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: tba

Nero Budur (Leuven): tba
Abstract: tba

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: tba

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): Some remarks on the Bernstein functional equation of Thom-Sebastiani maps
Abstract: tba

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): Equivariant D-modules on Binary Cubic Forms
Abstract: tba

Thomas Reichelt (Heidelberg): Global mirror symmetry
Abstract: tba

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): On the irregular Hodge filtration
Abstract: tba

Mathias Schulze (Kaiserslautern): Duality of logarithmic differential forms
Abstract: tba

Yota Shamoto (Kyoto): Hodge-Tate conditions for Landau-Ginzburg models
Abstract: tba

Duco van Straten (Mainz): tba
Abstract: tba

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.