This year's Student Number Theory Seminar was co-organized by Shengmei An (an000008@umn.edu) and myself. Meetings were loosely weekly on Mondays 1-2pm central time in the fall, and every few weeks in the spring on Mondays at 3.30pm, all held over Zoom.
Titles and abstracts from the 2019-2020 seminar may be found at Andy Hardt's webpage: http://www-users.math.umn.edu/~hardt040/snt/snt19-20.html
5/3/21: Eric Nathan Stucky
Title: Universal Quadratic Forms for Number Fields
Abstract: In 1770, Lagrange published the first known proof of the four-squares theorem, which states that every positive integer may be written as the sum of four (or fewer) perfect squares. A natural idea is to try to extend this result to other number fields— but in fact, the rationals are very nearly the only number field with such a theorem. In this expository talk, we will begin by describing what "such a theorem" actually means, and discuss the state of the art for some more flexible generalizations.
3/22/21: Shengmei An
Title: Rankin-Selberg integral and exterior cube L-function for GL_6
Abstract: I will start with historic background of Langlands L-function conjecture, and use the example of GL_2 to explain the local-global L-function machine. And then we will discuss the exterior cube L-functions constructed by Ginzburg and Rallis.
2/1/21: Devadatta Ganesh Hegde
Title: Number theory and the foundations of mathematics: an episode from the 19th century.
Abstract: We will discuss the role of number theory in the rewriting of the foundations of mathematics that started in the 1870s with Dedekind's rewrite of the foundations of analysis. We will also give some related historical episodes from 19th century analysis and geometry.
12/7/20: Emily Tibor
Title: Classification of models via Hecke algebras
Abstract: A model of a group G is a space of functions that realizes a large class of representations of G. Models have proven useful in the study of automorphic forms and L-functions. We would like a system to classify models, both to explain current examples of models and to provide a source for new models with useful applications. As progress towards this goal, certain models have previously been linked to characters of the finite Hecke algebra in work by Brubaker, Bump, and Friedberg and separately by Chan and Savin. After discussing the basics of models and some of their applications, we discuss the results of those authors and how one might expand their framework. In the mathematical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem states that a representation of an algebraic group over the adeles is automorphic whenever the L-functions of various twists of it are well behaved.
11/23/20: Anh Hoang
Title: Geometric Analytic Number Theory
Abstract: We will give an overview of the central ideas in geometric analytic number theory, in particular arithmetic statistics. We will discuss the procedure of extrapolating problems in analytic number theory to problems and solutions in arithmetic topology, and some of the tools commonly used in the subject. If time permits, we will further demonstrate the geometric Malle’s conjecture and recent results by Ellenberg, Tran, and Westerland.
11/9/20: Katy Weber
Title: Colored Lattice Models for Iwahori Whittaker Functions (a.k.a. summary of this paper https://arxiv.org/abs/1906.04140)
Abstract: Last time, Andy gave an introduction to Ben Brubaker's research and described how (spherical) Whittaker functions can be identified with certain functions associated to lattice models. We will ramp this up and show that by adding colored paths to these lattice models, we obtain refinements called Iwahori Whittaker functions. Moreover, we see how aspects of the lattice model give us new insights into properties of these functions. (Note: knowledge of Andy's talk not required!)
10/25/20: Andy Hardt
Title: From automorphic forms to integrable systems (a.k.a. The research of Ben Brubaker)
Abstract: We take a journey, to link some important concepts in automorphic forms to the solvable lattice models that have long been studied in statistical mechanics. We'll give a (very) brief overview of automorphic forms, talk about certain analogues of Fourier coefficients, called Whittaker functions, and state the Casselman-Shalika formula, which says that these functions are given by representation theoretic data. Then we'll pivot to talk about lattice models, a combinatorial framework that can encode these functions and many others. The most important lattice models have a (quantum) Yang-Baxter equation, and are then called solvable or integrable. If we have a solvable lattice model for a particular functions, it allows us to conclude important representation theoretic, number theoretic, and geometric properties. We'll focus on three: branching, symmetry, and (dual) Cauchy identities. We're covering a lot of ground, so it might feel fast paced. But our main focus is on the "narrative" structure, rather than specific details. With this in mind, the talk is designed to be accessible to incoming graduate students.
10/19/20: May An
Title: Local Converse Theorem (and if we have time Gan-Gross-Prasad Conjecture) (aka What does Dihua Jiang do)
Abstract: Dihua’s research area is very broad, so I chose two of his important papers to discuss in our seminar. These two papers involve the local converse theorem and GGP conjecture.
10/12/20: Claire Frechette
Title: Is this....Number Theory?
Abstract: Once upon a time, mathematicians noticed a pattern in natural numbers and they wondered: can we prove that? Then they noticed another pattern, but the same proof didn't work, so they found a new one. And thus the field of Number Theory was born: yes, it looks like a massive mathematical hodge-podge, and you might wonder: how does any of this stuff interact? But, when we hold it up to the light, you'll see the shared set of questions that fly through it, like the world's best mathematical magpies, all tying back to the natural numbers. This talk aims to give a brief history of Number Theory, and then a brief introduction to Number Theory at the U. If you're newly interested in Number Theory, or if you just like a good story, come join us. The Student Number Theory Seminar is low-key, low-stress, and very friendly, so we hope to see you there!