We review Kontsevich's homological mirror symmetry conjecture and introduce the main players and different variants of the conjecture. This involves a discussion of the Fukaya category and about what it takes to relate this (a priori $\mathbb{Z}/2$-graded) $A_\infty$-category to the derived category of coherent sheaves of a mirror complex manifold. We take a look at Seidel's proof for the quartic surface and genus two curve and Sheridan's proof for higher dimensional hypersurfaces. We also discuss the Fukaya-Seidel category for a Lefschetz fibration, Orlov's category of singularities and Abouzaid's homological mirror symmetry proof for Fano manifolds. We then develop an approach to prove homological mirror symmetry for general Calabi-Yau manifolds based on T-duality and skeleta. The T-duality here involves another conjecture known as the Strominger-Yau-Zaslow conjecture. The new part of this approach consists of ongoing work by Matessi, Mak, Zharkov and myself that provides a symplectic manifold with Lagrangian torus fibration over any given integral affine manifold with discriminant in codimension two that is well-behaved in a suitable sense.

$D$-modules are modules over the sheaf of differential operators and over the years they found many applications in singularity theory. In these lectures I will discuss certain invariants of singularities (of hypersurfaces or, more generally, locally complete intersections) that can can be defined and studied using techniques from $D$-module theory (especially Saito's theory of Hodge modules), as well as certain classes of singularities characterized by these invariants, that refine the classical notions of rational and Du Bois singularities.

In this talk we will see that some facets of the theory of D-modules over polynomial rings can be extended to the case of rings of invariants of finite groups. A blend of different techniques allow us to define a notion of holonomicity in this setting, we can develop a theory of Bernstein-Sato polynomials, V-filtrations, Hodge ideals and we can study the de Rham cohomology of holonomic modules.

Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties.

This is joint work with Pierrick Bousseau (arXiv:2302.02068 and arXiv:arXiv:2308.07270).

We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration for which we give two equivalent constructions. The first one extends the classical Kato-Nakayama log-space, and the second one, more geometric, is based on a the real oriented version of the deformation to the normal cone.

In particular, we recover the topological Milnor fibration by quotienting the motivic Milnor fibration with suitable powers of $(0,+\infty)$. Conversely, we also show that the stratified topological Milnor fibration determines the classical motivic Milnor fibre. (Joint work with Goulwen Fichou and Adam Parusiński).

Let $(X,0)$ be a germ of a complex surface embedded in $\mathbb{C}^n$ having an isolated singularity at the origin and let $f:(X,0) \longrightarrow (\mathbb{C},0)$ be a germ of non-constant holomorphic function. The aim of the presentation is to compute the integral of the Gauss curvature on the Milnor fibers $f^{-1}(t) \cap \mathrm{B}_{\epsilon}$ of the function $f$ as $t$ and $\epsilon$ tends to $0$. More precisely, we will decompose the surface X into regions and describe those where the curvature concentrates asymptotically and those where the integral of the curvature tends towards $0$ via an infinite family of analytic invariants of metric nature associated to the function $f$ called "inner rates".

The celebrated Bogomolov--Tian--Todorov theorem states that the functor of infinitesimal smooth deformations of a smooth and proper Calabi--Yau variety $X$ is unobstructed, meaning that any infinitesimal deformation can be lifted along any thickening. The same is true when we deform not only a Calabi--Yau variety, but a pair $(X,\mathcal{L})$ of a Calabi--Yau variety together with a line bundle. In logarithmic geometry, we replace the smooth Calabi--Yau variety with a log smooth space over a log point $S_0$. By previous work, we know already that the log smooth deformation functor of a proper log Calabi--Yau is unobstructed; in this talk, I will report on work in progress showing that log smooth deformations of a pair of a log smooth log Calabi--Yau $f_0: X_0 \rightarrow S_0$ together with a line bundle $\mathcal{L}_0$ are unobstructed as well.

Let $f: U \to \mathbb{C}^*$ be an algebraic map from a smooth complex connected algebraic variety $U$ to the punctured complex line $\mathbb{C}^*$. Using f to pull back the exponential map $\mathbb{C} \to \mathbb{C}^*$, one obtains an infinite cyclic cover $U^f$ of the variety $U$, together with a $\mathbb{Z}$-action coming from adding $2 \pi i$ in $\mathbb{C}$. The homology groups of this infinite cyclic cover, with their $\mathbb{Z}$-actions, are the family of Alexander modules associated to $f$.

In previous work jointly with Eva Elduque, Christian Geske, Laurențiu Maxim and Botong Wang, we constructed a mixed Hodge structure on the torsion part of these Alexander modules. In this talk, we will talk about work in progress aimed at generalizing this theory to abelian covering spaces of algebraic varieties which arise in an algebraic way, i.e. from maps $f: U \to G$, where $G$ is a semiabelian variety. This is joint work with Eva Elduque.

In connection with Mustata’s lectures, I will discuss an application of D-modules due to Gennady Lyubeznik. Local cohomology is one of the central objects of local algebra, but it is very hard to work with as these modules are usually not finitely generated. However, Lyubeznik observed that local cohomology has a natural D-module structure, and it can be, for example over a polynomial ring, finitely generated as a D-module. In such cases, one can work with local cohomology as a finitely generated module over a non-commutative ring D. A novel result of the talk, taken from a joint paper with Yairon Cid-Ruiz, is a generic freeness theorem for local cohomology.

The trace operator plays a crucial role in the theory of partial differential equations, as it helps to find weak formulations of the problems. This theory, which is very satisfying on domains that have Lipschitz regular boundary, is much more challenging when singularities arise.

Our aim is therefore to develop all the material necessary to find weak formulations of basic problems of PDE on a subanalytic or semi-algebraic open subset of $\mathbb{R}^n$, such as for instance elliptic differential equations with Dirichlet boundary conditions.

We will start by giving some Poincar\'e type inequalities for the functions of Sobolev spaces of bounded subanalytic open subsets of $\mathbb{R}^n$. Then we will investigate the trace operator on the Sobolev space $W^{1,p}(M)$, where $M$ is a bounded subanalytic submanifold of $\mathbb{R}^n$, in the case where $p$ is large. This manifold $M$ may of course admit singularities in its closure which are not metrically conical.