This is the website for the student number theory seminar for the academic year (2022 Fall-2023 Spring).
This is the website for the student number theory seminar for the academic year (2022 Fall-2023 Spring).
Schedules for Fall 2022
Wednesday, Nov 9th, 2:30-3:20 pm at Amundson Hall 116
Title: Branching rules and their generalizations
Speaker: Cheng Chen
Abstract: I will introduce the branching rules that arise from symmetry breakings in Hamiltonian physics. They have generalizations to compact groups and classical groups over local fields. One refinement is called the local Gan-Gross-Prasad conjecture, which has global applications that relate L-functions to period integrals of automorphic forms.
Wednesday, Nov 16th, 2:30-3:20 pm at Amundson Hall 116
Title: Quantum mechanics and Weil representations
Speaker: Cheng Chen
Abstract: Werner Heisenberg is a famous physicist known for his uncertainty principle in quantum mechanics. There is also a notion called Heisenberg group that is named after him which is indeed related to his work in QM. The Weil representation of the Heisenberg group will rise when we consider the solutions of Schrodinger's equation. This lies the foundation for Theta correspondence.
Thanksgiving break
Wednesday, Nov 30th, 10:00-11:00am at Vincent 570
Title: Bessel functions on GL(2) (or GL(n))
Speaker: Xinchen Miao
Abstract: The study of Bessel functions plays an important role in number theory, automorphic forms, and the Langlands program. In my talk, we will focus on the Bessel functions over non-archimedean local fields. I will give the definition of Bessel functions and introduce some basic and important properties of Bessel functions.
Schedule for Spring 2023
Monday, Feb 13th, 2:30-3:20 pm at Vincent Hall 213
Title: An introduction to the Rankin-Selberg integrals
Speaker: Cheng Chen
Abstract: I will introduce a generalization of the analytic methods in Tate's thesis, called the Rankin-Selberg integral. This a theory well-established in the general linear cases, but limited progress is made in other situations. I will focus on the GL-cases. The applications in number theory and representation theory will be mentioned.
Monday, Feb 27th, 2:30-3:20 pm at Vincent Hall 213
Title: More about the Rankin-Selberg integrals
Speaker: Cheng Chen
Abstract: I will continue the introduction on the Rankin-Selberg integrals, I will focus on the proof for the absolute convergence and meromorphic continuation on the integrals. I will also talk about how it works in other "not both GL"-cases.
Spring break
Monday, March 13th, 2:30-3:20 pm at Vincent Hall 213
Speaker: Xinchen Miao
Title: The Subconvexity Problem
Abstract: The subconvexity problem aims at providing non-trivial (i.e. subconvex) bounds for central values of automorphic L-functions. The main conjecture in this area is the Generalized Lindeloef Hypothesis which itself is a consequence of the Generalized Riemann Hypothesis. This lecture will survey several advances that have been made on this question during the past one hundred years.
Monday, April 17th, 2:30-3:20 pm at Vincent Hall 213
Speaker: Alexis Suki Dasher
Title: Cohomology classes of Richardson varieties from solvable lattice models
Abstract: A growing body of literature indicates a connection between algebraic objects called quantum groups and polynomial functions which arise in Schubert calculus and its generalizations. The bridge between these topics is a combinatorial construction called a lattice model. In this talk, I will explain how these connections show that representations of quantum groups encode information about the cohomology of flag varieties, and how recently constructed lattice models further suggest a relationship between quantum groups and subvarieties of flag varieties called Richardson varieties.
Monday, April 24th, 2:30-3:20 pm at Vincent Hall 213
Speaker: Devadatta G. Hegde
Title: Eisenstein Series and Spectral Decomposition I
I will provide the historical context in which Langlands addressed the problem of the spectral decomposition of automorphic forms and sketch a proof in the simplest case. The aim is to make the role of the Eisenstein series and its poles clear.