Seminar Categories

Current Series

[View Past Series]

Mon Sep 16

Student Number Theory Seminar

3:35pm - Vincent Hall 1
A Brief Introduction to L-Functions
Joe Dickinson

Like Andy's talk last week, this week will be another introductory talk; the topic is L-functions. We will start with a discussion of Dirichlet's use of L-series to show the infinitude of primes in arithmetic progressions, then proceed to how L-functions have become a major area of investigation. We will discuss only a sampling of topics, with the goal of motivating interest and setting the stage for future talks.

Mon Sep 23

Student Number Theory Seminar

3:35pm - Vincent Hall 1
Elliptic Curves
Claire Frechette
Mon Sep 30

Student Number Theory Seminar

3:35pm - Vincent Hall 1
Leading up to Dirichlet’s class number formula
Dev Hegde

The talk will give a historical introduction to number theory leading up to Dirichlet’s class number formula which was one of the biggest achievements of analytic methods in number theory. No background is necessary to understand the talk.

Mon Oct 07

Student Number Theory Seminar

3:35pm - Vincent Hall 1
A Tour of the p-adic Representation Theory of GL_2
Katy Weber

We summarize the classification of representations of GL_2 over a p-adic field, emphasizing the relationship between these representations and automorphic forms and L-functions. Time permitting, we will also discuss Whittaker models of these representations and the Casselman-Shalika formula. This talk is meant to be a sketch of these results and how they fit into the bigger picture, and should be accessible even if you are not very familiar with representation theory.

Mon Oct 14

Student Number Theory Seminar

3:35pm - Vincent Hall 1
Arithmetic Speculation on a Combinatorial Lemma
Eric Stuckey

Reflection groups are an object in classical geometry with deep connections to Lie theory, representation theory, algebraic geometry, invariant theory, and combinatorics. The first half of the talk will give a quick introduction to various flavors of reflection groups. In the second half of the talk I will state a lemma about a restricted class of reflection groups, and discuss an idea for how incorporating cyclotomic fields may allow us to remove that restriction.

Mon Oct 21

Student Number Theory Seminar

3:35pm - Vincent Hall 1
"Representation Stability, Étale Cohomology and Combinatorics of Configuration Spaces over Finite Fields"
David DeMark

After introducing the theory of FI-modules in 2012, the collaborative unit consisting of Thomas Church, Jordan Ellenberg and Benson Farb applied their framework to asymptotically stable counting problems in a certain classes of FI-varieties over finite fields in their 2013 paper Representation stability in cohomology and asymptotics for families of varieties over finite fields. The paper serves as a proof-of-concept, unifying a number of previously-known combinatorial results. The key to their method is the Grothendieck-Lefschetz fixed-point theorem with twisted statistics, which relates the rational cohomology of an algebraic variety over the complex numbers with the trace of the Frobenius map applied to the étale cohomology with coefficients in an $\ell$-adic sheaf of that variety over a finite field. In this talk, we shall introduce the Grothendieck-Lefschetz formula and its associated machinery as well as FI-modules and representation stability, then use these ideas to give an exposition of some results of Church, Ellenberg and Farb as they relate to configuration spaces and the braid group.

Mon Oct 28

Student Number Theory Seminar

3:35pm - Vincent Hall 1
"Part 2: Representation Stability, Étale Cohomology and Combinatorics of Configuration Spaces over Finite Fields"
David DeMark

After introducing the theory of FI-modules in 2012, the collaborative unit consisting of Thomas Church, Jordan Ellenberg and Benson Farb applied their framework to asymptotically stable counting problems in a certain classes of FI-varieties over finite fields in their 2013 paper Representation stability in cohomology and asymptotics for families of varieties over finite fields. The paper serves as a proof-of-concept, unifying a number of previously-known combinatorial results. The key to their method is the Grothendieck-Lefschetz fixed-point theorem with twisted statistics, which relates the rational cohomology of an algebraic variety over the complex numbers with the trace of the Frobenius map applied to the étale cohomology with coefficients in an $\ell$-adic sheaf of that variety over a finite field. In this talk, we shall introduce the Grothendieck-Lefschetz formula and its associated machinery as well as FI-modules and representation stability, then use these ideas to give an exposition of some results of Church, Ellenberg and Farb as they relate to configuration spaces and the braid group.

