Seminar Categories

Current Series

[View Past Series]

Thu Oct 11

Student Number Theory Seminar

12:20pm - Vincent Hall 313
Student Number Theory
TBA
Thu Oct 18

Student Number Theory Seminar

12:20pm - Vincent Hall 313
Elliptic functions and elliptic curves in the 19th century
Devadatta Hegde, University of Minnesota

We will give an account of the work on Weierstrass and Jacobi proving a result due to Abel on meromorphic functions on the torus. These are results about complex points of elliptic curves which suggest attributes for rational points. These examples were later greatly extended at the hands of several mathematicians and reached a high-point with the GAGA principle by Serre. It's also the first example of a Riemann-Roch type theorem which were greatly extended by Grothendieck, Atiyah and Singer. Only some familiarity with Cauchy's theorem in complex analysis is needed to understand the talk.

Thu Oct 25

Student Number Theory Seminar

12:20pm - Vincent Hall 313
Cryptographic Multilinear Maps from Elliptic Curves
Mahrud Sayrafi, University of Minnesota

We will begin with defining cryptographic multilinear maps, briefly discussing some of their applications, and referencing one such map from Boneh-Silverberg '03. After that, we will extend a problem involving isogenies of elliptic curves into an open problem of finding cryptographic invariant maps from Boneh, et al. '18.stract:

Thu Nov 01

Student Number Theory Seminar

12:20pm - Vincent Hall 313
The Stone-von Neumann Theorem
Joe Dickinson, University of Minnesota

I plan to talk about some number theoretic implications of the Stone-von Neumann theorem. The Stone-von Neumann theorem is a uniqueness theorem about commutation relations between position and momentum operators. I will give a historical discussion about number theory results implied by Stone-von Neumann.

Thu Nov 08

Student Number Theory Seminar

12:20pm - Vincent Hall 313
Physics of the Riemann-zeta function
Adrienne Sands, University of Minnesota

I will give an overview of how the Riemann-zeta function plays a role in different areas of physics, from condensed matter theory to quantum mechanics.

Thu Nov 15

Student Number Theory Seminar

12:20pm - Vincent Hall 313
Physics of the Riemann-zeta function II
Adrienne Sands, University of Minnesota

We continue to discuss how the Riemann-zeta function plays a role in different areas of physics, from condensed matter theory to quantum mechanics

Thu Nov 22

Student Number Theory Seminar

12:20pm - Vincent Hall 313
Student Number Theory
TBA
Thu Nov 29

Student Number Theory Seminar

12:20pm - Vincent Hall 313
The Weil Conjectures: 0 to 100
Liam Keenan, University of Minnesota

The Weil conjectures are perhaps one of the most stunning achievements in arithmetic geometry in the 20th century. In this talk, I plan to introduce the necessary algebro-geometric language, state the conjectures, discuss the some of the tools used to prove them, and draw connections to analytic number theory.

Thu Dec 06

Student Number Theory Seminar

12:20pm - Vincent Hall 313
A Tour of the Modularity Theorem
Katy Weber, University of Minnesota

The Modularity Theorem (also known as the Taniyama-Shimura-Weil Conjecture) states, essentially, that every elliptic curve arises from a modular form. It is a special case of the Langlands correspondence and was a major part of Andrew Wiles' proof of Fermat's Last Theorem. In this talk, I will sketchily discuss the theorem in its various forms, leading up to the statement that the L-function of an elliptic curve agrees with the L-function of a modular form. Along the way, we will encounter some algebraic geometry words, e.g. "moduli space," "universal curve," and "sheaf."

Thu Dec 13

Student Number Theory Seminar

12:20pm - Vincent Hall 313
Galois Representations and the Modularity Theorem
Andy Hardt, University of Minnesota

I will continue Katy's talk from last week, and talk about chapter 9 of Diamond and Shurman's "A First Course in Modular Forms." We'll define Galois representations corresponding to both elliptic curves and modular forms. Then we'll state the modularity theorem, which asserts that every elliptic curve corresponds to a modular form, and their Galois representations are equal. If we have time, we'll relate this back to the version of the theorem from last week, which equated points on an elliptic curve to Fourier coefficients of a modular form. The proof of the theorem, however, will be left as an exercise to the listener.

