• Lecturer: Christian Sevenheck
• Date: Friday 11:00 - 12:30
• Room: MPI MiS, G3 10
• Language: English
• Target audience: MSc students, PhD students, Postdocs
• Keywords: Differential operators, Weyl algebra, vector bundles with connections, functorial properties, filtered D-modules, Hodge modules
• Prerequisites: Basic knowledge of algebraic and complex geometry: algebraic varieties, some commutative algebra (basics of dimension theory), complex manifolds, vector bundles, basics of homological algebra

Abstract

This lecture aims at giving a leisure introduction to the field of algebraic analysis, that is, the algebraic study of linear partial differential equations with polynomial coefficients. We will start with basics on differential operators and the Weyl algebra as well as on vector bundles with connections. Next we will discuss the notion of holonomicity and how this gives finiteness restrictions on the solutions of a D-module. Depending on time and audience, we will go into some details of direct and inverse images, give the statement of the Riemann-Hilbert correspondence, explain some facts about filtered D-modules as well as on the V-filtration and Bernstein-Sato polynomials. Finally, we may give a small outline on the theory of mixed Hodge modules.