9.30 Opening
9.45-10.45 Emmanuel Breuillard: Uniform spectral gaps for linear groups and applications
If G is a countable group acting non-amenably on a countable space X, one may ask for explicit Kazhdan sets and constants for this action. Typically these constants depend on the Kazhdan set. For a class of actions that we call algebraic, which includes all transitive actions of a countable linear group with algebraic isotropy group, we prove that a Kazhdan constant exists that is uniform over all sets and all such actions. We derive a number of consequences, e.g. a uniform spectral gap for toral automorphisms, a uniform exponential decay for the hitting probability of random walks on algebraic varieties, a non-abelian Littlewood-Offord theorem. Along the way we introduce a new ping-pong lemma that produces paradoxical decompositions with controlled number of pieces and we revisit the set of results going along the nickname of ``escape from subvarieties''. Finally, a key ingredient is the theory of dophantine heights and the Height hap theorem. This is joint work with Oren Becker.
11.00-12.00 Camille Horbez: Rigidity for graph product von Neumann algebras
Graph products of von Neumann algebras provide a construction that encompasses both tensor products and free products. I will present rigidity theorems for graph product von Neumann algebras associated to finite graphs and families of tracial von Neumann algebras. Applications include W* classification theorems for right-angled Artin groups, and for graph products of ICC groups, as well as the computation of the fundamental group and outer automorphism group of certain graph products of II_1 factors. I will draw parallels with results regarding the automorphisms of graph products of groups, and their classification up to measure equivalence. This is a joint work with Adrian Ioana.
14.00-15.00 Joe Thomas: Spectral gap for random hyperbolic surfaces
The first non-zero eigenvalue, or spectral gap, of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about a surface. In this talk, I will discuss the typical size of the spectral gap for a random surface with large genus sampled with respect to the Weil-Petersson probability measure. In particular, I will explain joint work with Will Hide and Davide Macera where we obtain a spectral gap with a polynomial error rate. Our result uses a fusion of the polynomial method used in recent breakthroughs on the strong convergence of group representations with the trace formula for hyperbolic surfaces.
15.15-16.15 Ben Hayes: A random matrix approach to the Peterson-Thom conjecture
The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by Peterson-Thom's results on L^{2}-Betti numbers and was stated as an open problem in several problem lists, and mini-course talks since 2010. I will explain how one can reduce this conjecture to a natural statement about strong convergence of random matrices. This statement has now been verified by several groups of authors: Belinschi+Capitaine, Bordenave+Collins, Chen+ Garza-Vargas+van Handel, de la Salle + Magee, and Parruad. Time permitting, I will explain some consequences for the structure of von Neumann algebras associated to free groups that we can show now that we know this result is true, as well as potential future directions.
16.30-17.30 [ZOOM] Ramon van Handel: The polynomial method
The polynomial method is a new approach for establishing optimal spectral gaps that has led to a series of new developments in the past year, including the resolution of a number of important problems on spectral gaps of hyperbolic surfaces and on strong convergence of group representations. My aim in this talk is to describe the basic ingredients of this method in a general setting.
18.00 Walk to Piran
9.45-10.45 Emmanuel Kowalski: Jacobian graphs
We will introduce examples of graphs related to generalized jacobians of algebraic curves over finite fields, and explain some of their properties, including (in some cases) precise control on the spectrum beyond the spectral gap. The proofs of these properties depend on equidistribution theorems for discrete Fourier transforms over finite fields, which will also be presented. Finally, some open questions and potential applications will be discussed. (Joint work with A. Forey, J. Fresán and Y. Wigderson)
11.00-12.00 Noema Nicolussi: The Arakelov–Bergman Laplacian on degenerating Riemann surfaces and graphs
The Laplace–Beltrami operator for the (Arakelov-)Bergman metric is an interesting operator in spectral theory on Riemann surfaces. Its importance stems from the associated Green function, which is relevant to arithmetic geometry. Since the early 90s, there has been great interest in understanding the behavior of the Laplacian, Green function and related objects when the underlying Riemann surface degenerates to a singular Riemann surface. In this talk, we discuss a recent approach which explains the degeneration behavior of some of these objects using analogous objects on graphs. Based on joint work with Omid Amini (Orsay).
14.00-15.00 Tianyi Zheng: Uniformly Lipschitz actions on L^1 and coarse embeddings
In this talk we discuss connections between notions in coarse geometry and uniformly Lipschitz affine maps on L^1. Relying on a result of Vergara on almost invariant conditionally negative definite (CND) kernels, we show that a finitely generated group acts properly by uniformly Lipschitz affine maps on a subspace of L^1 if and only if it coarsely embeds into L^1. When the group has Yu’s Property A, the space can be taken to be the Lipschitz free space over the group equipped with a changed metric, which in this case is isomorphic to l^1. We will discuss some questions in the converse direction. Joint work with Gartland and Vergara.
15.15-16.15 Martin Kassabov
16.30-17.30 Theo McKenzie: The spectral edge of regular graphs
The spectral theory of regular graphs plays a central role in theoretical computer science, statistical physics, and many areas of mathematics. Nevertheless, it has remained challenging to determine which spectra are feasible for arbitrary sizes and degrees. In this talk, I will present new techniques showing that graphs sampled from generic distributions exhibit desirable spectral properties. The first result establishes that a random regular graph is, with positive probability, an optimal spectral expander (a Ramanujan graph). The second result shows that we can construct graphs whose nontrivial eigenvalues realize any prescribed spectrum. To do this, we give a rigorous analysis of the Green's function of the adjacency operator under simple graph operations.
