Jessica Fintzen (University of Cambridge and Duke University)
Representations of p-adic groups

Abstract: A fundamental problem in representation theory is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups. A solution to this problem has applications far beyond the representation theory of p-adic groups. For example, it plays a central role in the advancement of an explicit Langlands program, a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory.

In my talks I will introduce p-adic groups and provide an overview of our understanding of their representations, with a focus on recent developments. I will also briefly discuss applications to other areas.

Mu-Tao Wang (Columbia)
Angular momentum and supertranslation in general relativity

Abstract: How does one measure the angular momentum carried away by gravitational radiation during the merger of a binary black hole? This has been a subtle issue since the 1960’s due to the discovery of “supertranslation ambiguity”: the angular momentums recorded by two distant observers of the same system may not be the same. In this talk, I shall describe how the theory of quasilocal mass and optimal isometric embedding identifies a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.

Jack Thorne (University of Cambridge)
L-functions and symmetric power functoriality

Abstract: L-functions and symmetric power functoriality I
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 distribution of the number of points modulo p on a fixed elliptic curve E 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. Connecting the arithmetic and the automorphic points of view is an essential step towards solving problems such as the Sato—Tate conjecture. I will discuss the history surrounding these ideas and explain the particular importance of symmetric power L-functions for the Ramanujan conjecture and the Sato—Tate conjecture.

Abstract: L-functions and symmetric power functoriality II
In 2020, James Newton and I proved that the symmetric power L-functions associated to holomorphic modular forms are automorphic, as predicted by the Langlands conjectures. Our proof is based on the connection with number theory, especially the theory of Galois representations and p-adic automorphic forms. I will introduce some of these concepts and explain their connection to the proof of our theorem, including the key observation that automorphy of the symmetric power L-function (which implies the existence of its complex analytic continuation) is a property which can be “p-adically analytically continued” in the right setting.

Ryan O’Donnell (Carnegie Mellon)
Learning and Testing Quantum States

Abstract: Suppose you have an unbalanced die with d faces. The probabilities p_1, …, p_d of the faces are unknown to you. Suppose you can afford to roll the die n times. How accurately can you estimate the p_i’s? How accurately can you estimate the maximum p_i? How accurately can you estimate the entropy of the die? Classical questions like these are well-studied (and often nontrivial).

The laws of quantum mechanics are nothing more than a generalization of the usual laws of probability! If you have a quantum computer with 5 qubits, its “state” rho is like a “quantum die” with d=2^5=32 “faces”. Mathematically, the state is defined by probabilities p_1, …, p_d (adding to 1) together with orthonormal vectors v_1, …, v_d. Now we can ask all of the analogous questions: Given n copies of rho, how accurately can you estimate the p_i’s and the v_i’s? What if you just want to estimate the p_i’s, or the entropy of the p_i’s, or their maximum? Quantum questions like these are being newly studied, and involve beautiful mathematical ideas from many areas.

In the first part of the talk, I will give an overview of the task of learning and testing quantum states. It will be broadly accessible, but knowledge of elementary linear algebra and probability will be helpful. In the second part of the talk, I will dive deeper into mathematical tools entering into the analysis, from the combinatorics of longest increasing subsequences, to the representation theory of the symmetric and unitary groups, to differential privacy.

Cynthia Vinzant (University of Washington)
Log-concavity in matroids and expanders

Abstract: Log-concavity is an important feature of many functions and discrete sequences appearing across mathematics, including combinatorics, algebraic geometry, convex analysis, and optimization. In this talk, I will discuss recent advances in our understanding of the real and combinatorial geometry underlying multivariate polynomials that are log-concave on the positive orthant. These real functions are closely related to combinatorial objects called matroids and can be used to understand the behavior of large random walks on associated simplicial complexes. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.