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Daily Overview |
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D-M3: D-Modules: Bridging Theory and Applications
Session Topics: D-Modules: Bridging Theory and Applications
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Logarithmic Gauss-Manin connections and nearby cycle functors TU Chemnitz, Germany Given a divisor on a complex manifold, we show that the monodromy automorphism of the associated logarithmic Gauss-Manin connection is given by taking residues of relative logarithmic forms along said divisor, extending a result of Steenbrink in the case of simple normal crossing divisors. When the divisor satisfies the twisted logarithmic comparison theorem, we show that this is in turn equal to the monodromy automorphism on the nearby cycles functor along the divisor. D-Modules from Symmetry: From Hypergeometric Systems to Homogeneous Varieties Otto von Guericke University Magdeburg, Germany Symmetry often turns complicated differential equations into structured and computable objects. In the language of algebraic analysis, this principle gives rise to a rich class of equivariant D-modules associated with group actions. Important examples include GKZ hypergeometric systems and tautological systems attached to homogeneous varieties such as Grassmannians and more general flag varieties. In this talk, I will present recent results showing that many of these symmetry-driven D-modules possess a deeper geometric meaning: they admit a natural mixed Hodge module interpretation and therefore encode topological information about the underlying spaces. This perspective not only clarifies the structure of their solution spaces but also suggests new tools for their study. I will explain how Lie algebroid methods provide a natural framework for understanding D-modules arising from group actions, revealing connections between symmetry, geometry, topology, and differential equations. In particular, they offer new tools for studying the duality theory of tautological systems and their Hodge-theoretic properties. Finally, I will discuss ongoing work on toric degenerations, which aims to connect highly symmetric geometric situations with more combinatorial models. This talk is based on joint works with Thomas Reichelt, Christian Sevenheck, Avi Steiner and Uli Walther. On the Hodge index theorem for singular varieties and characteristic classes University of Cantabria, Spain
The classical Hodge index theorem identifies the signature of a compact complex manifold with the value at $y=1$ of its Hirzebruch $\chi_y$-genus. This result has been extended to singular varieties. Moreover, Brasselet, Schürmann, and Yokura conjectured a characteristic class version of this theorem. In this talk, we will discuss recent progress on this conjecture. Based on joint work with Fernández de Bobadilla and Saito.
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