## Abstracts

Andrea D'Agnolo (Padova): A microlocal approach to the enhanced Fourier-Sato transform in dimension one
Abstract: Let M be a holonomic algebraic D-module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to M at irregular points. The stationary phase formula states that the non linear exponential factors of the Fourier transform of M are obtained by Legendre transform from the non linear exponential factors of M. We give a microlocal proof of this fact, by translating it in terms of enhanced perverse sheaves through the Riemann-Hilbert correspondence. (This is joint work with Masaki Kashiwara.)

Holger Brenner (Osnabrück): Asymptotic properties of differential operators on a singularity.
Abstract: For a local algebra $R$ over a field, we study the decomposition of the module of principal parts. A free summand of the $n$th module of principal parts is essentially the same as a differential operator $E$ of order $\leq n$ with the property that the differential equation $E(f) =1$ has a solution. The asymptotic behavior of the seize of the free part gives a measure for the singularity represented by $R$. We compute this invariant for invariant rings, monoid rings, determinantal rings and compare it with the $F$-signature, which is an invariant in positive characteristic defined by looking at the asymptotic decomposition of the Frobenius. This is joint work with Jack Jeffries and Luis Nuñez Betancourt.

Michael Dettweiler (Bayreuth): Convolution on elliptic curves
Abstract: The talk is about a joint work with Benjamin Collas. We study various aspects of elliptic curve convolutions from a Tannakian, motivic, and monodromic point of view.

Alicia Dickenstein (Buenos Aires): Algebraic A-hypergeometric Laurent series and residues
Abstract: A-hypergeometric systems of partial differential equations (introduced by Gelfand, Kapranov and Zelevinsky) have natural geometric solutions, with singularities on the associated (full) discriminant. We describe A-hypergeometric algebraic Laurent series associated with Cayley configurations of n lattice configurations in n space. We show that these algebraic series are generated by certain combinatorially defined sums of point residues, whose computation can be interpreted in terms of a toric degeneration. This is joint work with Eduardo Cattani and Federico N. Martinez.

Nero Budur (Leuven): Bernstein-Sato ideals and local systems
Abstract: We will talk about recent progress on a conjecture we proposed in 2012 relating Bernstein-Sato ideals with rank one local systems. This conjecture would be a generalization of the classical result of Malgrange and Kashiwara relating the roots of the Bernstein-Sato polynomial of a germ of a holomorphic function to the local monodromy eigenvalues.

Claus Hertling (Mannheim): Conjectures on spectral numbers for upper triangular matrices and isolated hypersurface singularities
Abstract: Certain upper triangular matrices S with integer entries, 1's on the diagonal, and eigenvalues of S^{-1}S^t of absolute value 1, arise from Seifert forms and distinguished bases of holomorphic functions with isolated singularities. Which matrices arise? Some idea how to distinguish these matrices builds on an idea of Cecotti and Vafa how to associate spectral numbers to such matrices. I will report on partial results for both ideas.

Anatoly Libgober (Chicago): Phases of N=2 theories
Abstract: Landau/Ginzburg correspondence is a relation between invariants of weighted homogenous singularities and invariants of Calabi-Yau hypersurfaces in weighted projective spaces. Invariants previously considered are the Gromov-Witten invariants, categories of matrix factorizations and derived categories, Arnold-Steenbrink spectrum and Hodge numbers. This talk discusses a generalization of LG/CY correspondence which in particular finds partners (hybrid models) for complete intersections in homogeneous spaces of reductive groups in which cases we obtain partners for certain Chern numbers associated with these complete intersections. This approach uses as geometric backgrounds, GIT quotients extending the class of symplectic quotients proposed by Witten as the background for the study of LG/CY correspondence (and which he viewed as phases of N=2 theories).

Viktor Levandovskyy (Aachen): Algorithms for D-modules and insights from them
Abstract: We present a thorough analysis of the reduction of the computation of global Bernstein-Sato polynomial to commutative calculations. This has been circulating in the folklore, but we give the precise description, which has not been known before. In particular, we give the lower a priori bound for the order of a Bernstein-Sato operator. Our approach extends to algebraic local Bernstein data as well. We also show that it is possible to compute a Bernstein-Sato operator of a minimal order; to find special operators from a subring or to conclude that no such exists. Some other recent findings will be presented as well. The results have been obtained jointly with Daniel Andres (Aachen), Christian Schilli (Aachen) and Jorge Martin Morales (Zaragoza).

