In this seminar, I will explore how the sum over topologies in pure AdS_3​ quantum gravity furnishes a consistent statistical interpretation of the boundary CFT_2. By formulating a statistical version of the conformal bootstrap, which combines crossing symmetry with typicality at high energies, we will discover a large set of non-handlebody topologies in the bulk (of which I will give some examples) that are needed for consistency of the boundary description. Interestingly, this set contains only on-shell (i.e. hyperbolic) 3-manifolds, but not all of them. This suggests that the full sum over all on-shell saddles in 3d gravity may be a highly non-minimal solution of the statistical bootstrap. Based on the recent work 2601.07906 with Belin, Collier, Eberhardt and Liska.

An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially discrete series representations of a connected reductive p-adic group is stable under twist by automorphisms of the complex numbers, and hence this class is defined over $\bar{\mathbb{Q}_\ell}$. Recent work of Solleveld, building on work of Kazhdan-Varshavsky-Solleveld, says that the same is true of the class of standard representations. Stefan Dawydiak will give a geometric proof of this result for the principal block, and use this to deduce a local proof of Clozel's theorem for the general linear group. Time permitting, Stefan will also give geometric formulas for certain Harish-Chandra Schwartz functions that help illustrate these results.

Professor Ana Djurdjevac will talk about; 'The Dean–Kawasaki Equation: Theory, Numerics, and Applications'

The Dean–Kawasaki equation provides a stochastic partial differential equation description of interacting particle systems at the level of empirical densities and has attracted considerable interest in statistical physics, stochastic analysis, and applied modeling. In this work, we study analytical and numerical aspects of the Dean–Kawasaki equation, with a particular focus on well-posedness, structure preservation, and possible discretization strategies. In addition, we extend the framework to the Dean–Kawasaki equation posed on smooth hypersurfaces. We discuss applications of the Dean–Kawasaki framework to particle-based models arising in biological systems and modeling social dynamics.

Speaker Estefania Loayza Romero will talk about:  A Riemannian Approach for PDE-Constrained Shape Optimization Using Outer Metrics

In PDE-constrained shape optimisation, shapes are traditionally viewed as elements of a Riemannian manifold, specifically as embeddings of the unit circle into the plane, modulo reparameterizations. The standard approach employs the Steklov-Poincaré metric to compute gradients for Riemannian optimisation methods. A significant limitation of current methods is the absence of explicit expressions for the geodesic equations associated with this metric. Consequently, algorithms have relied on retractions (often equivalent to the perturbation of identity method in shape optimisation) rather than true geodesic paths. Previous research suggests that incorporating geodesic equations, or better approximations thereof, can substantially enhance algorithmic performance. This talk presents numerical evidence demonstrating that using outer metrics, defined on the space of diffeomorphisms with known geodesic expressions, improves Riemannian gradient-based optimisation by significantly reducing the number of required iterations and preserving mesh quality throughout the optimisation process.

This talk is hosted at RAL.