The Arithmetic Seminar

TOPICS: Arithmetics 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).

ORGANIZERS: Marcin Mazur, Adrian Vasiu, Ding Ding, and Joel Robert Dodge.

Number theory at Binghamton University: Presently consists of two faculties (Marcin Mazur and Adrian Vasiu), five Ph.D. students (Ding Ding, Michael Fink, Jinghao Li, Slobodan Tanusevski, Daniel Williams), and one post-doc Joel Dodge.

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

Spring 2012

• SPRING 2012: 17 talks and 1 conference.


• February 7 (CROSS LISTING WITH THE ALGEBRA SEMINAR; SPECIAL DAY TUESDAY and TIME 2:50pm): Andrew Linshaw (Brandeis University)
  Title: Jet schemes and invariant theory.
  Abstract: Given a complex, reductive algebraic group G and a G-module V, the mth jet scheme G_m acts on the mth jet scheme V_m, for all m > 0. I will discuss the ring of G_m invariant functions on V_m, and its relationship to the ring of functions on (V//G)_m, where V//G is the categorical quotient. Time permitting, I will discuss some applications of our results to vertex algebras. This is a joint work with Gerald Schwarz (Brandeis) and Bailin Song (University of Science and Technology of China).



• February 10 : Marcin Mazur (Binghamton University)
  Title: Representations of analytic functions as infinite products and some arithmetic applications.
  Abstract: I will discuss a certain decomposition of analytic functions into infinite products and show some applications to numerical computations and arithmetic. This is a joint work with B. Petrenko.



• February 14 : Joel Dodge (Binghamton University)
  Title: Special values of L-functions and the Coates-Sinnott conjecture in function fields.
  Abstract: I will give a short introduction to the field of special values of L-function which can be viewed as a vast generalization of Dirichlet's class number formula. In order to suggest how these sorts of problems can be attacked, I will then sketch the recent proof of a refined form of the Coates-Sinnott conjecture in function fields.



• February 23 (CROSS LISTING WITH THE COLLOQUIUM -- Dean's Speaker Series in Geometry/Topology--; SPECIAL DAY THURSDAY and TIME 4:30pm): Dinesh Thakur (University of Arizona)
  Title: The arithmetic of function fields.
  Abstract: We will explain what would be analogues of e, pi, gamma(1/7), and zeta(3) if integers are replaced by polynomials, and what is known about relations between such special values. Note: This talk will be accessible to graduate students.



• February 24 (Dean's Speaker series in Geometry/Topology) : Dinesh Thakur (University of Arizona)
  Title: Diophantine approximation of algebraic quantities in finite characteristic.
  Abstract: After recalling results of Dirichlet, Liouville, Roth, and Schmidt almost settling the number field case, we will explain why the situation is not even conjecturally understood in finite characteristic, and explain the recent progress. Note: This talk will be accessible to graduate students.



• March 2 : Adrian Vasiu (Binghamton University)
  Title: On a motivic conjecture of Milne.
  Abstract: Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We report on a proof of a motivic conjecture of Milne that relates, in the case of abelian schemes over W(k), the \'etale cohomology with ZZ_p coefficients to the crystalline cohomology with integral coefficients, in the more general context of p-divisible groups endowed with arbitrary families of crystalline tensors over W(k). This extends a result of Faltings. As a main new tool we construct global deformations of p-divisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over W(k).



• March 9 : Dikran Karagueuzian (Binghamton University)
  Title: Title: A Short History of Matrix Multiplication Algorithms
  Abstract: The definition of matrix multiplication is an algorithm for multiplying matrices, which was the fastest method known until 1968. Since then, increasingly ingenious methods have been developed, but the speed of the best known algorithms is not close to the conjectured bound. I will give an exposition of the development of these algorithms, and attempt to explain the relationship to number theory. (The audience will be free to form their own conclusions on whether this relationship actually exists.) This will be an expository talk.



• March 13 (CROSS LISTING WITH THE ALGEBRA SEMINAR; SPECIAL DAY TUESDAY and TIME 2:50pm) : Antun Milas (SUNY-Albany)
  Title: Nahm sums, combinatorial bases and vertex algebras
  Abstract: Nahm sums are important n-fold hypergeometric q-series appearing in representation theory of infinite dimensional Lie algebras, conformal field theory, and in other areas. In the first part of my talk, I will explain how to interpret a large class of (generalized) Nahm sums as graded dimensions of certain vertex algebras associated to integral lattices (this is based on a joint work with M. Penn). In the second part, which is based mostly on low-rank examples, I will focus on various inclusion-exclusion type formulas, modularity issues, etc. This talk is meant to be elementary and accessible to graduate students.