Luis Narváez Macarro (Sevilla): Rings of differential operators as enveloping algebras of Hasse-Schmidt derivations in arbitrary characteristics
Abstract: Let us consider a commutative base ring $k$ and a commutative $k$-algebra $A$. When $k$ contains the rational numbers and $A$ is a polynomial or power series ring with coefficients in $k$, or more generally, when $A$ is a smooth $k$-algebra, the ring $D$ of differential operators of $A$ over $k$ not only is generated by $A$ and by the $k$-derivations of $A$, but also the following property holds: giving a (left) $D$-module structure on a given $A$-module $M$ is equivalent to giving an integrable connection on $M$. This well known and classical fact can be stated in a fancy form by saying that the ring $D$ is the enveloping algebra of the Lie-Rinehart algebra of $k$-derivations of $A$. The above result has no chance to hold when $k$ does not contain the rational numbers. In this talk I will recall what Hasse-Schmidt derivations are and how these object can be endowed with an algebraic structure, namely a (non-abelian) group structure and the action of substitution maps, in such a way that under convenable smoothness hypotheses, the ring of differential operators $D$ can be reconstructed as a kind of enveloping algebra of Hasse-Schmidt derivations.

Mihnea Popa (Northwestern): Hodge ideals
Abstract: I will present joint work with M. Mustata, in which we study a sequence of ideals arising naturally from M. Saito’s Hodge filtration on the localization along a hypersurface. The multiplier ideal of the hypersurface appears as the first step in this sequence, which as a whole provides a more refined measure of singularities. I will also discuss their natural generalization to Q-divisors, and give applications to the comparison between the Hodge filtration and the pole order filtration, adjunction, and the singularities of hypersurfaces in projective space and theta divisors on abelian varieties.

Claudiu Raicu (Notre Dame): Iterated local cohomology and Lyubeznik numbers for determinantal rings
Abstract: The space X of complex m x n matrices admits an action of the group GL=GL_m x GL_n via row and column operations. The invariant closed subsets are the closures of the orbits of fixed rank matrices, and they give rise to local cohomology functors that preserve the category of GL-equivariant D-modules on X. Taking advantage of the quiver description of this category, I will explain how to determine explicitly an arbitrary iteration of local cohomology functors applied to the structure sheaf of X, or more generally to any simple equivariant D-module. A special case of this calculation describes the Lyubeznik numbers of determinantal rings, answering a question of Hochster. Joint work with András Lörincz.

Thomas Reichelt (Heidelberg): Global mirror symmetry
Abstract: Conjecturally, global mirror symmetry connects the quantum cohomology of projective varieties which are birational. In this talk, I will focus on the simplest case of a (dis-)crepant blow-up and explain the construction of the corresponding global Landau-Ginzburg model.

Helge Ruddat (Mainz): An introduction to generalized theta functions
Abstract: Theta functions are canonical sections of a relatively ample line bundle near the MUM point, that is a maximal Calabi-Yau degeneration. The canonical base coordinates of such Calabi-Yau degenerations are Hodge theoretic flat coordinates. On the other hand, pencils in projective space (the global picture) provide an algebraic coordinate. It is related to the canonical coordinate by the notorious mirror map. The theta functions provide canonical embeddings in projective space, thus providing an algebraic coordinate intrinsically, linking local and global mirror symmetry (geometric vs differential equation point of view). We give an introduction to these theta functions.

Claude Sabbah (Palaiseau): Some properties and applications of Brieskorn lattices
Abstract: After reviewing the main properties of the Brieskorn lattice in the framework of tame regular functions on smooth affine complex varieties, we indicate a proof of a conjecture of Katzarkov-Kontsevich-Pantev in the toric case.

Mathias Schulze (Kaiserslautern): Duality of logarithmic differential forms
Abstract: The talk gives a survey on logarithmic differential forms and derivations along increasingly general spaces. The focus lies on relations with normal crossing properties, duality results and generalizations of the notion of a free divisor.

Yota Shamoto (Kyoto): Hodge-Tate conditions for Landau-Ginzburg models
Abstract: From the viewpoint of Homological Mirror Symmetry for Fano manifolds, Katzarkov-Kontsevich-Pantev proposed some conjectures on algebro-geometric properties on Landau-Ginzburg models. In this talk, I will explain a Hodge-theoretical sufficient condition for some versions of their conjectures. If time permits, I will also mention the relation to quantum D-modules of Fano manifolds.

Duco van Straten (Mainz): Motivic Differential Equations
Abstract: It is well-known that differential equations that arise as Picard-Fuchs equations from families of of varieties (say defined over the field of rational numbers) have very strong arithmetical properties. Conversely, it has been conjectured that all globally nilpotent differential operators are geometric. So one would expect to be able to attach motivic invariants like L-functions to such differential equations. We report on work in progress with P. Candelas and X. de la Ossa that realises this expectation for a certain class of operators. As an application one can identify conjecturally some points in the parameter space with special Hodge properties.

Bernd Sturmfels (Leipzig): Exponential Varieties
Abstract: Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Another special class, including Gaussian graphical models, are inverses of symmetric matrices satisfying linear constraints. We present a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. This is joint work with Mateusz Michalek, Caroline Uhler, and Piotr Zwiernik.

Jean-Baptiste Teyssier (Leuven): Moduli of Stokes torsors and singularities of differential equations
Abstract: The aim of this talk is to explain how the geometry of the Stokes phenomenon in any dimension sheds light on the interplay between the singularities of a linear system of differential equations and the singularities of its solutions.