Talks

 In this talk, I will describe recent work on holographic correspondences in spacetimes which interpolate from anti-de Sitter space in the deep bulk to asymptotic regions which share some properties with flat space. Examples include the linear dilaton throat in the F1-NS5 solution and the NCOS decoupling limit of the D1-D5 system. In both examples, null geodesics take infinite coordinate time to reach the boundary, the causal structure resembles that of Minkowski space, and we can sensibly study radiation near future null infinity. These spacetimes are good solutions of string theory and thus might be considered candidates for a top-down sort of celestial holography.

Review of Classical groups in general, and their classification over local and global fields; their parabolics and Levi subgroups, Whittaker models, degenerate Whittaker models, Bessel and Fourier-Jacobi models, the last will need a bit of the Weil representations.

In a joint work with H. Hida, we proved in the 90's that the anticyclotomic Katz p-adic L function associated to a p-adic CM type divides the characteristic power series of the Iwasawa module associated to this p-adic CM type. The goal of these two talks is to sketch this proof. Note that we couldn't treat the divisibility at the prime p of the Iwasawa algebra. This has been treated in subsequent works by H. Hida.

The arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, expresses the degrees of certain 0-cycles on integral models of unitary Shimura varieties in terms of the nondegenerate Fourier coefficients of the central derivative of an Eisenstein series.
Feng-Yun-Zhang proved a higher derivative version of this arithmetic Siegel-Weil formula in the function field setting, now expressing degrees of 0-cycles on moduli spaces of unitary shtukas to the nondegenerate Fourier coefficients of higher central derivatives of an Eisenstein series.
The goal of my lecture series is (1) to explain all of this background, (2) extend the results of Feng-Yun-Zhang to include some degenerate coefficients, and (3) deduce from this extension an arithmetic application: the nonvanishing of higher central derivatives of certain Langlands L-functions implies the nonvanishing of classes in the Chow groups of moduli spaces of shtukas.
All of the new results are joint work with Tony Feng and Mikayel Mkrtchyan.

Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.

The speaker will try to give an introduction to the GGP conjectures, keeping in mind that he will be speaking to a very mixed audience some of whom may be seeing representation theory of groups over local fields for the first time. I will try not to presume much beyond a basic introduction to representation theory of finite groups over complex numbers, and familiarity with p-adic fields, and p-adic groups. There will be four lectures whose outline I give below.
Lecture 1: Branching laws illustrated with some finite dimensional examples, emphasizing the need of a parametrization, Gelfand pairs, strong Gelfand pairs. Automorphic representations and period integrals, Local-global principle, L-functions.
Lecture 2: Review of Classical groups in general, and their classification over local and global fields; their parabolics and Levi subgroups, Whittaker models, degenerate Whittaker models, Bessel and Fourier-Jacobi models, the last will need a bit of the Weil representations.
Lecture 3: A bit of representation theory of groups over local fields, parabolic induction, cuspidal representations. Review of the Local Langlands correspondence, L-functions and epsilon factors. L-packets, the Jacquet-Langlands correspondence, The GGP conjectures: both local and global conjectures.
Lecture 4: Spill-over from the last lecture, and finish with some low dimensional examples, including the fundamental work of Waldspurger; illustrative examples from finite fields.
References:
• D. W. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge Univ. Press, Cambridge, 1997; MR1431508
• C. J. Bushnell and G. M. Henniart, The local Langlands conjecture for GL(2), Grundlehren der mathematischen Wissenschaften, 335, Springer, Berlin, 2006; MR2234120
• Automorphic forms, representations and L-functions. Part 1, Proceedings of Symposia in Pure Mathematics, XXXIII, American Mathematical Society, Providence, RI, 1979; MR0546586
• Automorphic forms, representations, and L-functions. Part 2, Proceedings of Symposia in Pure Mathematics, XXXIII, American Mathematical Society, Providence, RI, 1979; MR0546606
• W. T. Gan, B. H. Gross and D. Prasad, Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups, Ast´erisque No. 346 (2012), 1–109; MR3202556
• W. T. Gan, B. H. Gross and D. Prasad, Restrictions of representations of classical groups: examples, Ast´erisque No. 346 (2012), 111–170; MR3202557

Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.

