Tuesdays at 10:15
CM 1 113
Program
Date  Title  Speaker 

15.09.2017 PH H3 33 
Separable and Galois extensions in tensor triangulated categories  Bregje Pauwels Australian National University 
19.09.2017 CM 012 
Configuration spaces of products  Kathryn Hess EPFL 
03.10.2017 CM 012 
Homotopically rigid Sullivan algebras and their applications  David Méndez EPFL & Universidad de Málaga 
17.10.2017 CM 012 
Towards the dual motivic Steenrod algebra in positive characteristic  Martin Frankland Universität Osnabrück 
24.10.2017 CM 012 
Understanding rational equivariant commutative ring spectra  Magdalena Kedziorek MPIM Bonn 
CM 012 
EPFL 

7.11.2017 CM 012 
The spectrum of equivariant stable homotopy theory  Beren Sanders EPFL 
14.11.2017 CM 012 
Motivic infinite loop spaces  Adeel Khan Universität Regensburg 
14.11.2017 ELE 111 
Dessins d’enfants and the generalized Hurwitz numbers  George Shabat Moscow State University 
21.11.2017 CM 012 
Injective and projective model structures on enriched diagram categories  Lyne Moser EPFL 
28.11.2017 CM 012 
Genuine equivariant conjugation  Jérôme Scherer EPFL 
05.12.2017 CM 012 
A model for Motivic stable homotopy in classical homotopy  Nicolas Ricka Université de Strasbourg 
12.12.2017 CM 012 
Real topological Hochschild homology  Irakli Patchkoria Universität Bonn 
06.02.2018 CM 1 113 
Equivariant Monoidal structures on stable categories  Denis Nardin Université Paris 13 
13.02.2018 CM 1 113 
Multiplicativity of the idempotent splittings of the Burnside ring and the Gsphere spectrum  Benjamin Böhme University of Copenhagen 
13.02.2018 CM 012 @ 13:00 
The Dennis trace map  Aras Ergus EPFL 
20.02.2018 CM 1 113 
Classical and noncommutative Voevodsky’s conjecture for cubic fourfolds and GushelMukai fourfolds  Laura Pertusi University of Milan 
17.04.2018 CM 1 113 
Infinityoperads via symmetric sequences  Rune Haugseng University of Copenhagen 
24.04.2018 CM 1 113 
Six operations formalism for generalized operads  Benjamin Ward Stockholm University 
01.05.2018 CM 1 113 
C*superrigidiity of nilpotent groups  Sven Raum EPFL 
08.05.2018 CM 1 113 
Stratified homotopy theory  Sylvain Douteau Université de Picardie 
15.05.2018 CM1 113 @ 09:15 
Combinatorial models for stable homotopy theory  Matija Bašić University of Zagreb 
15.05.2018 CM 1 113 @ 10:15 
Galois actions, purity and formality with torsion coefficients  Joana Cirici University of Barcelona 
22.05.2018 CM 1 113 
On Mackey 2functors  Ivo Dell’Ambrogio Université de Lille 
05.06.2018 CM 1 104 
Volume variation for representations of 3manifold groups in SL_n(C)  Wolfgang Pitsch Universitat Autònoma de Barcelona 
12.06.2018 MA A1 12 @ 11:15 
From elementary abelian pgroup actions to pDG modules  Marc Stephan University of British Columbia 
12.06.2018 MA A1 12 @ 15:15 
Rational Parametrised Stable Homotopy Theory  Vincent BraunackMayer University of Zurich 
19.06.2018 CM 1 113 
Topology of robot motion planning  Michael Farber Queen Mary University of London 
20.06.2018 CM 1 113 
Intersection cohomology and spectra  David Chataur Université de Picardie Jules Verne 
03.07.2018 CM 1 113 
Lifting Gstable endotrivial modules  Joshua Hunt University of Copenhagen 
(See also the program of the topology seminar in 2011/12, 2010/11, 2009/10, 2008/09, 2007/08, 2006/07, and 2005/06.)
Abstracts
Bregje Pauwels – Separable and Galois extensions in tensor triangulated categories
I will consider separable and Galois extensions of commutative monoids in tensor triangulated categories, and show how they pop up in various settings. In stable homotopy theory, separable extensions of commutative Salgebras have been studied extensively by Rognes. In modular representation theory, restriction to a subgroup can be thought of as extension along a separable monoid in the (stable or derived) module category. In algebraic geometry, separable monoids correspond to étale extensions of schemes, alowing us to define a generalized étale site for any tensor triangulated category.
