C3

Convex geometry has long been influenced by the study of dualities and extremal inequalities, with origins in classical affine geometry and functional analysis. In this talk, Kasia Wyczesany will explore an abstract concept of duality, focusing on the classical idea of the polar set, which captures the duality of finite-dimensional normed spaces. This notion leads to fundamental questions about volume products, inspiring some of the most famous inequalities in the field. Whilst Mahler’s influential 1939 conjecture regarding the minimiser of the volume product will be mentioned, the emphasis will be on the Blaschke–Santaló inequality, which identifies the maximiser, along with its modern extensions. Main new results are joint work with S. Artstein-Avidan and S. Sadovsky, and S. Artstein-Avidan and M. Fradelizi. 

to follow

to follow

In 1962, Horn raised the following problem: Let A and B be n-by-n Hermitian matrices with respective eigenvalues a_1,...,a_n and b_1,...,b_n. What can we say about the possible eigenvalues c_1,...,c_n of A + B?

The deterministic perspective is that the set of possible values for c_1,...,c_n are described by a collection of inequalities known as the Horn inequalities.

Free probability offers the following alternative perspective on the problem: if (A_n) and (B_n) are independent sequences of n-by-n random matrices with empirical spectra converging to probability measures mu and nu respectively, then the random empirical spectrum of A_n + B_n converges to the free convolution of mu and nu.

But how are these two perspectives related?

In this talk Samuel Johnston will discuss approaches to free probability that bridge between the two perspectives. More broadly, Samuel will discuss how the fundamental operations of free probability (such as free convolution and free compression) arise out of statistical physics mechanics of corresponding finite representation theory objects (hives, Gelfand-Tsetlin patterns, characteristic polynomials, Horn inequalities, permutations etc).

This talk is based on joint work with Octavio Arizmendi (CIMAT, Mexico), Colin McSWiggen (Academia Sinica, Taiwan) and Joscha Prochno (Passau, Germany).

This talk by Tim Austin, at the University of Warwick, will be an introduction to "almost periodic entropy".  This quantity is defined for positive definite functions on a countable group, or more generally for positive functionals on a separable C*-algebra.  It is an analog of Lewis Bowen's "sofic entropy" from ergodic theory.  This analogy extends to many of its properties, but some important differences also emerge.  Tim will not assume any prior knowledge about sofic entropy.

After setting up the basic definition, Tim will focus on the special case of finitely generated free groups, about which the most is known.  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.

In this talk, Konstantin Recke, University of Oxford,  will report on some results pertaining to the interplay between geometric group theory, operator algebras and probability theory. Konstantin will introduce so-called invariant percolation models from probability theory and discuss their relation to geometric and analytic properties of groups such as amenability, the Haagerup property (a-T-menability), $L^p$-compression and Kazhdan's property (T). Based on joint work with Chiranjib Mukherjee (Münster).

The simplex of traces of a unital C*-algebra has long been regarded as a central invariant in the theory. Likewise, from the group-theoretic perspective, the simplex of traces of a discrete group (namely, the simplex of traces of its maximal C*-algebra) is a fundamental object in harmonic analysis, and the study of this simplex led to many applications in recent years.

Itamar Vigdorovich , UCSD, will discuss several results describing the simplex of traces in concrete and significant cases. These include Property (T) groups and especially higher rank lattices, for which the simplex of traces is as tame as possible. In contrast, for free products, the simplex is typically as wild as possible, yet still admits a canonical and universal structure—the Poulsen simplex. In ongoing work, an analogous result is obtained for the space of traces on the fundamental group of a closed surface of genus g≥2.

Itamar presents these results, outlines the main ideas behind the proofs, and gives an overview of the central concepts. The talk is based on joint works with Gao, Ioana, Levit, Orovitz, Slutsky, and Spaas.

In this talk, I will give an account of the measure of large values where |ζ(1/2 + it)| > exp(V), with t ∈ [T,2T] and V ∼ αloglogT. This is the range that influences the moments of the Riemann zeta function. I will present previous results on upper bounds by Arguin and Bailey, and new lower bounds in a soon to be completed paper, joint with Louis-Pierre Arguin, and explain why, with current machinery, the lower bound is essentially optimal. Time permitting, I will also discuss adaptations to other families of L-functions, such as the central values of primitive characters with a large common modulus.

The Deligne-Beilinson conjecture predicts that the special values of many L-functions are related to the ranks of certain Ext groups in the category of mixed Hodge structures. In this talk, we present Skinner’s constructions of certain extensions that are extracted from the cohomology of the modular curve using CM points and the Eisenstein series. Through an explicit analytic calculation, which is performed in the adelic setting using (g,K)-cohomology and Tate’s zeta integrals, we obtain a formula relating the non-triviality of these extensions to the well-known non-vanishing at s=1 of the L-functions associated to Hecke characters of imaginary quadratic fields. These constructions have natural analogs in the category of p-adic Galois representations which are useful for Euler systems.

For families of Calabi-Yau threefolds, we derive an explicit formula to count the number of points over $\mathbb{F}_{q}$ in terms of the periods of the holomorphic three-form, illustrated by the one-parameter mirror quintic and the 5-parameter Hulek-Verrill family. The formula holds for conifold singularities and naturally incorporates p-adic zeta values, the Yukawa coupling and modularity in the local zeta function. I will give a brief introduction on the physics motivation and how this framework links arithmetic, geometric and physics.