Oberseminare und sonstige Vorträge

Given a six functor formalism, one can parametrise the ways to twist the f_!-functors a bit without changing f^*. Using the same principles, one obtains general constructions of monodromic and gerbe-twisted sheaves. The usual construction of six functor formalisms via factorisation into classes I and P by Liu-Zheng assumes that morphism in the intersection of I and P are truncated. In joint work in progress with Will Fisher, we show that without this assumption, one can still construct a twisted six functor formalism.

A Riemannian metric is said to be Einstein if it has constant Ricci curvature. In dimensions 2 or 3, this is actually equivalent to requiring the metric to have constant sectional curvature. However, in dimensions 4 and higher, the Einstein condition becomes significantly weaker than constant sectional curvature, and this has rather dramatic consequences. In particular, it turns out that there are high-dimensional smooth closed manifolds that admit pairs of Einstein metrics with Ricci curvatures of opposite signs. After explaining how one constructs such examples, I will then discuss some recent results that explore the coexistence of Einstein metrics with zero and positive Ricci curvatures.

We establish a version of Poincaré duality for a class of (topological) groups called profinite groups, which provides isomorphisms from the cohomology groups to certain homology groups of a profinite group. The main goal of the talk is to outline the proof techniques for this result, namely condensed mathematics and derived categories. The former is a novel theory promising to act as an alternative fundament of geometry while the latter constitutes an approach of doing homological algebra. The talk will conclude by a rough sketch of the main argument of the proof.

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Abstract: We study the centralizer of a parabolic subalgebra in the Hecke algebra associated with an arbitrary (possibly infinite) Coxeter group. While the center and cocenter have been extensively studied in the finite and affine cases, much less is known in the indefinite setting. We describe a basis for the centralizer, generalizing known results about the center. Our approach combines algebraic techniques with geometric tools from the Davis complex, a CAT(0)-space associated to the Coxeter group. As part of the construction, we classify finite partial conjugacy classes in infinite Coxeter groups and define a variant of the class polynomial adapted to the centralizer.