is infamous for being extraordinarily tricky, as exemplified by a large number of incorrect proofs by respectable mathematicians. Despite that, it is actually a beautiful area of research, with a number of interesting results obtained by diverse methods. In this talk I will review some of these results and present yet another approach to this conjecture, using ideas of modern birational algebraic geometry.
May 1 (SPECIAL DAY THURSDAY AND TIME 4:30 IN LN 2205): Florian Pop (University of Pennsylvania)
Title: Lifting covers of curves.
Abstract: The result I want to present generalizes both: (a) The so called Oort conjecture, which asserts that every cyclic possibly ramified cover of curves Y->X in characteristic p can be lifted to a cyclic cover of smooth curves in (generic) characteristic zero; (b) The Grothendieck's Specialization Theorem of tame fundamental groups. The result is in print at the Annals of Math.
May 2 : Florian Pop (University of Pennsylvania)
Title: Pro-l birational anabelian phenomena.
Abstract: The anabelian conjectures (both in the Grothendieck's arithmetical setting and/or in Bogomolov's geometrical setting require that "enough" geometric Galois theoretical information is present. There is though evidence that less is needed, namely that everything could be done in a pro-l setting. I plan to explain the terms and give some evidence for my claims. This is work in progress.
May 9 : Andrew Schultz (Wellesley College)
Title: Ice models, the Yang-Baxter equation, and a deformation of the Weyl Character Formula.
Abstract: By assigning polynomial weights to certain symmetry classes of alternating sign matrices, Okada gave deformations of the Weyl denominator formula in types $B$, $C$ and $D$. In this talk we generalize these results by replacing Okada's families of alternating sign matrices with more general families of ice models from statistical mechanics. This allows us to use some local conservation rules --- particularly the Yang-Baxter equation --- to evaluate the corresponding partition functions. We will see that a certain specialization of these polynomials returns the Weyl character formula.
Departmental home page.