Tomoyuki Abe

Title: A remark on trace formalism

Abstract: The trace formalism for cohomology theory is fundamentally important. This is because in many cases, we through algebro-geometric data into the cohomological formalism via trace formalism. One of the most general trace formalism was developed by Grothendieck (or Deligne) in SGA 4. I would like to discuss a very small improvement of it, with which I hope to have better perspective on the trace formalism.

Andrea D'Agnolo

Title: Enhanced nearby and vanishing cycles in dimension one

Abstract: Enhanced ind-sheaves provide a setting for the target category of the irregular Riemann-Hilbert correspondence. In the talk we will illustrate some features of this correspondence by discussing a topological construction of moderate nearby and vanishing cycles in dimension one. This is from joint work with Masaki Kashiwara.

Hélène Esnault

Title: An obstruction to lifting to characteristic 0

Abstract: We introduce a new obstruction to lifting smooth proper varieties in characteristic p>0 to characteristic 0. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of étale fundamental groups. Joint with Vasudevan Srinivas and Jakob Stix.

Sam Gunningham

Title: On quantum holomorphic Lagrangian intersections with applications to skein theory

Abstract: Given a pair of holomorphic Lagrangian submanifolds of a holomorphic symplectic manifold (plus some additional orientation data), we may consider the following two objects. On the one hand, Bussi (based on work with Brav, Dupont, Joyce, and Szendroi) assign to such data a perverse sheaf (over Z) supported on the Lagrangian intersection. On the other hand, the theory of deformation quantization (as developed by d’Agnolo, Kashiwara, Polesello, Schapira, and others) assigns to such data a dg-category and a pair of objects, whose internal RHom is known to be a perverse sheaf (over C((h))). I will describe some work in progress with Pavel Safronov in which we compare these objects, and time permitting, will explain the applications to skein theory.

Clemens Koppensteiner

Title: Logarithmic D-modules

Abstract: I will report on joint work with Mattia Talpo on D-modules on logarithmic spaces. This includes and generalizes D-modules which are allowed to have simple poles along a simple normal crossings divisor. The theory of logarithmic D-modules differs from that of usual D-modules in subtle ways. We will discuss some of the new features, especially with regards to holonomic modules. Finally, we discuss how to generalize ideas by Kato-Nakayama and Ogus on logarithmic connections in order to formulate a logarithmic version of the Riemann--Hilbert correspondence for logarithmic D-modules.

Thomas Krämer

Title: Big monodromy theorems on abelian varieties

Abstract: This talk is motivated by new applications of D-modules that arise from the work of Lawrence and Sawin on the Shafarevich conjecture for hypersurfaces on abelian varieties over number fields. A key ingredient of their approach is a big monodromy theorem for the cohomology of hypersurfaces twisted by rank one local systems, which amounts to a statement about the Tannaka groups of holonomic D-modules on abelian varieties. After a brief survey of known results about these groups, I will discuss recent progress towards big monodromy theorems for subvarieties of higher codimension; this is ongoing work with Ariyan Javanpeykar, Christian Lehn and Marco Maculan.

Tatsuki Kuwagaki

Title: Exact WKB analysis and sheaf quantization

Abstract: Exact WKB analysis is a specific way to solve differential equations parametrized by \hbar.  In this talk, I will explain that how one can put exact WKB analysis into a Riemann--Hilbert correspondence-style statement. In the course of the explanation, we will see how the theory makes contact with Fukaya/Floer theory, cluster algebra, and deformation quantization.

Claudia Polini

Title: Simple D-module components of local cohomology modules

Abstract: A long standing problem in algebraic geometry and commutative algebra is to determine whether every irreducible curve in projective three-space is a set-theoretic complete intersection. One way to approach this problem is via the study of local cohomology modules. As modules over the ring, local cohomology modules are huge (neither finitely generated nor Artinian), hence intractable. However, as modules over the Weil algebra D they can be filtered by simple objects and become manageable. Hence an important task is to understand the D-module structure of local cohomology modules. In this talk we describe their composition series. We obtain this as a corollary of a characterization of the dimension of certain de Rham cohomology groups of a holonomic D-module as the number of specific D-linear maps associated with the holonomic D-module. This is joint work with Robin Hartshorne. If time allows I will report on a generalization of this work by Nicholas Switala and Wenliang Zhang in which they prove a duality theorem for the cohomology groups of graded D-modules, under the assumption that these are finite-dimensional.

Luca Prelli

Title: Sheaves on T-topologies

Abstract: Let T be a suitable family of open subsets of a topological space X stable under unions and intersections. Starting from T we construct a (Grothendieck) topology on X and we consider the associated category of sheaves. This gives a unifying description of various constructions in different fields of mathematics.