Tuesday, 31 January 2023, 4-5pm Location: 24 Gordon Square Room 105

• Speaker: Jack Thorne (Cambridge)
  
  Title: Equidistribution and reciprocity in number theory
  
  Abstract:
A famous result in number theory is Dirichlet’s theorem that there exist infinitely many prime numbers in any given arithmetic progression a, a + N, a + 2 N, … where a, N are coprime. In fact, a stronger statement holds: the primes are equidistributed in the different residue classes modulo N. In order to prove his theorem, Dirichlet introduced Dirichlet L-functions, which are analogues of the Riemann zeta function which depend on a choice of character of the group of units modulo N. More general L-functions appear throughout number theory and are closely connected with equidistribution questions, such as the Sato—Tate conjecture (concerning the number of solutions to y^2 = x^3 + a x + b in the finite field with p elements, as the prime p varies). L-functions also play a central role in both the motivation for and the formulation of the Langlands conjectures in the theory of the automorphic forms. I will give a gentle introduction to some of these ideas and discuss some recent theorems in the area.

Tuesday, 21 February 2023, 4-5pm Location: 24 Gordon Square Room 105

• Speaker: Maciej Zworski (UC Berkeley)
  
  Title: Mathematics of magic angles.
  
  Abstract: Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwan ath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators, and analytic hypoellipticity. Recent mathematical progress also includes the proof of existence of infinitely many generalized magic angles, of classically forbidden regions for eigenstates and computer assisted proofs of existence of real ones (Luskin--Watson, 2021). The results will be illustrated by colourful numerics which suggest many open problems (joint work with S Becker, M Embree, J Wittsten, 2020 and S Becker, T Humbert 2022, and with M Hitrik and S Becker 2023).

Tuesday, May 16 4-5pm Location: 25 Gordon St. Room 505

• Speaker: Holly Krieger (Cambridge)
  
  Title: A transcendental birational dynamical degree
  
  Abstract:In the study of a discrete dynamical system defined by polynomials, we hope as a starting point to understand the growth of the degrees of the iterates of the map. This growth is measured by the dynamical degree, an invariant which controls the topological, arithmetic, and algebraic complexity of the system. I will discuss the history of this question and the recent surprising construction, joint with Bell, Diller, and Jonsson, of a transcendental dynamical degree for a birational map, and how our work fits into the general phenomenon of power series taking transcendental values at algebraic inputs.

---

You'll find the old colloquium schedules here.