**Tuesdays at 10:15**

**CM 113**

### Program

Date | Title | Speaker |
---|---|---|

27.09.16 | Characteristic classes as obstructions | Martina Rovelli EPFL |

25.10.16 (16.15-17.05) Room BI A0 448 |
Automorphisms of fusion and linking systems of finite simple groups (CENTRE INTERFACULTAIRE BERNOULLI seminar) |
Bob Oliver Université Paris 13 |

1.11.16 | Fibrancy of Categories | Viktoriya Ozornova University of Bonn |

2.11.16 (9.15-10.30) Room GC A1 416 |
Fully extended twisted field theories | Claudia Scheimbauer MPIM Bonn |

8.11.16 | The quasi-category of homotopy coherent monads in a (∞, 2)-category. | Dimitri Zaganidis EPFL |

22.11.16 | Hermitian K-theory and trace methods | Emanuele Dotto University of Bonn |

29.11.16 | Towards an abstract stabilization theory | Moritz Groth University of Bonn |

5.01.17 (10.15am in MA12) | On the surgery kappa-classes and a theorem of Madsen | Diarmuid Crowley University of Aberdeen |

5.01.17 (2.15pm in MA12) | 2-Segal sets and the Waldhausen construction | Martina Rovelli EPFL |

10.01.17 (11.00am in MA12) | La dualité de Poincaré | Valentin Roy EPFL |

19.01.17 (10.15am in MA12) | Basic localizers and derivators | Lyne Moser EPFL |

6.02.17 (10.15am in MA10) | Factorization homology and topological Hochschild cohomology of Thom spectra | Inbar Klang Stanford University |

7.02.17 (10.15am in MA10) | On triangulated categories of module spectra | Ivo Dell’Ambrogio Université de Lille |

7.02.17 (14.15am in MA12) | Generalized Bestvina-Brady groups via branched covers | Ian Leary University of Southampton |

21.02.17 | On an integral analog of Quillen’s rational homotopy theorem | Jacobson Blomquist Ohio State University |

28.02.17 | From Lie algebras to configuration spaces | Benjamin Knudsen Harvard University |

07.03.17 | The gravity operad | Aras Ergus University of Bonn |

21.03.17 | Cofinality Properties of Categories of Chain Complexes | Wojtek Chacholski KTH |

28.03.17 | An algebraic analogue of a theorem by Dwyer-Hess | Haoqing Wu ETHZ |

25.04.17 | A generalized Blakers-Massey theorem and Goodwillie calculus | Georg Biedermann Université Paris 13 |

09.05.17 | Cohomology of symmetric and alternating groups | Dev Sinha University of Oregon |

16.05.17 | Calculus of functors and knot theory | Dev Sinha University of Oregon |

30.05.17 | Cellular covers of local groups | Jérôme Scherer EPFL |

01.06.17 | Knotoids and virtual knot theory | Louis Kauffman University of Illinois-Chicago |

19.06.17 10h15 CM 012 |
Lie algebras and coalgebras via the configuration pairing | Ben Walter METU |

19.06.17 11h30 CM012 |
Quillen stratification for tensor triangulated categories with applications to the Steenrod algebra | Beren Sanders 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

**Rovelli:** Characteristic classes are invariants for principal bundles that take values in the cohomology of the base space. Every characteristic class captures different geometric features of principal bundles. It is known, for instance, that the first Stiefel-Whitney class of an O(n)-bundle vanishes if and only if the bundle is orientable, in which case the structure group can be reduced to SO(n). In the first part of this talk, we propose a uniform treatment to interpret most of the studied characteristic classes as an obstruction to group reduction. By plugging in the correct parameters, the method recovers several classical theorems. Afterwards, I will explain how the main result leads to the construction of a long exact sequence of abelian groups for any principal bundle. This sequence involves the cohomology of the base space and the group cohomology of the structure group, and the connecting map is deeply related with the characteristic classes of the bundle.

**Ozornova****:** In 1980, Thomason showed that there is a model structure on the category of small categories which makes them Quillen equivalent to topological spaces and simplicial sets. In a joint work with Lennart Meier, we study fibrant objects in this model structure. We provide a sufficient criterion for fibrancy and using it, a new class of examples, including certain categories coming from model categories.

**Scheimbauer:** After giving an introduction to functorial field theories I will explain a natural generalization thereof, called “twisted” field theories by Stolz-Teichner. The definition uses the notion of lax or oplax natural transformations of strong functors of higher categories for which I will sketch a framework. I will discuss the fully extended case, which gives a comparison to to Freed-Teleman’s “relative” boundary field theories. Finally, I will explain some examples, one of which is given by factorization homology and whose target is the higher Morita category of E_n-algebras, bimodules, and intertwiners, and generalizations thereof.

**Zaganidis:** Homotopy coherent diagrams in a simplicial category K can be encoded as simplicial functors C → K, where C is a well chosen simplicial category. This idea goes back at least to Cordier and Porter (Math. Proc. Cambridge Philos. Soc. 1986) and originated in earlier work of Vogt (Math. Z. 134, 1973) on homotopy coherent diagrams. For instance, the homotopy coherent nerve is constructed in this way. In Riehl and Verity’s paper (Adv. Math. 2016), C is the universal 2-category containing the object of study, either a monad or an adjunction. For instance they define homotopy coherent monads as simplicial functors Mnd → K, where K = qCat∞ , the category of quasi-categories enriched over itself, and where Mnd is the universal 2-category containing a monad.

