EPFL Topology Seminar 2013/2014

Fridays at 14:15

MA 12


Date Title Speaker
MA 31
Cellular approximations of classifying spaces of compact Lie groups Alberto Gavira-Romero
UA Barcelona
20.09.13 Cellular properties of nilpotent spaces Jérôme Scherer
25.10.13 A new approach to the bar-cobar constructions Matthieu Anel
MA 31
The degrees of maps between (2n-1)-Poincare complexes Jelena Grbic
MA 31
Loop space decompositions of simply-connected four-manifolds Stephen Theriault
01.11.13 A small model for the cohomology of some principal bundles Samuel Tinguely
08.11.13 A 2-categorical approach to quasi-categories Dimitri Zaganidis
15.11.13 Direct products and Ext1 contravariant functors Claudiu Valculescu
22.11.13 Batalin-Vilkovisky algebra via the double cobar construction Alexandre Quesney
29.11.13 Towards new topological models of neural processes  Sophie Raynor
06.12.13 Categories of fibrant objects and higher stacks  Ezra Getzler
13.12.13 Geometric categories related to factorization homology  Ricardo Andrade
Paris XIII
07.02.14 Homotopic Hopf-Galois extensions of commutative dga’s

Varvara Karpova



Master thesis oral defense 9h45, 10h45, 11h45; 14h15, 15h15 Eiichi Piguet, Mathias Constantin, Ismaël Jecker, Coralie Spahn, Rachel Jeitziner
28.02.14 Loop space homology of a p-local group

Ran Levi

21.03.14 Computational aspects of (persistent) homology. From discrete to continuous and back. Pawel Dlotko
University of Pennsylvania
04.04.14 Detection property in stable equivariant homotopy theory Nicolas Ricka
Université de Strasbourg
03.06.14  08h30 Semester project presentation, MA30 Carù, Slamitz, Ho Thanh, Espic, Hunkeler
06.06.14  13h30 Semester project presentation, MA30 Hong, Boys, Scheurer
12.06.14 10h15 Introduction to Categorical Geometry Eric Bunch
Kansas State University
20.06.14 Some properties of the category of quandles Valérian Even
Université de Louvain-la-Neuve

16, 22-24.07.2014
MA 10

Working group in topological data analysis
Members of the
Homotopy Theory Group
MA 30
Spaces of commuting elements in Lie groups Mentor Stafa
Tulane University

(See also the program of the topology seminar in 2011/12,  2010/11,  2009/102008/09,  2007/08, 2006/07, and 2005/06.)


Gavira-Romero: Abstract can be downloaded here.


Scherer: This is joint work with Wojciech Chachólski, Emmanuel Dror Farjoun, and Ramón Flores. One property of classifying spaces of discrete groups is that the pointed mapping space map*(BG, BG) is (homotopically) discrete, more precisely the fundamental group functor induces an equivalence with Hom(G, G). Which other spaces have this property? For nilpotent spaces, only classifying spaces do!

To attack this problem we study cellular properties of nilpotent spaces. We show in particular that a nilpotent space X “constructs” any of its Postnikov sections, hence K(pi1 X, 1). I will explain our strategy of this cellularity statement and how it implies the claim about mapping spaces.


Anel: This is joint work with André Joyal. Our main result is that the category of dg-coalgebras is symmetric monoidal closed and that the category of dg-algebras is enriched, bicomplete and monoidal over that of dg-coalgebras. This structure produces six operations on algebras and coalgebras which can be used to construct many adjunctions between algebras and coalgebras: de Rham complex, jet algebras, Sweedler duality, and the bar-cobar adjunction.


Grbic: I shall describe results on degrees of maps between (n-2)-connected (2n-1)-dimensional Poincare complexes which have torsion free integral homology. In particular, necessary and sufficient algebraic conditions for the existence of map degrees between such Poincare complexes will be established and the calculation of the set of all map degrees between certain Poincare complexes will be illustrated. Time permitting, we shall see how the knowledge of possible degrees of self maps can be used to classify, up to homotopy, torsion free (n-2)-connected (2n-1)-dimensional Poincare complexes for low n.


Theriault:  Let M be a simply-connected four-manifold. We give an explicit homotopy decomposition of the based loops on M, which depends only on the rank of the degree two cohomology of M, and whose factors are spheres and loops on spheres. Some consequences will also be discussed.


Tinguely: Let G be a compact, connected and simply connected Lie group, and ΩG the space of the loops in G based at the identity. I will present a way to compute the cohomology of the total space of a principal ΩG-bundle over a manifold M, from the cohomology of G, the differential forms on M and the characteristic classes of the bundle. If times permits, I will also cover the equivariant case.


Zaganidis: In a recent paper, Emily Riehl and Dominic Verity have shown that the (strict) 2-category of quasi-categories captures already a lot of information about the category theory of quasi-categories. In this talk, we will present some of their constructions and results. We will discuss weak 2-limits and in particular weak comma objects. We will also define adjunctions, equivalences, terminal objects and limits in a 2-categorical way.


Valculescu: Using inequalities between infinite cardinals, I will show that, if R is an hereditary ring, then the contravariant derived functor Ext1R(−,G) commutes with direct products if and only if G is an injective R-module.


