Exact categories were introduced by Quillen in 1970s as part of his seminal work on algebraic K-theory. Exact categories provide a suitable enlargement of the class of abelian categories (for example, an extension-closed subcategory of an abelian category inherits the structure of an exact category) in which one "can do homological algebra". Recently, motivated also by questions in algebraic K-theory, Barwick introduced the class of exact infinity-categories, relying on the newly-developed theory of infinity-categories developed by Joyal, Lurie and others. This new class of mathematical objects includes not only the exact categories in the sense of Quillen but also the stable inftinty-categories in the sense of Lurie (the latter are to be regarded as refinements of triangulated categories in the sense of Verdier). The purpose of this lecture series is to motivate the theory of exact infinity-categories and sketch some of its applications. Familiarity with the theory of infinity-categories is not expected.[-]

Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some of the properties of this category, I explain how to define the motive of certain algebraic stacks. I will then focus on defining and studying the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers. If there is time, I will give a conjectural formula for this motive, and explain how this follows from a conjecture on the intersection theory of certain Quot schemes. This is joint work with Simon Pepin Lehalleur.[-]

On the occasion of the 1998 ICM in Berlin, Boris Dubrovin (1950-2019) conjectured an intriguing connection between the enumerative geometry of a Fano variety $X$ with algebraic and geometric properties of exceptional collections in the derived category $D^{b}(X)$. The aim of Dubrovin's conjecture is twofold.
In its qualitative formulation, the conjecture asserts the equivalence of the semisimplicity condition of the quantum cohomology $Q H(X)$ and the existence of full exceptional collections in $D^{b}(X)$.
In its quantitative formulation, the conjecture prescribes explicit formulas for local invariants of $Q H(X)-$ the so-called "monodromy data" - in terms of characteristic classes of exceptional collections.
The central object for the study of these conjectural relations is a family of linear ODEs labeled by points of $Q H(X)$, called the "quantum differential equation" of $X$ ( $q D E$, for short).
The $q D E$ is a rich invariant of $X$. First, it encapsulates information on the Gromov-Witten theory of $X$. Second, it also defines local moduli invariants for the Frobenius manifold structure on $Q H(X)$. Moreover, the asymptotics and monodromy of its solutions conjecturally rule the topology and complex geometry of $X$. The study of $q D E$ s represents a challenging active area in both contemporary geometry and mathematical physics: it is continuously inspiring the introduction of new mathematical tools, ranging from algebraic geometry, the realm of integrable systems, the analysis of ODEs, to the theory of integral transforms and special functions.
In the first talk, the speaker will give a gentle introduction to the isomonodromic approach to Frobenius manifolds and quantum cohomology. In addition, a historical overview of Dubrovin's conjecture (from its origin to its recent refinements) will be presented.
In the second talk, after surveying known positive results on Dubrovin's conjecture, the speaker will discuss several further research directions including:
- analytical refinements of the theory of isomonodromic deformations to coalescing irregular singularity
- results evoking an equivariant analog of Dubrovin's conjecture - integral representations of solutions for the $q D E \mathrm{~s}$.
These talks will be based on several works of the speaker, partially joint with B. Dubrovin, D. Guzzetti, and A. Varchenko.[-]

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].[-]

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].[-]

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].[-]

There is a very general story, due to Joyce and Kontsevich-Soibelman, which associates to a CY3 (three-dimensional Calabi-Yau) triangulated category equipped with a stability condition some rational numbers called Donaldson-Thomas (DT) invariants. The point I want to emphasise is that the wall-crossing formula, which describes how these numbers change as the stability condition is varied, takes the form of an iso-Stokes condition for a family of connections on the punctured disc, where the structure group is the infinite-dimensional group of symplectic automorphisms of an algebraic torus. I will not assume any knowledge of stability conditions, DT invariants etc.[-]