In a joint work with H. Hida, we proved in the 90's that the anticyclotomic Katz p-adic L function associated to a p-adic CM type divides the characteristic power series of the Iwasawa module associated to this p-adic CM type. The goal of these two talks is to sketch this proof. Note that we couldn't treat the divisibility at the prime p of the Iwasawa algebra. This has been treated in subsequent works by H. Hida.

Given two nonincreasing n-tuples of real numbers l_n, m_n, the Horn problem asks for a description of all nonincreasing n-tuples of real numbers u_n such that there exist Hermitian matrices X_n, Y_n and Z_n respectively with these spectra such that X_n+Y_n=Z_n. There is also a randomized version of this problem where X_n and Y_n are sampled uniformly at random from orbits of Hermitian matrices arising from the conjugacy action by elements of the unitary group. One then asks for a description of the probability measure of the spectrum of the sum Z_n. Both the original Horn problem and its randomized version have solutions using the hives introduced by Knutson and Tao. In an asymptotic sense, as n tends to infinity, large deviations for the randomized Horn problem were given in joint work with Sheffield in terms of a notion of surface tension for hives. In this talk, we discuss upper and lower bounds on this surface tension function. We also obtain a closed-form expression for the total entropy of a surface tension minimizing continuum hive with boundary conditions arising from GUE eigenspectra. Finally, we give several empirical results for random hives and lozenge tilings arising from an application of the octahedron recurrence for large n and a numerical approximation of the surface tension function. This is a joint work with Aalok Gangopadhyay.

The online convex paging problem models a broad class of cost  functions for the classical paging problem. In particular, it naturally  captures fairness constraints: e.g., that no specific page (or groups of  pages) suffers an ``unfairly'' high number of evictions by considering
$\ell_p$ norms of  eviction vectors for $p>1$. The case of the $\ell_\infty$ norm has also been of special interest, and is called  min-max paging.

We give tight upper and lower bounds for the convex paging problem for a  broad class of convex functions.  Prior to our work, only fractional algorithms were known for this general setting. Moreover, our general result also improves on prior works for special cases of the problem.   For example, it implies that the randomized competitive ratio of the  min-max paging problem is $\Theta(\log k\log n)$; this improves both the  upper bound and the lower bound given in prior work by logarithmic factors. It also shows that the randomized and deterministic competitive
ratios for $\ell_p$-norm paging are $\Theta(p\log k)$ and $\Theta(pk)$  respectively.

This is joint work with Anupam Gupta and Debmalya Panigrahi.

The theory of probabilistically checkable proofs (PCPs) shows how to encode a proof for any theorem into a format where the theorem's correctness can be verified by making only a constant number of queries to the proof. The PCP Theorem [ALMSS] is a fundamental result in computer science with far-reaching consequences in hardness of approximation, cryptography, and cloud computing. A PCP has two important parameters: 1) the size of the encoding, and 2) soundness, which is the probability that the verifier accepts an incorrect proof, both of which we wish to minimize.

In 2005, Dinur gave a surprisingly elementary and purely combinatorial proof of the PCP theorem that relies only on tools such as graph expansion, while also giving the first construction of 2-query PCPs with quasi-linear size and constant soundness (close to 1). Our work improves upon Dinur's PCP and constructs 2-query, quasi-linear size PCPs with arbitrarily small constant soundness. As a direct consequence, assuming the exponential time hypothesis, we get that no approximation algorithm for 3-SAT can achieve an approximation ratio significantly better than 7/8 in time 2^{n/polylog n}.

In this talk, I will introduce PCPs and discuss the components that go into our proof. This talk is based on joint work with Dor Minzer and Nikhil Vyas, with an appendix by Zhiwei Yun.