The Dehn function of a finitely presented group provides a quantitative measure for the difficulty of detecting if a word in its generators represents the trivial element of the group. By work of Gersten, Holt and Riley the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$. However, we are still far from determining the precise Dehn functions of all nilpotent groups. In this talk, I will explain recent results that allow us to determine the Dehn functions of large classes of nilpotent groups arising as central products. As a consequence, for every $k>2$, we obtain many pairs of finitely presented $k$-nilpotent groups with bilipschitz asymptotic cones, but with different Dehn functions. This shows that Dehn functions can distinguish between nilpotent groups with the same asymptotic cone, making them interesting in the context of the conjectural quasi-isometry classification of nilpotent groups.  This talk is based on joint works with García-Mejía, Pallier and Tessera.

I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of S-duality translates to conjectural symmetries between these contributions.  One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a certain quiver variety, the instanton moduli space of torsion-free framed sheaves on $\mathbb{P}^2$. Using a recent identity of Kuhn-Leigh-Tanaka, we deduce constraints on Vafa-Witten invariants conjectured by Göttsche-Kool-Laarakker. One consequence is a formula for the contribution of the vertical component to refined Vafa-Witten invariants in rank 2.

Finite quotient singularities have a long history in mathematics, intertwining algebraic geometry, hyperkähler geometry, representation theory, and integrable systems.  I will highlight the correspondences at play here and how they culminate in Nakajima quiver varieties, which continue to attract interest in geometric representation theory and physics.  I will motivate some recent work of G. Bellamy, A. Craw, T. Schedler, H. Weiss, and myself in which we show that, remarkably, all of the resolutions of a particular finite quotient singularity are realized by a certain Nakajima quiver variety, namely that of the 5-pointed star-shaped quiver.  I will place this work in the wider context of the search for McKay-type correspondences for finite subgroups of $\mathrm{SL}(n,\mathbb{C})$ on the one hand, and of the construction of finite-dimensional-quotient approximations to meromorphic Hitchin systems and their integrable systems on the other hand.  The Hitchin system perspective draws upon my prior joint works with each of J. Fisher and L. Schaposnik, respectively. Time permitting, I will speculate upon the symplectic duality of Higgs and Coulomb branches in this setting.

Given a 4-dimensional Einstein orbifold that cannot be desingularized by smooth Einstein metrics, we investigate the existence of an ancient solution to the Ricci flow coming out of such a singular space. In this talk, we will focus on singularities modeled on a cone over $\mathbb{R}P^3$ that are desingularized by gluing Eguchi-Hanson metrics to get a first approximation of the flow. We show that a parabolic version of the corresponding obstructed gluing problem has a  smooth solution: the bubbles are shown to grow exponentially in time, a phenomenon that is intimately connected to the instability of such orbifolds. Joint work with Tristan Ozuch.

In this talk, I will introduce the notion of $n$-morphisms between two $A_\infty$-algebras. These higher morphisms are such that 0-morphisms correspond to standard $A_\infty$-morphisms and 1-morphisms correspond to $A_\infty$-homotopies. Their combinatorics are encoded by new families of polytopes,  which I call the $n$-multiplihedra and which generalize the standard multiplihedra.
Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of $A_\infty$-algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.

The SYZ conjecture is a geometric way of understanding mirror symmetry via the existence of dual special Lagrangian fibrations on mirror Calabi-Yau manifolds. Motivated by this conjecture, it is expected that $G_2$ and $Spin(7)$-manifolds admit calibrated fibrations as well. I will explain how to construct examples of a weaker type of fibration on compact $Spin(7)$-manifolds obtained via gluing, and give a hint as to why the stronger fibrations are still elusive. The key ingredient is the stability of the weak fibration property under deformation of the ambient $Spin(7)$-structure.

I will discuss a joint work with Olivier Biquard about degenerating conic Kähler-Einstein metrics by letting the cone angle go to zero. In the case where one is given a smooth anticanonical divisor $D$ in a Fano manifold $X$, I will explain how the complete Ricci flat Tian-Yau metric on $X \smallsetminus D$ appears as rescaled limit of such conic KE metrics.