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Events
A Hypergeometric Day

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

05.05.23

A Hypergeometric Day

On this seminar day, the speakers will present different viewpoints on hypergeometric systems. Recently, such systems have gained interest in particle physics, due to their relation to Feynman integrals. The talks will feature an introduction to D-modules and GKZ systems, a functorial description of these systems, Euler and Feynman integrals and twisted (co-)homology.

Program

09:30 - 09:45	Welcome
09:45 - 10:45	Anna-Laura Sattelberger (KTH Royal Institute of Technology) Introduction to D-modules and GKZ systems I give an introduction into the algebraic theory of linear differential equations. Algebraically, systems of linear PDEs with polynomial coefficients are encoded as modules over the Weyl algebra, denoted D. I am going to introduce important concepts such as the holonomic rank, the characteristic variety, and the singular locus, and am going to explain what meaning they have for the holomorphic solutions to the system of PDEs encoded by the D-module. One particular example will be GKZ systems, also called "A-hypergeometric systems” - the main player of this seminar day.
10:45 - 11:00	Coffee Break
11:00 - 12:00	Avi Steiner (TU Chemnitz) Fourier transformation and resonance It is often fruitful to study GKZ systems via what is called their Fourier(--Laplace) transform. I will introduce this transform and explain how to view the transformed GKZ system as an object living on a toric variety. Separately, I will discuss various notions of resonance, which track whether or not the GKZ system in question satisfies various "nice" properties.
12:00 - 13:30	Lunch Break
13:30 - 14:30	Claudia Fevola (Max Planck Institute for Mathematics in the Sciences) Generalized Euler integrals A basis of the space of local solutions to a GKZ system can be represented by integrals of a particular form. Such integrals have gained recent interest because of their important role in particle physics and statistics. I will introduce these integrals and show that they satisfy the GKZ differential equations for a particular matrix A. I will describe the hypersurface on which this construction of a basis fails, called the principal A-determinant. This is the singular locus of the GKZ system.
14:30 - 15:00	Coffee Break
15:00 - 16:00	Interactive Discussion
16:00 - 17:00	Saiei-Jaeyeong Matsubara-Heo (Kumamoto University) Twisted cohomology for GKZ systems Generalized Euler integrals naturally give rise to a cohomology group, called twisted cohomology. Its dimension is the same as the Euler characteristics of a particular very affine variety. The D-module structure on the twisted cohomology is a Gauss-Manin connection, which is isomorphic to a GKZ system. The isomorphism tells us how to compute twisted cohomology in terms of the Weyl algebra.

Participants

Daniele Agostini

Rida Ait El Manssour

Zaïneb Bel-Afia

Barbara Betti

Henry Dakin

Elena Derunova

Nicholas Early

Claudia Fevola

Leitenberger Frank

Luca Giovenzana

Franco Giovenzana

Paul Görlach

Efe Güleroğlu

Claudio Iuliano

Emeryck Marie

Saiei-Jaeyeong Matsubara-Heo

Marta Panizzut

Dmitrii Pavlov

Silke Prohl

Kemal Rose

Anna-Laura Sattelberger

Christian Sevenheck

Elima Shehu

Peter Spacek

Avi Steiner

Bertrand Teguia Tabuguia

Simon Telen

Maximilian Wiesmann

Scientific Organizers

Simon Telen

Max Planck Institute for Mathematics in the Sciences

Christian Sevenheck

Technische Universität Chemnitz

Administrative Contact

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail