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Barcelona Topology Workshop 2016

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Schedule
Schedule
Friday, 16 December
10:30-11:00
Welcome Coffee
11:00-11:50
Sarah Whitehouse
Derived $A_{\infty}$ algebras and their homotopies (Abstract)
Derived $A_{\infty}$ algebras are a generalisation of $A_{\infty}$ algebras, relevant to the study of differential graded algebras over a commutative ring up to quasi-isomorphism. We give two new interpretations of derived $A_{\infty}$ algebras. We introduce and explore a hierarchy of notions of homotopy for such algebras. For each natural number $r$, $r$-homotopy underlies $E_r$ equivalence, where this is defined via the $r$-th page of an associated spectral sequence. This is joint work with Joana Cirici, Daniela Egas Santander and Muriel Livernet.
12:00-12:50
Sune Precht Reeh
Characters, free loops and fusion systems (Abstract)
Every saturated fusion system (or $p$-local finite group) has an associated characteristic idempotent in the double Burnside ring of the underlying $p$-group. This can be leveraged in every $p$-local cohomology theory to express the cohomology of groups and fusion systems as retracts of the cohomology of $p$-groups. In a joint project with Tomer Schlank and Nat Stapleton, we apply these ideas to chromatic homotopy theory and certain generalizations of the usual character map from a representation ring to class functions, and we generalize results by Hopkins-Kuhn-Ravenel and Stapleton from groups to abstract fusion systems. The actual work involves studying the free loop spaces $L(BG)$ and $L(BF)$ for groups and fusion systems, and constructing transfer maps from $L(BG)$ to $L(BH)$ when $H$ is a subgroup of $G$.
13:00-14:30
Lunch
14:30-15:20
Javier Gutiérrez
Towers and fibered products of localized model structures (Abstract)
Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products of model categories. As applications we consider Postnikov towers of model categories and Bousfield arithmetic squares of spectra. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization. This is a joint work with C. Roitzheim.
15:30-16:20
Drew Heard
The chromatic splitting conjecture for Noetherian commutative ring spectra (Abstract)
We formulate a version of Hopkins' chromatic splitting conjecture for an arbitrary structured ring spectrum $R$, and prove it whenever $\pi_*R$ is Noetherian. Our approach relies on a novel decomposition of the local cohomology functors constructed previously by Benson, Iyengar, and Krause as well as a generalization of Brown--Comenetz duality. As an application, these results provide a new local-to-global principle in the modular representation theory of finite groups.
16:20-17:00
Coffee Break
17:00-17:50
Jérôme Scherer
Acyclic Blakers-Massey inequalities (Abstract)
This is based on joint work with W. Chacholski and K. Werndli, and also on some results taken from the thesis of the latter. Classically homotopy excision is stated in terms of connectivity. Our point of view is that cellular or acyclic inequalities are better, and I will explain why. We establish such an inequality for commutative squares. It explains how certain homotopy fibers allow us to estimate the difference between the initial object of the square and the homotopy pullback of that (punctured) square. Moving to higher dimensional cubes is difficult, and I will explain how K. Werndli manages to deal with 3-dimensional cubes.
Saturday, 17 December
10:00-10:50
Stephen Theriault
The dual polyhedral product, cocategory and nilpotence (Abstract)
The notion of a dual polyhedral product is introduced as a generalization of Hovey's definition of Lusternik-Schnirelmann cocategory. Properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if $X$ is a simply-connected space then the weak cocategory of $X$ equals the homotopy nilpotence class of the based loops on $X$.
11:00-11:30
Coffee Break
11:30-12:20
Alejandro Adem
Homotopy Group Actions and Group Cohomology (Abstract)
Let $G$ denote a finite group and $X$ a CW-complex. A homotopy group action is defined as a homotopy class of maps B$G$ $\to$ BAut($X$). In this talk we will analyze these actions using techniques from group cohomology. We will show how they relate to geometric actions and how they can be used to construct new and somewhat exotic examples.This is joint work with Jesper Grodal.
13:30
Lunch