Organisers: Prof. M. Dettweiler with M. Hien (Ausburg, DE), Thomas Reichelt (Heidelberg, DE) and C. Sevenheck (TU Chemnitz, DE) -- SISYPH.
13.10 - 17.10.14
Organisers: Prof. M. Dettweiler with L. Funar (UJF Grenoble, FR), P. Lochak (UPMC Paris 6, FR) and I. Marin (Amiens, FR).
02.02. - 08.02.2014
Organisers: Prof. M. Dettweiler with Jochen Heinloth (Essen, GE) and Zhiwei Yun (Stanford, US).
14.09. - 16.09.2011
Organisers: Prof. M. Dettweiler.
Every Thursday: Program of the Colloquium of the Mathematic Institute.
The oberseminar of the group is a weekly joint Oberseminar on Arithmetische Geometrie with Prof. Dr. Stoll. It takes place every Friday from 12:30 to 14:00 in room S82-NW II and is followed by a Tea in our seminar room (3.2.02.748).
The goal of this Seminar is to give the keys to current and dynamic research topics of Arithmetic-Geometry. It is intended as a research seminar to acquire a precise and clear view of the field, and welcomes students eager of developing their understanding of mathematical theories in a research situation.
The joint Arithmetische Geometrie Oberseminar now has its own webpage. See here for current programme, schedule and updates.
Archive previous Semesters [Org. Prof. Stoll]
- Dr. Alexander Ivanov - Paris 6 UPMC University, France - February 2 - 3
- Prof. Stefan Wewers - Universität Ulm, Germany - January 15 - 20
- Msc. Ran Azouri - Tel Aviv University, Israel - April 1 - June 23
- Prof. Pierre Dèbes - Université Lille, France - June 19 - 30
- Dr. Geoffroy Horel - Max Planck Institute Bonn, Germany - November 16 - 18
- Prof. U. Kühn - Universität Hamburg, Germany - January 22 - 25
- Dr. Francois Legrand - Tel Aviv University & the Open University of Israel - June 28 - July 4
- Nils Matthes - Universität Hamburg, Germany - July 14 - 15
The SISYPH project consists of a French and a German partnership on the following topics:
1. Mirror symmetry as an effective tool for the computation of Gromov-Witten invariants of various kinds of smooth algebraic varieties or orbifolds,
2. Irregular singularities of linear differential systems in any dimension, either from the point of view of holonomic D- modules or from that of isomonodromy deformations,
3. Hodge theoretic aspects of such differential systems.
The project consists on the development of interplays between algebraic geometry, non commutative Hodge theory, singularity theory and D-modules, symplectic geometry, with in the background some motivations and conjectures formulated by physicists.
A central object of interest will be the generalized hypergeometric systems of linear differential equations (GKZ systems) as models for the quantum D-module of toric manifolds or orbifolds. These GKZ systems also provide a large class of examples of holonomic D-modules with irregular singularities, where conjectures and preliminary results can be tested.
The understanding of the geometry of different types of moduli spaces like those for isolated hypersurface singularities, for curves, or more generally for stable mappings (entering in the very definition of Gromov-Witten invariants), and for meromorphic connections on vector bundles, is one of the most important motivations of the whole project. Although the first ones are known to be essential for mirror symmetry, a basic question will be to make understand the notion of mirror symmetry for the moduli spaces of irregular singular connections on Riemann surfaces.
The Stokes phenomenon, which is a fundamental property of irregular singularities of differential equations, is a basic object to be understood in the context of either Gromov-Witten theory or Landau-Ginzburg models and their extensions to singularity theory. Its relationship with Hodge-theoretic properties (in particular their non-commutative aspects) will allow the analysis of moduli spaces of singularities.
Differential equations are fundamental objects within mathematics and especially within algebraic geometry. Often, fundamental aspects of a geometric problem can be described in terms of a system of differential equations. Let us mention here the principle of monodromy (analytic continuation of solutions) and the notion of a variation of periods, describing the integration of differential forms (i.e. the Hodge theory) on a family of varieties. These concepts both play an important role in the field of Mirror Symmetry of Calabi-Yau varieties with many applications to physics and enumerative mathematics. It is the aim of this proposal to develop and implement two computer algebra packages which make the above mentioned concepts accessible for explicit computation.
The first package shall deal with computational aspects of the Hodge theory of families of varieties with a special attention to the important case of Calabi-Yau varieties and related convolutions of variations of Hodge structures.
The second shall deal with the computation of the monodromy of integrable differential equations (integrable connections) on quasiprojective varieties. Thereafter, these packages shall be applied to various aspects, e.g., determination of new differential operators of Calabi-Yau type, computation of Instanton numbers or the computation of uniformizing differential equations.
The usual Hadamard product f * g of two power series f and g is well known to be related to the convolution on the multiplicative group of the complex numbers. If f and g are motivic power series in the sense that they describe the variation of periods of cohomology groups of two family of varieties, then also f * g is motivic by twisting the original families over the square of the multiplicative group. In this way one obtains families of Hadamard product motives which are important for many applications in mathematics and physics. The following motives occur as Hadamard products and illustrate the importance of this class: Motives of generalized hypergeometric differential equations, playing a role in the two-body problem in quantum mechanics, and motives of families of Calabi-Yau-varieties occurring in the mirror symmetry conjecture of string theory. The latter motives also play an important role in the recent proof of the Sato-Tate conjecture. The aim of this proposal is the development and the implementation of algorithms for the computation of the monodromy, Hodge Type and Galois representations for Hadamard product motives. The resulting programs should be applied among others on modularity questions of (rigid) Calabi-Yau varieties and the study of polylogarithms and Feynman integrals.