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. 

I will construct the differential geometric, gauge-theoretic, and duality covariant model of classical four-dimensional Abelian gauge theory on an orientable four-manifold of arbitrary topology. I will do so by implementing the Dirac-Schwinger-Zwanziger (DSZ) integrality condition in classical Abelian gauge theories with general duality structure and interpreting the associated sheaf cohomology groups geometrically. As a result, I will obtain that four-dimensional Abelian gauge theories are theories of connections on Siegel bundles, namely principal bundles whose structure group is the generically non-abelian disconnected group of automorphisms of an integral affine symplectic torus. This differential-geometric model includes the electric and magnetic gauge potentials on an equal footing and describes the equations of motion through a first-order polarized self-duality condition for the curvature of a connection. This condition is reminiscent of the theory of four-dimensional Euclidean instantons, even though we consider a two-derivative theory in Lorentzian signature. Finally, I will elaborate on various applications of this differential-geometric model, including a mathematically rigorous description of electromagnetic duality in Abelian gauge theory and the reduction of the polarized self-duality condition to a Riemannian three-manifold, which gives as a result a new type of Bogomolny equation.

We describe an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined "virtual Morse-Bott index" at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when it is a singular point in the moduli space. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.

Pluripotential theory studies plurisubharmonic functions and complex Monge-Ampère equations on complex manifolds, and has played a key role in recent progress on Kähler-Einstein and constant scalar curvature Kähler metrics. This theory admits a non-Archimedean analogue over Berkovich spaces, that can be used to study K-stability. The purpose of this talk is to provide an introduction to this circle of ideas, and to discuss more specifically recent joint work with Mattias Jonsson studying Green's functions in this context.