9.45-10.45 Rostislav Grigorchuk: A Hunt for Spectral Gaps
11.00-12.00 Mikael de la Salle: Superexpanders and nonlinear spectral gap from lattices in simple Lie groups
Expander graphs are graphs that have some kind of extremal behaviour for maps into Hilbert spaces. The notion of nonlinear spectral gap and superexpanders, introduced by Mendel and Naor, is a version for maps into other metric spaces. It is particularily meaningful for Banach spaces, and fundamental examples have been provided by Lafforgue and Liao coming from algebraic groups over non-archimedean local fields. I will explain that higher-rank simple Lie groups such as SL_n(R) for n\geq 3 have strong forms of fixed point properties for actions on Banach spaces, and that this implies that finite quotients of lattices in them are superexpanders. This is a joint work with Tim de Laat from Kiel.
14.00-15.00 Ewan Cassidy: Spectral gaps for random Schreier graphs
I will discuss a generalization of Friedman's theorem, focusing on the spectral gap of random regular Schreier graphs depicting the action of the symmetric group S_n on k_n tuples of elements in {1, . . . , n}. I will also discuss elements of the proof, which relies on the new approach to strong convergence of Chen, Garza Vargas, Tropp and van Handel based on `differentiation with respect to 1/n', whereby one can exploit properties of the expressions for the expected irreducible characters of random permutations obtained via word maps. A key ingredient in the proof is a new asymptotic bound on such an expression, given in terms of the dimension of the corresponding representation.
15.15-16.15 Young researchers
Adrian Beker: Limiting spectral laws for sparse random circulant matrices
Lawford Hatcher: Hot spots in infinite cones
Elena Maini: On the Gap Conjecture for residually soluble groups
Luca Sabatini: Groups that provide expander graphs
Santiago Radi Severo: Random subgroups of branch groups
16.30-? Problem session
9.45-10.45 Peter Keevash: The diameter of somewhat dense Cayley graphs on A_n
The expansion of Cayley graphs on groups may be studied algebraically via spectral gaps or geometrically via product theorems. A fundamental open problem on the geometry of Cayley graphs is Babai's Diameter Conjecture, which states that the diameter of any connected Cayley graph on a nonabelian finite simple group G is at most polylogarithmic in |G|. A natural extremal variant, also open in general, asks for the maximum possible diameter given the density of the generating set. In this talk, we consider the alternating permutation groups A_n, for which Helfgott and Seress showed that the diameter of any Cayley graph is at most quasipolynomial in n. We will present an essentially optimal upper bound on the diameter when the density is at least 2^{-O(n)}. Our proof combines combinatorial, analytic and algebraic arguments, with the key ingredient being a new sharp hypercontractive inequality in S_n. Our methods have several other applications to extremal problems for permutations, including an analogue of Polynomial Freiman-Ruzsa for somewhat dense sets. This is joint work with Noam Lifshitz.
11.00-12.00 Miklos Abert
19.00 Conference dinner
9.45-10.45 Harald Helfgott: Diameter bounds for linear algebraic and permutation groups
Let G be a finite simple group and A a subset of G generating G. We would like to bound the diameter of the Cayley graph \Gamma(G,A) independently of A. The best bounds in the literature are of the type O\left((\log |G|)^{(\log n)^{3+\epsilon}}\right) (Helfgott-Seress, 2011--14), for G=\Alt(n), and O((\log |G|)^{O(n^4)}) (Bajpai-Dona-Helfgott), for classical simple groups of rank n. As one can see, for classical simple groups such as \PSL_{n+1}(\mathbb{F}), we are still exponentially behind: we would like to have O(\log n)^C in the exponent, not n^4.
The talk will have two halves: (a) we will go over the current, improved version of Bajpai-Dona-Helfgott, (b) we will discuss some parallels involving buildings and the theory of the field with one element.
11.00-12.00 Vadim Kaimanovich
14.00-15.00 Tanya Brailovskaya: Matrix concentration and spectral gaps
Proving that a particular type of random graph is, with high probability, a near-optimal expander is, in general, a challenging and model-specific problem. In this talk, I will present a set of generic tools from random matrix theory, known as matrix concentration inequalities, that will allow us to immediately establish a Friedman’s theorem for sufficiently rapidly growing degree graphs. Additionally, our methodology allows us to show that certain random Cayley graphs of quasirandom groups are near-optimal expanders, complementing a classical result of Alon and Roichman. This talk is based on joint work with Ramon van Handel.
15.15-16.15 Tim Austin: Entropy and large deviations for random unitary representations
This talk will be an introduction to "almost periodic entropy". This notion of entropy applies to positive definite functions on a countable group, and more generally to positive functionals on separable C*-algebras. It is an analog of Lewis Bowen's notion of "sofic entropy" from ergodic theory. This analogy extends to many of its properties, but some important differences also emerge.\\ After setting up the basic definition, I will focus on the special case of finitely generated free groups, about which the most is known. (No knowledge of sofic entropy will be assumed.) For free groups, results include a large deviations principle in a fairly strong topology for uniformly random representations. This, in turn, offers a new proof of the Collins—Male theorem on strong convergence of independent tuples of random unitary matrices, and a large deviations principle for operator norms to accompany that theorem.
Closing