Mon Nov 04

Student Number Theory Seminar

3:35pm - Vincent Hall 1
Rankin-Selberg Method
May Shengmei
Mon Nov 11

Student Number Theory Seminar

3:35pm - Vincent Hall 1
Introduction to Rankin-Selberg Method
Shengmei An

Rankin-Selberg method has been one of the most powerful techniques for studying the Langlands program. In this talk, we will start with the original simplest example of the Rankin-Selberg method, and then come to a more general case of the Rankin-Selberg method on GL_m*GL_n where we can reduce the global integral to the more accessible lovely local integrals so that we can establish some of the important analytic properties of the L-functions.

Mon Nov 18

Student Number Theory Seminar

3:35pm - Vincent Hall 1
The Casselman-Shalika Formula for GL_2
Emily Tibor

This talk will focus on the Casselman-Shalika formula for GL_2 over a non-Archimedean local field, which is an explicit formula for the values of the spherical Whittaker function. A good amount of time will be dedicated to explaining the necessary background including Whittaker models and spherical vectors, which come together to form the spherical Whittaker function. We will then be ready to discuss the formula, Casselman's method of calculating it, and its significance.

Mon Nov 25

Student Number Theory Seminar

3:35pm - Vincent Hall 1
Crystalline cohomology and Katz's conjecture
Shengkai Mao

Crystalline cohomology is a type of Weil cohomology theory that fills in the gap at $p$ in the family of $l$-adic cohomologies. It's introduced by Alexander Grothendieck and developed by Pierre Berthelot. We will briefly discuss what is crystalline cohomology and why we need it. With the help of Frobenius action, we can define a semi-linear morphism on crystalline cohomology which provides a Newton polygon. We will state the Katz's conjecture (which is proved by Mazur and Ogus) (slogan: Newton polygon lies above Hodge polygon) and show some applications (if time permits).

Mon Dec 02

Student Number Theory Seminar

3:35pm - Vincent Hall 1
Student Number Theory Seminar
Henry Twiss
Mon Feb 03

Student Number Theory Seminar

3:25pm - Vincent Hall 570
Self-adjoint operators and zeta function
Paul Garrett

One hundred years ago, when the theory of self-adjoint operators was
just being developed, Hilbert and Polya independently speculated on
the possibility of using self-adjoint operators (the fact that all
eigenvalues are real) to prove the Riemann Hypothesis. In 1977, a
flawed numerical computation of eigenvalues of the invariant
Laplace-Beltrami operator on the modular curve seemed to indicate that
zeros of zeta appeared as spectral parameters.

Mon Feb 10

Student Number Theory Seminar

3:25pm - Vincent Hall 570
The Conditional Probability That an Elliptic Curve Has a Rational Subgroup of Order 5 or 7
Meagan Kenney

Let E be an elliptic curve over the rationals. Divisibility of the set of rational points on E by some integer m can occur locally or globally. If E has global divisibility by m, then E has local divisibility by m; however, work of Katz shows that the converse is only guaranteed up to isogeny. Cullinan and Voight showed that the probability than an elliptic curve has global divisibility by an integer m is non-zero for all integers m allowed by Mazur's classification of rational torsion on elliptic curves. In this talk, I will discuss the probability that E has global divisibility by 5 or 7, given that E has local divisibility by 5 or 7, respectively.

Mon Feb 24

Student Number Theory Seminar

3:35pm - Vincent Hall 570
The Satake equivalence I: The classical formulation
John O'Brien

When studying the representation theory of reductive groups, one runs into a mysterious phenomenon: a certain duality between certain groups. In 1963, Ichir? Satake gave one of the first attempts of
explaining this duality--the Satake isomorphism between the spherical Hecke algebra of one group over a local field with the complexified representation ring of the Langlands dual group. In this duo of talks,
we will discuss two formulations of the Satake equivalence: the classical formulation in terms of algebras, and the more recent geometric formulation in terms of categories related to the affine Grassmannian, an infinite-dimensional space. For this first talk, we will give an overview of the classical formulation.

Mon Mar 02

Student Number Theory Seminar

3:25pm - Vincent Hall 570
The Satake equivalence II: The geometric formulation
John O'Brien

We continue our discussion of the Satake equivalence and Langlands dual groups with an introduction to the geometric Satake equivalence. The classical Satake isomorphism establishes an algebra isomorphism between the spherical Hecke algebra of one group G and the Grothendieck group of the category of representations of the dual. We wish for a stronger statement--an equivalence of categories between a categorical analogue of the spherical Hecke algebra of G and the category of representations of the dual of G. The geometric Satake isomorphism establishes this equivalence, using the geometry of the affine Grassmannian of G to construct a suitable "spherical Hecke category" of G. In this talk, we discuss the affine Grassmannian and introduce the tools needed to understand the geometric Satake equivalence.