Termine

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.

Abstract: In Appendix C of his "Anyons" paper, Kitaev introduced the notion of a "generalized Chern number" for a 2-dimensional system by diving the system in three ordered parts and measuring a signed rotational flux. This construction has since been used by several authors to measure topological non-triviality of a physical system. In recent work with Guo Chuan Thiang, we observe that the recipe provided by Kitaev can be interpreted in coarse geometry as the pairing of a K-theory class with a coarse cohomology class. A corresponding index theorem then provides a proof that the set of values of this "Kitaev pairing" is always quantized, as already argued by Kitaev. In our work, we generalize Kitaev?s definition and the corresponding quantization result to arbitrary dimensions.

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 start by demonstrating that the interplay between advection and diffusion in the incompressible porous media equation with diffusion -- a dissipative version of the classical active scalar problem -- can lead to enhanced dissipation. Subsequently, we derive a scaling limit that perfectly balances these two physical mechanisms. The high degeneracy of the limiting equation prevents us from proving existence of weak solutions in the distributional form. To address this challenge, we use the gradient flow structure of the equation to define weak solutions within a robust "geometric" framework and show that the solution space is compact. The talk is based on the joint work with Felix Otto and Bian Wu.

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.