• March 16 : Mahdi Asgari (Oklahoma State University and Cornell University)
  Title: Counting Cusp Forms
  Abstract: How many cusp form are there on SL(2), SL(n), or a more general (reductive or semisimple) linear algebraic group? Until a few years ago it was not known that there are infinitely many cusp forms on a group such as SL(n) beyond very small values of n. Weyl's law refers to an asymptotic formula for the number of cusp forms on a given connected reductive group, in particular establishing their infinitude. I will discuss some work-in-progress, joint with Werner Mueller of University of Bonn, establishing Weyl's law with remainder terms for classical groups. Without remainder terms, Weyl's law was recently established by Lindenstrauss and Venkatesh in a rather general setting.



• March 30 : Daniel Williams (Binghamton University)
  Title: Quotients of varieties by finite groups.
  Abstract: If X is an algebraic variety and G is a group of automorphisms of X then even in very simple cases we can have that a quotient variety X/G will fail to exist. We will prove, however, that when G is finite and X satisfies some reasonable hypotheses then X/G is a variety in a natural way. We will discuss the implications of this for abelian varieties. If time permits we will give a closer discussion of the relationship between sheaves of modules over X and sheaves of modules over X/G.



• April 11 (Dean's Speaker Series in Geometry/Topology --; SPECIAL DAY AND LOCATION: WEDNESDAY 3:30pm in AA G21): Cristian Popescu (UC San Diego)
  Title: An equivariant main conjecture in Iwasawa theory and applications
  Abstract: I will discuss the statement and proof of an Equivariant Main Conjecture (EMC) in the Iwasawa theory of arbitrary global fields. This will be followed by applications of the EMC (via Iwasawa co-descent) towards proving various well-known conjectures on special values of global L-functions, such as the Brumer-Stark, Coates-Sinnott and the Equivariant Tamagawa Number conjectures. In the process, an important role will be played by an explicit construction of ell-adic Tate sequences, which will be explained in some detail. This is based on joint work with Cornelius Greither.



• April 12 (CROSS LISTING WITH THE COLLOQUIUM -- Dean's Speaker Series in Geometry/Topology--; SPECIAL DAY AND TIME 4:30 pm) : Cristian Popescu (UC San Diego)
  Title: Generalized Class Number Formulas
  Abstract: The well-known analytic class number formula, linking the special value at s=0 of the Dedekind zeta function of a number field to its class number and regulator has been the foundation and prototype for the highly conjectural theory of special values of L-functions for close to two centuries. We will discuss generalizations of the class number formula to the context of equivariant Artin L-functions, which capture refinements of the Brumer-Stark and Coates-Sinnott conjectures. The generalized formulas relate various algebraic-geometric invariants associated to a global field, e.g. its Quillen K-theory and \'etale cohomology, to various special values of its Galois-equivariant L-functions. This is based on joint work with Banaszak, Dodge and Greither.


April 17 (CROSS LISTING WITH THE ALGEBRA SEMINAR; SPECIAL DAY Tuesday AND TIME 2:50pm): Viji Thomas (TIFR, India)
Title: Bazzoni-Glaz Conjecture
Abstract: We will introduce the notion of Gaussian rings and Prufer domains. We will look at Kaplansky's conjecture, and we will look at the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Time permitting, I will give an outline of a proof of the Bazzoni-Glaz conjecture. This is joint work with Guram Donadze.


• April 20 : Jinghao Li (Binghamton University)
  Title: Constancy theorems for F-crystals over k[[T]]
  Abstract: I will first define nilpotent integral connections and F-crystals over k[[T]]. Then I will introduce the Newton-Hodge and Newton filtrations for such F-crystals. Finally I will define a constant F-crystal over k[[T]] and prove constancy theorems for F-crystals over k[[T]].



• April 28--29 : Second Annual Upstate Number Theory Conference
  Web page: A-DAY
  Place: UR Mathematics, 915 Hylan Building, University of Rochester, Rochester, NY 14627



• May 11 : Ding Ding (Binghamton University)
  Title: Lifts and automorphisms of truncated Barsotti-Tate groups
  Abstract: In this talk we discuss the paper "Dimensions of group schemes of automorphisms of truncated Barsotti-Tate groups" by O. Gabber and A. Vasiu. Various invariants of truncated Barsotti-Tate groups will be reviewed. Time permitting, we will also construct a suitable group action whose set of orbits parametrizes the set of isomorphism classes of lifts of Barsotti-Tate groups of level m to level m+1.