Quesney: Let X be a simplicial set. H.-J. Baues constructed an explicit coproduct on the cobar construction of X. This coproduct has two well-known properties : 1- the resulting double cobar construction of X is a model for the chain complex of the double loop space of |X|; 2- it corresponds to an E_2-coalgebra structure on the chain complex of X. By using this E_2-coalgebra structure, we establish a criterion for obtaining a BV-algebra structure up to homotopy (à la Gerstenhaber-Voronov) on the double cobar construction of X. We obtain such a BV-algebra structure up to homotopy when X is an iterated simplicial suspension. We also discuss the particular case when the ground ring is a characteristic zero field.


Raynor: Together with Kathryn Hess and Ran Levi, and in collaboration with the Blue Brain Project here at the EPFL, I am working to develop new topological methods for neural modelling. Though the hope is that topology will provide useful insights for neuroscience, as mathematicians we are really motivated by the potential for new mathematics inspired by neuroscientific questions. I will introduce the category we are using to model brain plasticity and learning processes, and some ideas for how we are actually going to do this, highlighting some of the topological features that are emerging from our investigations. Since this is work in its infancy, the focus will be on developing the context, and discussing questions meriting further investigation.


Getzler:  We explain a simple method of constructing categories of fibrant objects (which generalizes the theory of simplicial sheaves to simplicial objects in more general categories than a topos). As an example, we present the category of derived stacks.


Andrade:  This talk will provide a brief overview of factorization homology, and its relations to stratified spaces and configuration spaces. I will begin with a construction of Hochschild homology in terms of a category of configurations on a circle. Factorization homology generalizes Hochschild homology and is obtained by taking configurations of points on an arbitrary manifold. I will give a detailed description of these categories of configurations using stratified spaces. The connection to stratified spaces is the basis for proving some results concerning factorization homology.


Karpova: After giving a brief overview of (homotopic) (Hopf-)Galois theory and explaining its motivations, we define the concept of homotopic Hopf-Galois extensions in the category of non-negatively graded chain complexes over a field. We will then discuss the behavior of homotopic Hopf-Galois extensions under cobase change (i.e., under pushout). Finally, we will formulate and establish one direction of a homotopic Hopf-Galois correspondence in this context.
This work takes its roots and inspiration in the works of Kathryn Hess (2009, 2010) and John Rognes (2008).


Levi: For a finite group G the homology of the loop space of  BG^p with coefficients in a field of characteristic p was shown by Benson to admit a definition by purely algebraic terms. We present an approach which generalises Benson’s construction to p-local group theory motivated by the aim of providing a tool to distinguish the family of p-compact groups among all p-local compact groups. Background, general motivation, and terminology for the subject will be provided.


Dlotko: Computational topology is a branch of a computational mathematics – a new discipline emerging in between mathematics and computer science and aiming in obtaining mathematical rigorous results with a help of computer. Recently computational topology gain a lot of attention in various fields outside mathematics. I will start this talk by summarizing some of those applications. Later the concept of
persistent homology will be explained and the algorithm(s) to compute it for a finite filtered cell complex will be given. Having this knowledge we will consider the problem of rigorous computations of level sets (up to homotopy type) and persistent homology of sufficiently smooth functions f : Rn-> Rn.

Ricka: The computation of generalised cohomology of groups is an interesting and difficult problem in general. In this talk, we will explain how the ideas of Ossa, which led to a computation of connective complex k-theory of elementary abelian groups generalises to the Z/2-equivariant setting. The approach we will describe leads to a computation of the connective k-theory with reality of abelian 2-groups. This object of equivariant nature contains in particular both real and complex connective k-theory of such groups.

Bunch: Given a complete and cocomplete closed symmetric monoidal category C, consider CMon(C), the category of commutative monoids in C. We wish to define a spectrum that takes R in CMon(C) and gives Spec(R), a topological space together with a structure sheaf that takes values in CMon(C). To do this, we make use of the (bi)fibration Mod –> CMon(C), and instead of defining directly Spec(R), we will actually define Spec(R-mod). In the case when C = Z-mod, this spectrum recovers the prime spectrum of a ring. If time permits, I will discuss thoughts on the case when C = S-mod; modules over the sphere spectrum, and CMon(C) is commutative ring spectra. In this case, we do not wish to distinguish between two Quillen equivalent ring spectra, and thus must modify the definition of Spec(R). The case when C = S-mod is the subject of my research this summer under the guidance of Dr. Hess.

Even: The purpose of this talk is to clarify the relationship between the algebraic notion of quandle covering introduced by M. Eisermann and the categorical notion of covering arising from Galois theory. A crucial role is played by the adjunction between the category of quandles and the category of trivial quandles. We will also compare the factorisation system induced by this adjunction with the factorisation system discovered by E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter.

Stafa: We study the spaces of ordered pairwise commuting n-tuples in a compact and connected Lie group G, denoted Hom(Zk,G). We introduce an infinite dimensional topological space denoted Comm(G), reminiscent of a Stiefel variety, that assembles the spaces Hom(Zk,G) into a single space. This construction admits stable decompositions which allow the study of the spaces Hom(Zk,G) individually, and the Hilbert-Poincare series is also calculated using Molien’s theorem. The cohomology of Comm(G) is given in terms of the tensor algebra generated by the reduced homology of the maximal torus. This is joint work with Fred Cohen.