GKZ system with non-resonant parameters enjoys an integral representation, which in turn gives rise to a relation to intersection theory of twisted (co)homology groups. We show how the secondary structure (regular triangulations, secondary fan) is encoded in the intersection theory and discuss some consequences. This talk will include joint works with Yoshiaki Goto and Nobuki Takayama.
The GKZ discriminant locus is in general rather complicated, however in some cases it is a hyperplane arrangement. Then the perverse sheaf corresponding to the GKZ hypergeometric system has a concrete combinatorial description due to Kapranov and Schechtman. We will look at those GKZ perverse sheaves, and describe their categorification, i.e. perverse schobers. This is joint work with Michel Van den Bergh.
We present the current understanding of the variation with respect to parameters of the rank and solution space of an A-hypergeometric system. This will include joint work with my collaborators Barrera, Fernández-Fernández, Forsgård, and Matusevich.
Since almost 100 years of research, quantum field theories (QFTs) have provided an incredible agreement between theoretical predictions and experimental data, giving us deep insights into nature at its smallest scales. On the other hand, this subject with its tremendous complicated structures challenges both physicists and mathematicians. A problem that occurs in almost every QFT is the understanding and evaluation of so-called Feynman integrals. It turns out that these Feynman integrals are A-hypergeometric, as even Gelfand, Kapranov and Zelevinsky noted in 1989. However, this connection was not pursued further in the 1990s and has recently been rediscovered.
In the talk I will motivate why physicists are interested in Feynman integrals and give a proper definition of them. In addition, I will outline some conclusions for this special class of integrals, which can be derived from the point of view of A-hypergeometric functions.
I shall talk about a generalization of the GKZ systems called tautological systems, and explain several of the applications to the study of period integrals with motivation coming from the B-side of mirror symmetry.
GKZ hypergeometric systems were introduced by Gelfand, Kapranov and Zelevinsky as a generalization of Gauss hypergeometric differential equation. It can be shown that for certain parameters the GKZ-systems carry the structure of an irregular mixed Hodge module, a category recently defined by Claude Sabbah. We will discuss the Hodge and weight filtration of these D-modules.