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D-M2: D-Modules: Bridging Theory and Applications
Session Topics: D-Modules: Bridging Theory and Applications
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Convolution of D-modules and applications TU Chemnitz, Germany
On any commutative group variety the category of holonomic D-modules comes with a natural convolution product given by pushforward under the sum map. In the talk I will survey some recent applications of the resulting Tannakian description to the study of monodromy groups in algebraic geometry.
On ramified formal series solutions of GKZ-hypergeometric systems 1: University of Seville, Spain; 2: Tohoku University, Japan $A$-hypergeometric systems, also called GKZ systems, are systems of linear partial differential equations in several complex variables that generalize classical hypergeometric differential equations in one variable. They were introduced by Gel'fand, Kapranov and Zelevinsky as the $D$-module counterpart of a toric variety and they are associated with a full rank matrix $A = (a_1 \cdots a_n)$, $ a_i \in \mathbb{Z}^{d}$, $d<n$, and a parameter vector $\beta \in \mathbb{C}^d$. Many fundamental properties of these systems are known and can be described in terms of the combinatorics of the columns of the matrix $A$, especially when $\beta$ is generic. We study the space of ramified formal series solutions of an $A$-hypergeometric system along a coordinate hyperplane $Y=\{x_n=0\}$, which is closely related to the irregularity of the system along $Y$. Under some assumptions, we show that this space coincides with a certain space of formal Nilsson solutions of the GKZ system, which can be described in combinatorial terms. In this talk, we will focus on examples to illustrate our results. This is joint work in progress with Francisco-Jesús Castro-Jiménez and Saiei-Jaeyeong Matsubara-Heo. The first two authors were partially supported by PID2024-156912NB-I00 and PID2020-117843GB-I00, financed by MICIU/AEI/10.13039/501100011033 and FEDER, UE. The third author was financially supported by Grants-in-Aid for Scientific Research (KAKENHI) grant number 26K06838. | ||
