Topology Seminar 16 February, 14:15 - 15:15, Room 734

Clover May (NTNU): Structure theorem for RO(G)-graded equivariant cohomology

Abstract: For spaces with an action by a group G, one can compute an equivariant analogue of singular cohomology called RO(G)-graded Bredon cohomology. Computations in this setting are often challenging and not well understood, even for G = C_2, the cyclic group of order 2. In this talk, I will start with an introduction to RO(G)-graded cohomology and describe a structure theorem for RO(C_2)-graded cohomology with (the equivariant analogue of) Z/2-coefficients. The structure theorem describes the building blocks for the cohomology of C_2-spaces and makes computations significantly easier. It shows the cohomology of a finite C_2 space depends only on the cohomology of two types of spheres, representation spheres and antipodal spheres. I will give some applications and talk about work in progress generalizing the structure theorem to other settings.

Topology Seminar 2 March, 14:15 - 15:15

Mikala Ørsnes Jansen (University of Copenhagen): The reductive Borel-Serre compactification and unstable algebraic K-theory

Abstract: The reductive Borel-Serre compactification, introduced by Zucker in 1982, is a stratified space which is well-suited for the study of L2-cohomology of arithmetic groups and which has come to play a central role in the theory of compactifications of locally symmetric spaces. We determine its stratified homotopy type (exit path ∞-category) to be a 1-category defined purely in terms of parabolic subgroups, their unipotent radicals and a conjugation action. This category makes sense in a much more general setting; in fact, we can associate such a category to any ring R and any integer n and interpret it as a "compactification" of GLn(R). We propose these categories as models for unstable algebraic K-theory and investigate them in detail for finite fields and commutative local rings with infinite residue field. This is joint work with Dustin Clausen.

Topology Seminar 23 March, Room 734, 14:15 - 15:15

Niall Taggart (Utrecht University): Homological localizations of orthogonal functor calculus

Abstract: Orthogonal calculus is a version of functor calculus that sits at the interface between geometry and homotopy theory; the calculus takes as input functors defined on Euclidean spaces and outputs a Taylor tower of functors reminiscent of a Taylor series of functions from differential calculus. The interplay between the geometric nature of the functors and the homotopical constructions produces a calculus in which computations are incredibly complex. These complexities ultimately result in orthogonal calculus being an under-explored variant of functor calculus.

On the other hand, homological localizations are ubiquitous in homotopy theory. They are employed to split `integral' information into `prime' pieces, typically simplifying both computation and theory.

In this talk, I will describe a 'local' version of orthogonal calculus for homological localizations, and survey several immediate applications.

Topology Seminar 30 March, 14:15 - 15:15

William Hornslien (NTNU): Understanding the homotopy groups of motivic spheres

Abstract: A fundamental problem in algebraic topology is the study of homotopy group of spheres. Motivic homotopy theory is the homotopy theory of smooth schemes over some field. It turns out there are motivic spheres, and we can study the homotopy groups of those. Due to the algebraic nature of schemes, we can describe all maps between certain spheres algebraically which in turn make them easier to study.

In this talk we will use algebraic methods to explicitly describe a group structure on some homotopy groups of motivic spheres. This is joint work with Viktor Barth, Gereon Quick, and Glen Wilson.

Topology Seminar 27 April, 14:15 - 15:15

Peter Kropholler (University of Southampton): Applications of condensed mathematics to the study of finiteness conditions in Galois cohomology

Abstract: We’ll show how to use the Clausen-Scholze condensed mathematics to prove new theorems in classical Galois cohomology. As a first application, we prove that every solvable pro-p group of type FP_\infty has finite virtual cohomological dimension. The proof becomes possible because condensed mathematics creates an abelian category closed under limits and colimits that has enough projectives. Having enough projectives is a Holy Grail for anyone interested in using Farrell-Tate cohomology. We’ll see that arguments first designed with discrete groups in mind can now be generalised to topological groups and for certain hierarchically decomposable groups that include all solvable groups there is a beautiful vanishing theorem in Farrell-Tate cohomology. This suggests that condensed mathematics will have a transformative impact on the way we understand Galois cohomology. I would like to thank Lukas Brantner, Ged Corob Cook, Rob Kropholler, Dustin Clausen and Peter Scholze for valuable discussions and for teaching me many important insights.

Topology Seminar 11 May, 14:15 - 15:15, Room 734

Louis Martini (NTNU): Introduction to condensed mathematics

Abstract: Doing algebra with objects that carry a topology can be challenging as the categories of interest are usually not nice enough in order to apply the usual homological or homotopical machinery. The condensed/pyknotic formalism developed independently by Clausen-Scholze and Barwick-Haine suggests a fix to this problem by replacing topological spaces with condensed sets. In this talk I will give an introduction to this novel framework. I will set up the basic theory of condensed sets and explain in what way they can serve as a convenient category of topological spaces. More generally, I will discuss how condensed objects in an arbitrary (infinity-)category can be used to capture topological features and in what way such objects arise in practice.

Topology Seminar 8 June, 14:15 - 15:15, Room 734

Morten Brun (University of Bergen): Homology basis: a tool for computation of persistent homology

Abstract: I will present some elementary considerations on homological algebra leading to a strategy for use of the homology long exact sequence to compute homology of chain complexes. I will go on to explain how this strategy can be implemented on a computer and explain why is can be more efficient than classical matrix reduction techniques for computation of (persistent) homology. I will end the talk by comparing the performance of the proposed strategy to publically available software for computation of homology.

Topology Seminar Thursday 16 June, 14:15 - 15:15, Room 656

Knut Haus (NTNU): Geometric Hodge filtered complex cobordism

Abstract: I will present a geometric description of Hopkins—Quick “Hodge filtered complex cobordism” (HFCC). The Hopkins—Quick theory is defined in terms of hom-sets in the homotopy category of spectra of simplicial presheaves on the site of complex manifolds. I will outline the construction of an isomorphism from the Hopkins—Quick theory, to the geometric model for HFCC. Using the geometric model we construct a pushforward along proper holomorphic maps. We thereby obtain a cycle map. There is a canonical map from HFCC to Deligne cohomology, which we describe in our geometric framework. Precomposing with our cycle map we recover the more classical cycle map to Deligne-cohomology. In particular we obtain a new Abel—Jacobi invariant, refining that of Griffith.