Bigness of the cotangent bundle is a negativity property of the curvature which has important complex analytic consequences, such as on the Kobayashi hyperbolicity properties and the GGL-conjecture for surfaces. We present a birational criterion for a surface to have big cotangent bundle that takes in account the singularities present in the minimal model and describe how it improves upon other criterions. The criterion allows certain geographic regions of surfaces of general type to have big cotangent bundle, that other criterions can not reach. In this spirit, we produce the examples with the lowest slope $c_1^2/c_2$ having big cotangent bundle that are currently known.

Given a principal bundle over a compact Riemannian 4-manifold or special-holonomy manifold, it is natural to ask whether a uniform gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. As applications, we recover a theorem of Taubes on path-connectedness of instanton moduli spaces on the 4-sphere, and obtain a method to construct instantons on quaternion-Kähler manifolds with positive scalar curvature.

The talk is based on joint work in progress with Anuk Dayaprema (UW-Madison).

Famous results of N. Hitchin establish existence of an integrable structure of the Hitchin moduli spaces. The goal of this talk will be to discuss a more explicit approach known in the integrable models literature as separation of variables, how it can be applied to the quantisation of the Hitchin system, and how the result is related to the analytic Langlands correspondence studied by Etingof, Frenkel and Kazhdan.

Partly based on arXiv:1707.07873

I will talk about geometric compactifications of moduli spaces of K3 surfaces, similar in spirit to the Deligne-Mumford moduli spaces of stable curves. Constructions borrow ideas from the tropical and integral-affine geometry and mirror symmetry. The main result is that in many common situations there exists a geometric compactification which is toroidal, and many of these compactifications can be described explicitly using tropical spheres with 24 singular points. Much of this talk is based on the joint work with Philip Engel.

What does it mean for a category to be endowed with a compatible differentiable structure? In this talk, we will discuss the interplay of a categorical structure with that of a smooth manifold, and show how to describe such categories infinitesimally, similarly as to how we construct the Lie algebra of a Lie group. We will generalise the notion of rank from linear algebra to morphisms of Lie categories, and introduce the notion of an extension of a Lie category to a groupoid. Examples of Lie categories arising in differential geometry and in physics will be highlighted.