Barcelona Topology Workshop 2018

The category of G-spectra, for any compact Lie group G is very interesting, but at the same time very complicated. A big part of the interesting information comes from the internally encoded group action while one of the main complications comes from working over the integers. The first step on our understanding is to simplify this category by working over rationals. This removes the complexity coming from ordinary stable homotopy theory, while leaving much of the information about the group G. In this talk I will present recent work with David Barnes and John Greenlees on an algebraic model for a toral part of rational G-spectra for any compact Lie group G. This is a way of understanding rational G-spectra with geometric isotropy in the maximal torus T of G purely in terms of algebraic data.

Since the works of Segal and Rezk simplicial spaces have been used to encode homotopically coherent associative structures. Dyckerhoff--Kapranov and Galvez-Carrillo--Kock--Tonks independently introduced the notion of a 2-Segal space, that is, a simplicial space satisfying a 2-dimensional analogue of the Segal conditions. Both sets of authors introduced these spaces as a unifying framework for understanding Hall algebras and related constructions. Their key insights were that all of these algebras arise by 'linearising' the so-called universal Hall algebra of a 2-Segal space: a multivalued, coherently associative product on its space of 1-simplices. A classic result due to Green asserts that the Hall algebra of an abelian category can be extended to a bialgebra. In this talk I will show how Green's theorem is in fact a shadow of the universal Hall bialgebra of a double 2-Segal space. The latter is a bisimplicial space which is 2-Segal in each variable, examples of which are given by variations on Waldhausen's S-construction.

Given a manifold M, one can study the configuration space of n points on the manifold: the subspace of M^n in which two points cannot be in the same position. Despite their apparent simplicity such configuration spaces are remarkably complicated; even the homology of these spaces is reasonably unknown, let alone their (rational) homotopy type. This classical problem in algebraic topology has much impact in more "modern" mathematics, namely in the Goodwillie-Weiss embedding calculus and in factorization homology, where one is interested not only in configuration spaces of points, but also in natural actions from the little discs operad. In this talk, I will give an introduction to the problem of understanding configuration spaces and present a combinatorial/algebraic model of these spaces using graph complexes that, under some conditions, model the right operadic actions. We will explore some applications and see how these models allow us to answer fundamental questions about the dependence on the homotopy type of M.

Khovanov homology is a link invariant introduced by Mikhail Khovanov in 1999 as a categorification of Jones polynomial, and nicely reinterpreted by Viro in terms of Kauffman states. While conceptually simple, this definition becomes impractical when increasing the number of crossings of a link diagram. In this talk we present an alternative approach to Khovanov homology of semiadequate links by constructing a presimplicial set such that its geometric realization is homotopy equivalent to the almost-extreme Khovanov complex of the link. Moreover, we determine the homotopy type of the presimplicial set obtained when the link is strongly A-adequate. We show explicitly the particular cases of trefoil and figure-8 knots. This is a joint work with Józef H. Przytycki.

We will start by giving a proof of a theorem of Powell, which asserts that mapping class groups of finite-type surfaces of genus at least 3 have trivial abelianization. We will then explain how to construct nontrivial integer-valued homomorphisms from infinite-type mapping class groups. Moreover, we will give a complete description of all the possible ways in which these arise. This is joint work with Priyam Patel and Nick Vlamis.

We will survey some recent works of Sylvain Douteau and joint works with Joana Cirici and with Martin Saralegui and Daniel Tanré whose goal is to lay down the foundations of "Intersection homotopy theory". Intersection homotopy theory aims to describe the homotopical background hidden underlying Goresky-MacPherson's intersection cohomology of stratified spaces. The development of such background provides new topological invariants for singular spaces and reveal new topological properties of algebraic varieties.

Tom Leinster has introduced various notions of "magnitude", including the magnitude of an enrichedcategory, the magnitude of a metric space, and the magnitude of a graph. The main part of my talk will introduce the magnitude of a graph, and then explain how to define the "magnitude homology" of a graph. This new invariant categorifies the magnitude in the same sense that Khovanov homology categorifies the Jones polynomial. Moreover, many of the known properties of the magnitude are explained by "categorified" properties of the homology. This part is joint work with Simon Willerton. Finally, very recent work of Leinster and Shulman gives a vast generalisation of our theory from graphs to enriched categories, and I will conclude with a brief discussion of this.

In this talk, I will discuss the spectrum of the G-equivariant stable homotopy category, for G a finite group. In joint work with P. Balmer, we were able to describe this space, as a set, for all finite groups and gain a lot of information about its topology, obtaining a complete answer for groups of square-free order. We also reduced the problem of understanding the topology for arbitrary finite groups to understanding a specific question about the topology for p-groups. Understanding this unresolved question for p-groups boils down to understanding an interesting phenomenon --- that the Tate construction performs a chromatic "blue shift". Recently, T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel and N. Stapleton have clarified such behavior, showing that an earlier "blue shift" conjecture of ours was too naive, and have thereby succeeded in computing the spectrum for all finite abelian groups. The task that remains is to find and prove a correct "blue shift" conjecture for nonabelian p-groups. I will discuss these ideas and, if time permits, also discuss some recent work with I. Patchkoria on the spectrum of the derived category of Mackey functors, which can be regarded as a kind of "linearization" of the equivariant stable homotopy category.