I will talk about homological enumerative invariants for vector bundles on algebraic curves. These invariants were defined by Joyce, and encode rich information about the moduli space of semistable vector bundles, such as its volume and intersection numbers, which were computed by Witten, Jeffrey and Kirwan in previous work. I will define a notion of regularization of divergent infinite sums, and I will express the invariants explicitly as such a divergent sum in a vertex algebra.

We show that the category of vector bundles over the vertices of a locally finite $\tilde{A}_{n-1}$ building $\Delta$, $Vec(\Delta)$, has the structure of a $Tilt(SL_{2k+1})$ module category. This module category is the $q$-analogue of the $Tilt(SL_{2k+1})$ action on vector bundles over the $sl_n$ weight lattice.  Our construction of the $Tilt(SL_{2k+1})$ action on $Vec(\Delta)$ extends to $Vec(\Delta)^{G}$, its equivariantization, which gives us a class of non-standard $Tilt(SL_{2k+1})$ module categories. When $G$ acts simply transitively, this recovers the fiber functors of Jones.

Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, “Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?” We’ll discuss work that shows the answer is yes when the soliton is asymptotically conical. No symmetry or Kahler assumption is required, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity. We’ll also discuss applications to the (non-)uniqueness of weak Ricci flows through singularities.

Labourie proved that every Hitchin representation into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the Hitchin component admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmüller space.

In this talk, we will define Hitchin representations, Higgs bundles, and minimal surfaces, and give the background for the Labourie conjecture. We will then explain that the conjecture fails for n at least 4, and point to some future questions and conjectures.

We study the problem of determining when a compact group can be realized as the group of isometries of a norm on a finite dimensional real vector space.  This problem turns out to be difficult to solve in full generality, but we manage to understand the connected groups that arise as connected components of isometry groups. The classification we obtain is related to transitive actions on spheres (Borel, Montgomery-Samelson) on the one hand and to prehomogeneous spaces (Vinberg, Sato-Kimura) on the other. (joint work with Martin Liebeck, Assaf Naor and Aluna Rizzoli)

Mathematics has always been part of the fabric of culture. References to mathematics in literature go back at least as far as Aristophanes, and encompass everyone from Dostoevsky to Oscar Wilde. In this talk I’ll explore some of the ways that literature has engaged with mathematical ideas, from the 17^th and 18^th century obsession with the cycloid (the “Helen of Geometry”) to the 19^th century love of the fourth dimension.

For positive integers $k$ and $n$ let $\sigma_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erd\H{o}s and Kac conjectured that, for every $k$, the number $\alpha_k = \sum_{n\geq 1} \frac{\sigma_k(n)}{n!}$ is irrational. This is known conditionally for all $k$ assuming difficult conjectures like the Hardy-Littlewood prime $k$-tuples conjecture. Before our work it was known unconditionally that $\alpha_k$ is irrational if $k\leq 3$. We discuss some of the ideas in our recent proof that $\alpha_4$ is irrational. The proof involves sieve methods and exponential sum estimates.

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas.

I will describe a procedure, known as holomorphic twist, to isolate protected quantities in supersymmetric quantum field theories. The resulting theories are holomorphic, interacting and have infinite dimensional symmetries, analogous to the holomorphic half of a 2D CFT. I will explain how to study quantum corrections to these symmetries and other  higher operations.
As a surprise, we find a novel UV manifestation of
confinement, dubbed "holomorphic confinement," in the example of pure
SU(N) super Yang-Mills.

After reviewing different aspects of thermalization and chaos in holographic quantum systems, I will argue that universal aspects can be captured using an effective field theory framework that shares similarities with hydrodynamics. Focusing on the quantum butterfly effect, I will explain how to develop a simple effective theory of the 'scramblon' from path integral considerations. I will also discuss applications of this formalism to shockwave scattering in black hole backgrounds in AdS/CFT.