TBA

The so called Drinfeld conjecture states that the complement to very stable bundles has pure codimension one in the moduli space of vector bundles. In this talk I will explain a constructive proof in rank three, and discuss if/how it generalises to wobbly fixed points of the nilpotent cone as defined by Hausel and Hitchin. This is joint work with Pauly (Nice).

The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different curve graphs and other geometric complexes.   

In this talk I will review the “holomorphic modular bootstrap,” i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of PSL(2, Zn), we describe a method to classify allowed central charges and weights (c, hi) for theories with any number of characters d. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of (c, hi). In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed d ≤ 5.

Donaldson-Thomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will explain the role of d-critical locus structure in the definition of motivic DT invariant, following the definition by Bussi-Joyce-Meinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by Ricolfi-Savvas, but techniques used here are different. 

Dolbeault hypertoric manifolds are hyperkahler integrable systems generalizing the Ooguri-Vafa space. They approximate the Hitchin fibration near a totally degenerate nodal spectral curve. On the other hand, Betti hypertoric varieties are smooth affine varieties parametrizing microlocal sheaves on the same nodal spectral curve. I will review joint work with Zsuzsanna Dansco and Vivek Shende (arXiv:1910.00979) which constructs a diffeomorphism between the Dolbeault and Betti hypertorics, and proves that it intertwines the perverse and weight filtrations on their cohomologies. I will describe our main tool : deletion-contraction sequences arising from either smoothing a node of the spectral curve or separating its branches. I will also discuss some more recent developments and open questions.

Locally analytic representations of $p$-adic Lie groups are of interest in several branches of number theory, for example in the theory of automorphic forms and in the $p$-adic local Langlands program. To better understand these representations, Ardakov-Wadsley introduced a sheaf of infinite order differential operators $\overparen{\mathcal{D}}$ on smooth rigid analytic spaces, which resulted in several Beilinson-Bernstein style localisation theorems. In this talk, we discuss the current research on analogues of a Riemann-Hilbert correspondence for $\overparen{\mathcal{D}}$-modules, and what this has to do with complete convex bornological vector spaces.

In the search for a complete description of quantum mechanical and
gravitational phenomena, we are inevitably led to consider observables on
boundaries at infinity. This is the common mantra that there are no local
observables in quantum gravity and gives rise to the tantalising possibility
of a purely boundary--or holographic--description of physics in the
interior. The AdS/CFT correspondence provides an important working example
of these ideas, where the boundary description of quantum gravity in anti-de
Sitter (AdS) space is an ordinary quantum mechanical system-- in particular,
a Lorentzian Conformal Field Theory (CFT)--where the rules of the game are
well understood. It would be desirable to have a similar level of
understanding for the universe we actually live in. In this talk I will
explain some recent efforts that aim to understand the rules of the game for
observables on the future boundary of de Sitter (dS) space. Unlike in AdS,
the boundaries of dS space are purely spatial with no standard notion of
locality and time. This obscures how the boundary observables capture a
consistent picture of unitary time evolution in the interior of dS space. I

will explain how, despite this difference, the structural similarities
between dS and AdS spaces allow to forge relations between boundary
correlators in these two space-times. These can be used to import
techniques, results and understanding from AdS to dS.

High-dimensional approximation of Hamilton-Jacobi-Bellman PDEs – architectures, algorithms and applications

Hamilton-Jacobi Partial Differential Equations (HJ PDEs) are a central object in optimal control and differential games, enabling the computation of robust controls in feedback form. High-dimensional HJ PDEs naturally arise in the feedback synthesis for high-dimensional control systems, and their numerical solution must be sought outside the framework provided by standard grid-based discretizations. In this talk, I will discuss the construction novel computational methods for approximating high-dimensional HJ PDEs, based on tensor decompositions, polynomial approximation, and deep neural networks.