Kathryn Hess – Configuration spaces of products
I will explain the construction of a new model for the configuration space of a product of two closed manifolds in terms of the configuration spaces of each factor separately. The key to the construction is the lifted BoardmanVogt tensor product of modules over operads, developed earlier in joint work with Dwyer.
(Joint work with Bill Dwyer and Ben Knudsen.)
David Méndez – Homotopically rigid Sullivan algebras and their applications
The group of homotopy selfequivalences of a space is rarely trivial. Kahn was the first to obtain an example of one such space with nontrivial rational homology in the seventies. Later, Arkowitz and Lupton came across an example of a Sullivan algebra (equivalently, a rational homotopy type) with trivial homotopy selfequivalences. This algebra was used by Costoya and Viruel to solve Kahn’s group realisability problem for finite groups, thus obtaining for any finite group G a rational space X whose group of homotopy selfequivalences is isomorphic to G. This construction also provide a way to obtain an infinite amount of homotopically rigid spaces. However, they all share their level of connectivity with the example of Arkowitz and Lupton.
The objective of this talk is to illustrate that
(i) Homotopically rigid spaces are not as rare as they were though to be. We are able to obtain an infinite family of homotopically rigid spaces, showing a level of connectivity as high as desired.
(ii) Building blocks other than the example of Arkowitz and Lupton can be used to solve Kahn’s realisability problem.
We can also apply the obtained results to differential geometry by enlarging the class of inflexible manifolds existing in literature and building new examples of strongly chiral manifolds.
Reference: C. Costoya, D. Méndez, A. Viruel, Homotopically rigid Sullivan algebras and their applications, arXiv:1701.03705 [math.AT].
Martin Frankland – Towards the dual motivic Steenrod algebra in positive characteristic
Several tools from classical topology have useful analogues in motivic homotopy theory. Voevodsky computed the motivic Steenrod algebra and its dual over a base field of characteristic zero. Hoyois, Kelly, and Ostvaer generalized those results to a base field of characteristic p, as long as the coefficients are mod l with l \neq p. The case l = p remains conjectural.
In joint work with Markus Spitzweck, we show that over a base field of characteristic p, the conjectured form of the mod p dual motivic Steenrod algebra is a retract of the actual answer. I will sketch the proof and possible applications. I will also explain how this problem is closely related to the HopkinsMorelHoyois isomorphism, a statement about the algebraic cobordism spectrum MGL.
Magdalena Kedziorek – Understanding rational equivariant commutative ring spectra
Recently, there has been some new understanding of various possible levels of commutative ring Gspectra. In this talk I will recall these possibilities and discuss the most naive (or trivial) commutative ring Gspectra. Then I will sketch the main ingredients coming into the proof that if G is finite and we work rationally these objects correspond to (the usual) commutative differential algebras in the algebraic model for rational Gspectra. This is joint work with David Barnes and John Greenlees.
Kathryn Hess – Motivic homotopical Galois extensions
(Joint work with Agnès Beaudry, Magdalena Kedziorek, Mona Merling, and Vesna Stojanoska) I will sketch a formal framework for homotopical Galois extensions, motivated by the case of commutative ring spectra developed by Rognes, within which we can prove invariance of Galois extensions under extension of coefficients and the forward part of a Galois correspondence. I will explain why both motivic spaces and motivic spectra fit into this framework, then provide explicit examples of motivic homotopical Galois extensions, some of which have no classical analogue.
Beren Sanders – The spectrum of equivariant stable homotopy theory
In this talk, I will discuss the spectrum of the Gequivariant 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 squarefree order. We also reduced the problem of understanding the topology for arbitrary finite groups to understanding a specific question about the topology for pgroups. Understanding this unresolved question for pgroups 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 pgroups.
Adeel Khan – Motivic infinite loop spaces
Given a topological space X, May’s recognition principle says that a structure of infinite loop space on X is equivalent to a grouplike $E_\infty$monoid structure. We will discuss an analogous result in motivic homotopy theory which says that a structure of infinite P^1loop space on a motivic space X is equivalent to a homotopy coherent system of certain transfer maps. This is based on an analogue of the PontrjaginThom construction which identifies stable homotopy groups of spheres with groups of framed bordisms. This is joint work with E. Elmanto, M. Hoyois, V. Sosnilo and M. Yakerson.