Thu Dec 20

Student Number Theory Seminar

12:20pm - Vincent Hall 313
Student Number Theory
TBA
Thu Feb 14

Student Number Theory Seminar

11:00am - Vincent Hall 213
Counting Number Fields, Second Order Asymptotics, and Hurwitz Spaces
Greg Michel, University of Minnesota

I'll introduce well-known counting conjectures related to number fields of bounded discriminants, and then I'll talk about some semi-related topological ideas that may allow us to strengthen these conjectures in certain situations.

Thu Feb 28

Student Number Theory Seminar

11:00am - Vincent Hall 213
Finite Hecke Algebras and Their Characters
Andy Hardt, University of Minnesota

We explore some of the major results in the study of finite Hecke algebras and their character tables. These algebras are useful in the study of representations of finite Chevalley groups, and also appear in the study of quantum groups and knot/link invariants. We'll run through some equivalent definitions of this versatile object, and then talk about a couple approaches to its character theory. In particular, Starkey's rule is a combinatorial formula for the character table of the type A finite Hecke algebra. We'll briefly sketch a proof of this result and talk about the possibility for extension to other types. Starkey's rule allows us to calculate the weights of Ocneanu traces, which are important invariants in knot theory relating to type A Hecke algebras, and a type B version of Starkey's rule would give us insights into the type B analogue to Ocneanu traces.

Thu Mar 07

Student Number Theory Seminar

11:00am - Vincent Hall 213
Knots and Primes
Eric Stucky, University of Minnesota

In this talk we will outline the basic premise for the analogy between knots (in manifolds) and primes (in number fields). This analogy involves some rather heavy definitions; we will review the topological background as needed, while taking a more intuitive angle on the arithmetic machinery. Time permitting, we will briefly sketch an extension of the Frobenius automorphism which is a major tool in understanding the analogy between the Legendre symbol and the linking number.

Thu Mar 14

Student Number Theory Seminar

11:00am - Vincent Hall 213
Knots and Primes Part II: The Linking Number and Legendre Symbol
Katy Weber, University of Minnesota

We review the analogy between knots in 3-manifolds and prime ideals in number rings, and push it further to realize the Legendre symbol as the analogue of the (mod 2) linking number. This talk should be accessible even if you did knot attend seminar last week.

Thu Mar 28

Student Number Theory Seminar

11:00am - Vincent Hall 213
On congruence properties of Ramanujan tau function
Dev Hegde, University of Minnesota

Serre and Swinnerton-Dyer reproved many congruence relations around 1970 for the Ramanujan tau function by reinterpreting the relations in terms of l-adic representations. We will give an introduction to this topic.

Thu Apr 04

Student Number Theory Seminar

11:00am - Vincent Hall 213
On congruence properties of Ramanujan tau function
Dev Hegde, University of Minnesota

Serre and Swinnerton-Dyer reproved many congruence relations around 1970 for the Ramanujan tau function by reinterpreting the relations in terms of l-adic representations. We will give an introduction to this topic.

Thu Apr 18

Student Number Theory Seminar

11:00am - Vincent Hall 213
More Yang-Baxter Equations for Metaplectic Ice
Claire Frechette, University of Minnesota

In preparation for my oral exam, I will extend the existing connections between quantum groups and the study of spherical Whittaker models on metaplectic covering groups of GL(r,F), for F a nonarchimedean local field. Brubaker, Buciumas, and Bump showed that for a certain metaplectic n-fold cover of GL(r,F) a set of Yang-Baxter equations govern the behavior of the Whittaker functions and that these equations arise from a Drinfeld twist of a quantum affine Lie superalgebra. I will extend their results to all metaplectic covers of GL(r,F), showing that the same Yang-Baxter equations underlie the scattering matrix for the Whittaker functions over an nQ-fold metaplectic cover, where nQ is an integer determined by the cover, and that these equations arise from Drinfeld twists of quantum groups. (I will also define what most of these words mean.)

Thu Apr 25

Student Number Theory Seminar

11:00am - Vincent Hall 213
Applying to Jobs in Number Theory
Panel, University of Minnesota

Come hear recent graduates and current grad students discuss their experiences navigating the job market. This panel is hosted by the Student Number Theory seminar, but all graduate students are welcome to attend.

Thu May 02

Student Number Theory Seminar

11:00am - Vincent Hall 213
Introduction to Langlands philosophy
Dev Hedge, University of Minnesota

We will give less precise but hopefully very motivational introduction to Langlands philosophy with many examples. A serious background in anything is not essential.