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Texas A&M University
Mathematics

Number Theory Seminar

Spring 2018

 

Date:January 24, 2018
Time:1:15pm
Location:BLOC 220
Speaker:Matt Young, Texas A&M University
Title:Equidistribution of Eisenstein series
Abstract:I will discuss recent work on the behavior of Eisenstein series on the modular surface, restricted to geodesic segments.

Date:January 31, 2018
Time:1:15pm
Location:BLOC 220
Speaker:Andrew Bridy, Texas A&M University
Title:The cycle structure of unicritical polynomials in finite fields
Abstract:Let f(x) = x^k+a in Z[x] for k \geq 2. Consider the family of dynamical systems given by the action of f on F_p as p varies among primes. The question of how and in what sense this family approximates a random family of dynamical systems has been studied extensively, motivated in part by Pollard's "rho" algorithm for integer factorization. We show that for most choices of a, the cycle structure in this family is "as random as possible" in an appropriate sense. As a corollary, we show that most members of these families have many cycles. This is joint work with Derek Garton.

Date:February 28, 2018
Time:1:15pm
Location:BLOC 220
Speaker:Wei-Lun Tsai, Texas A&M University
Title:Analytic formulas for Stark units in quadratic extensions of totally real cubic fields
Abstract:In this talk, we will explain how Stark units in certain quadratic extensions of totally real cubic fields can be evaluated explicitly in terms of values of the Barnes triple Gamma function at algebraic arguments. This is joint work with Adrian Barquero-Sanchez and Riad Masri.

Date:March 7, 2018
Time:1:15pm
Location:BLOC 220
Speaker:Souvik Goswami, Texas A&M University
Title:Higher arithmetic Chow groups
Abstract:We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over a number field, which is similar to Gillet and Soulé's definition of arithmetic Chow groups. We also give a compact description of the intersection theory of such groups. A consequence of this theory is the definition of a height pairing between two higher algebraic cycles, of complementary dimensions, whose real regulator class is zero. This description agrees with Beilinson's height pairing for the classical arithmetic Chow groups. We also give examples of the higher arithmetic intersection pairing in dimension zero that, assuming a conjecture by Milnor on the independence of the values of the dilogarithm, are non zero. This is a joint work with José Ignacio Burgos-Gil from ICMAT, Spain.

Date:April 11, 2018
Time:1:15pm
Location:BLOC 220
Speaker:Alan Haynes, University of Houston
Title:Bounded remainder sets for rotations on the adelic torus
Abstract:Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied, especially in the context of Diophantine approximation, for over 100 years. In the last few years there have been a number of results which culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable volumes. In this talk we are going to explain these results, and then explain how to generalize them to give explicit constructions of bounded remainder sets for rotations on the adelic torus. Our method of proof combines ideas from harmonic analysis on the adeles, dynamical systems, and the theory of mathematical quasicrystals.

Date:April 18, 2018
Time:1:15pm
Location:BLOC 220
Speaker:Shuhui Shi, University of Rochester
Title:Multiple zeta values over F_q[t]
Abstract:

Classical multizeta values (abbrev. mzv) were first introduced by Euler when studying zeta values. Recently, these values have drawn people's interest because of their appearance in many different fields. Seeing these connections, Dinesh Thakur introduced mzv analogously in the function field case.

In this talk, we first give a general introduction of the analogues of zeta and mzv in the classical and function field cases. We then talk about our study of these values, including zeros of mzv at negative integers, conjectured Hopf algebra structure and truncated mzv and their linear Fq(t)-relations.


Date:May 2, 2018
Time:1:15pm
Location:BLOC 220
Speaker:Solly Parenti, University of Wisconsin
Title:Unitary CM fields and the Colmez conjecture
Abstract:In 1993, Pierre Colmez conjectured a relation between the Faltings height of a CM abelian variety and certain log derivatives of L-functions associated to the CM type, generalizing the classical Chowla-Selberg formula. I will discuss how we can extend the known cases of the conjecture to a certain class of unitary CM fields using the recently proven average version of the conjecture.