George Shabat – Dessins d’enfants and the generalized Hurwitz numbers
A multigraph, embedded into a closed connected oriented surface, is called a ‘dessin d’enfant’ if its complement is homeomorphic to a disjoint union of cells. The (appropriately defined) category of dessins d’enfants turned out to be equivalent to the category of ‘Belyi pairs’ that belongs to arithmetic geometry; a first part of the talk will be devoted to the foundations of this theory.
Then the certain problems of enumeration of dessins (and their generalizations) will be introduced. The particular cases of these problems, e.g., recursions for the socalled ‘Hurwitz numbers’, were studied intensively during the last decades; the corresponding recent results will be mentioned. However, the general case is currently out of reach, and during the second part of the talk a certain project, based on the above category equivalence, will be presented. Hopefully, realization of this project will promote the understanding of general case.
Lyne Moser – Injective and projective model structures on enriched diagram categories
K. Hess, M. Kedziorek, E. Riehl, and B. Shipley have developed methods to induce model structures from an adjunction which they apply to prove the existence of injective and projective model structures on categories of diagrams in accessible model categories. In this talk, I will explain how to adapt their proof to an enriched setting, in order to prove the existence of injective and projective model structures on some enriched diagram categories. I will talk in particular about the case of enriched diagrams from a small simplicial category to the category of pointed simplicial sets, and then generalize it by replacing the category of pointed simplicial sets by other symmetric monoidal categories, which are locally presentable bases and accessible model categories.
Jérôme Scherer – Genuine equivariant conjugation
This is joint work with Wolfgang Pitsch and Nicolas Ricka. I will first introduce conjugation spaces as they have originally been defined by Haussmann, Holm, and Puppe. Roughly speaking they are spaces equiped with a nice action by a cyclic group of order 2, so that the fixed points look like “half the space’’ through the eyes of mod 2 cohomology (think of real projective spaces as fixed points of complex ones under conjugation). I will then explain how tools from (genuine) equivariant stable homotopy theory allow us to give a more conceptual characterization of conjugation spaces in terms of purity. A priori one only looks at the mod 2 cohomology of a conjugation space as a graded vector space. Our approach highlights the compatibility with the cup product and the action of the Steenrod algebra.
Nicolas Ricka – A model for Motivic stable homotopy in classical homotopy
MorelVoevodsky’s motivic stable category of schemes plays a crutial role in today’s stable homotopy. For instance, a recent work of IsaksenWang push our knowledge of stable stems up to dimension 93. In this joint work with Achim Krause and Bogdan Gheorghe, we construct a new model category, equivalent to the cellular motivic stable category, but whose construction lies entierly in the realm of classical stable homotopy theory. Moreover, this construction emphasizes the close relationship between cellular motivic homotopy theory and the theory of BPresolutions. If time permits, I will talk about a new computation of the motivic dual Steenrod algebra using this new model, as well as the perspectives opened by similar constructions.
Irakli Patchkoria – Real topological Hochschild homology
This talk will define the real topological Hochschild homology (THR), introduced by Hesselholt and Madsen. THR is an invariant for rings with antiinvolution and is a genuine Z/2equivariant refinement of the classical topological Hochschild homology. THR approximates the real algebraic Ktheory KR and hence on fixed points one gets an approximation for the Hermitian Ktheory. In this this talk we will concentrate on foundations of THR. We will compare different models and discuss tools for computations. Along the way we will introduce necessary equivariant homotopy theory background. At the end we will compute the group of components of THR, THR of finite fields and the geometric fixed points of THR of integers. This work is joint with Dotto, Moi and Reeh.
Denis Nardin – Equivariant Monoidal structures on stable categories
Many important objects in homotopy theory can be endowed with genuine equivariant structures: algebraic Ktheory, Thom spectra etc. In this talk we will explore how this interacts with the monoidal structures and what is one possible definition of an “equivariant monoidal structure”. As examples we will present how to put a Galoiscommutative ring structure on algebraic Ktheory and how to recover the S^1equivariant structure on THH.
