Moderately Discontinuous Algebraic Topology

Abstract: An algebraic or complex analytic subset in C^n has 2 natural metrics: the outer metric (restriction of the euclidean metric) and the inner metric (natural extension of the riemannian metric on the smooth part). These metrics considered up to bilipschitz mappings are analytic invariants, that is, they do not depend on the complex analytic embedding. Recently there is an intense activity in bilipschitz geometry of germs and degenerations, enriching and providing finer information on problems that were studied previously from the topological viewpoint (for multiplicity invariance of the germ or certain equisingularity notions). In the works [1] and [2] we develop a new metric algebraic topology, called the Moderately Discontinuous Homology and Homotopy, in the context of subanalytic germs in R^n (with a supplementary metric structure) and more generally of (degenerating) subanalytic families. This theory captures bilipschitz information, or in other words, quasi isometric invariants, and aims to codify, in an algebraic way, part of the bilipschitz geometry. A subanalytic germ is topologically a cone over its link and the moderately discontinuous theory captures the different speeds, with respect to the distance to the origin, in which the topology of the link collapses towards the origin. Similarly, in a degenerating subanalytic family, it captures the different speeds of collapsing with respect to the family parameter. The MD algebraic topology satisfies all the analogues of the usual theorems in Algebraic Topology: long exact sequences for the relative case, Mayer Vietoris and Seifert van Kampen for special coverings... In this talk, I will present the most important concepts in the theory and some results or applications that we got until the present. [1] (with J. Fernandez de Bobadilla, S. Heinze, E. Sampaio) Moderately discontinuous homology. To appear in Comm. Pure App. Math.. Available in arXiv: 1910.12552 [2] (with J. Fernández de Bobadilla, S. Heinze) Moderately discontinuous homotopy. Submitted. Available in ArXiv:2007.01538

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

The seminar takes place on Tuesdays and Thursdays via Zoom. Zoom passwords are given via mailing list on Fridays. To join the mailing list go to the website.

If you use a calendar system, you can see the individual seminars at bit.ly/zag-seminar-calendar

Times vary to accommodate speakers time zones but times will be announced in GMT time.