Benjamin Böhme – Multiplicativity of the idempotent splittings of the Burnside ring and the Gsphere spectrum
I provide a complete characterization of the equivariant commutative ring structures of all the idempotent summands of the Gequivariant sphere spectrum, including their HillHopkinsRavenel norms, where G is any finite group. My results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, my analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, I obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category.
Aras Ergus – The Dennis trace map
Algebraic Ktheory is an invariant of rings (or in general, ring spectra) which appears in many areas of mathematics. An important approach to understanding algebraic Ktheory is relating it to certain other invariants which are easier to compute. One such invariant is topological Hochschild homology, the analogue of the usual Hochschild homology in higher algebra. The aim of this talk is to give an exposition of the Dennis trace map from algebraic Ktheory to topological Hochschild homology which is the starting point of the above mentioned “trace methods” in the study of algebraic Ktheory.
Laura Pertusi – Classical and noncommutative Voevodsky’s conjecture for cubic fourfolds and GushelMukai fourfolds
Rune Haugseng – Infinityoperads via symmetric sequences
A useful description of operads is that they are associative monoids in symmetric sequences. I’ll discuss an analogous description of (enriched) infinityoperads; this gives rise to a barcobar adjunction for infinityoperads, with potential applications to Koszul duality.
Benjamin Ward – Six operations formalism for generalized operads
In this talk I will fill in an analogy between Verdier duality for sheaves and Koszul duality for algebras over operads. To make this analogy precise we will consider the underlying categorical structure present in both situations. Time permitting I will explain how understanding Koszul duality for modular operads from this perspective can be used to do computations in graph homology.
Sven Raum – C*superrigidiity of nilpotent groups
It is a classical problem to recover a discrete group from various rings or algebras associated with it, such as the integral group ring (cf. the Whitehead group and the Whitehead torsion). By analogy, in an operator algebraic framework we want to recover torsionfree groups from certain topological completions of the complex group ring, such as the reduced group C*algebra. Groups for which this is possible are called C*superrigid. In recent joint work with Caleb Eckhardt, we could prove C*superrigidity for arbitrary finitely generated, torsionfree, 2step nilpotent groups by combining Ktheoretic methods with certain bundle decompositions of C*algebras.
In this talk, I will introduce the relevant notion of reduced group C*algebras and put it in the context of the more familiar complex group algebra. Further, I will provide a first motivation to study C*superrigidity. Then I will discuss our result with Caleb Eckhardt, focusing on methods that have analogies in topology such as bundle decompositions and topological Ktheory. The talk will finish with a persepective on relevance of C*superrigidity in other areas of mathematics.
Sylvain Douteau – Stratified homotopy theory
Stratified spaces appear as natural objects in singularity theory. Goresky and MacPherson introduced intersection cohomology to extend cohomological properties of closed manifolds to stratified spaces, and it proved to be a powerful tool to study those objects. However, intersection cohomology is not homotopy invariant, rather it is invariant with respect to homotopies that “preserve” the stratification. This begs the question : does there exist a model category of stratified spaces which reflects this stratified notion of homotopy, and if so, is intersection cohomology representable in it?
We answer the first part of this question using a simplicial model category of filtered simplicial sets. As a category, it is only the category of simplicial sets over the classifying space of some fixed poset, but as a presheaf category, it inherits a model structure constructed using a natural cylinder object. We show that this category is simplicial, then we get stratified versions of Kan complexes and of homotopy groups that characterise fibrations and weak equivalences.
Matija Bašić – Combinatorial models for stable homotopy theory
We will recall the definition of dendroidal sets as a generalization of simplicial sets, and present the connection (Quillen equivalence) to connective spectra which gives a factorization of the socalled Ktheory spectrum functor from symmetric monoidal categories to spectra. We will present a common generalization of two results of Thomason: 1) posets model all homotopy types; 2) symmetric monoidal categories model all connective spectra. We will introduce a notion of multiposets (special type of coloured operads) and of the subdivision of dendroidal sets which can be used to show that multiposets model all connective spectra. If time permits we will mention homology of dendroidal sets as it provides a means to define equivalences of multiposets in an internal combinatorial way.
Joana Cirici – Galois actions, purity and formality with torsion coefficients
In joint work with Jeremiah Heller, we extend this connection to all primes using pDG modules as developed in the Hopfological algebra framework by Khovanov and Qi. In this talk, I will provide an introduction to Hopfological algebra and explain its relevance for our work.