In this talk, we define a cosimplicial object Mnd[-] in 2-categories which induces a nerve N_Mnd : sCat → sSet. When K is a 2-category, N_Mnd(K) = N(Mnd(K)), where Mnd(K) is the 1-category of monads in K, as defined by Street in (JPAA 1972). We will sketch the proof that when K is enriched in quasi-categories and sufficiently complete, NMnd (K) is a quasi-category whose objects are the homotopy coherent monads in K.

**Dotto:** The Hermitian K-theory of a ring with anti-involution is the group-completion of its space of Hermitian forms and isometries. In recent work Hesselholt and Madsen describe this space as the Z/2-fixed-points of an involution on the algebraic K-theory spectrum of the underlying ring. Conjecturally the geometric fixed-points of this Z/2-spectrum should be equivalent to the symmetric L-theory spectrum of the ring. I will discuss ongoing work on Hermitian and L-theoretical versions of topological Hochschild homology and on the corresponding trace map from Hermitian K-theory.

**Groth:** In this talk we discuss classical and more recent characterizations of abstract stable homotopy theories. Since spectra are obtained by stabilizing (pointed) spaces, every such characterization describes defining features of the passage from (pointed) spaces to spectra. A characterization in terms of weighted (co)limits opens the door for more abstract stabilizations, and this is the starting point for an on-going project with Mike Shulman.

**Crowley:** Let SG be the topological monoid of stable self-homotopy equivalences of the sphere. The surgery kappa-classes lie in the mod 2 cohomology of the space SG and correspond to the Kervaire invariant in dimensions 4k+2 when evaluated on the homotopy groups of SG.

The space SG is an infinite loop space and in his PhD Thesis Madsen proved that the surgery kappa-classes cannot be delooped three times, whereas Madsen and Milgram proved these classes deloop twice. I will discuss an observation relating to Madsen’s theorem and applications.

**Roy:** Cette présentation donnera une preuve du théorème de dualité de Poincaré concernant les groupes d’homologie et de cohomologie de variétés orientables. Le concept d’homologie à support compact sera ainsi introduit.

**Rovelli:** In order to provide a common framework to model several flavours of Hall algebras, Dyckerhoff and Kapranov introduced the notion of a 2-Segal object, which is a generalisation of an ordinary Segal object. An important example of 2-Segal object is the Waldhausen construction of an exact category. The Waldhausen construction makes sense for a more general input, and the goal of the talk is to explain that, in the discrete setting, the Waldhausen construction is in fact quite exhaustive. More precisely, it induces an equivalence between the category of stable pointed double categories and the category of reduced unital 2-Segal sets. This is joint work with Bergner, Osorno, Ozornova and Scheimbauer.

**Moser:** The notion of a basic localizer was first introduced by Grothendieck and then developed by Maltsiniotis. In particular, one can define a basic localizer associated to each derivator, for example to the represented derivator of a cocomplete and complete category, which can be computed explicitly. In this talk, I use these results to find a characterization of initial functors in terms of comma categories. Furthermore, I compute the basic localizer associated to the homotopy derivator of the category of simplicial sets equipped with the Quillen model structure, which allows us to also find a characterization of the homotopy initial functors in terms of comma categories.

**Klang:** By a theorem of Lewis, the Thom spectrum of an n-fold loop map to BO is an E_n-ring spectrum. I will give a description of the factorization homology and the higher topological Hochschild cohomology of such Thom spectra, and talk about some consequences, such as computations and new structured ring spectra which are Thom spectra. This talk will include an exposition to factorization homology and higher Hochschild cohomology.

**Dell’Ambrogio:** In stable homotopy theory, one of the motivations for considering higher structures is that the category of (naive) modules over a (naive) ring spectrum is rarely a triangulated category, which is very annoying for computational purposes. So instead, one looks at highly structured modules over a highly structured ring spectrum (for any of the various possible meanings of “highly structured”), and then it is typically no trouble to show that their homotopy category is triangulated. Nonetheless, sometimes even naive modules form a triangulated category, for instance over separable ring spectra, or over certain Eilenberg-Mac Lane ring spectra. After recalling all these known facts, I will present a result obtained jointly with Beren Sanders: over any (highely structured) ring spectrum, naive modules form a triangulated category if and only if the naive and highly structured modules coincide. This clarifies somewhat the meaning of the word “rarely” used above, and is obtained as a corollary of a more general result about monads on triangulated categories.

**Leary:** In the late 1990’s Bestvina and Brady constructed non-finitely presented groups of type FP, resolving a notorious question that had been open for 30 years. I will define type FP and discuss my generalization of the Bestvina-Brady construction and some of its corollaries.

**Blomquist: ** In his landmark 1969 Annals paper, Quillen showed that the rational homotopy type of a simply connected space could be detected at the level of its singular rational chains, and furthermore, that rational chains fit into a derived equivalence with cocommutative dg-coalgebras over the rationals, after restricting to 1-connected objects. In 1977 Sullivan subsequently proved the analogous result in the case of rational cochains and commutative dg-algebras over the rationals. Since then topologists have worked on attempting to establish analogous results for finite fields (Kriz, Goerss, Mandell), and more recently some partial results have been established in the integral chains case (Mandell, Karoubi). Nevertheless, establishing that integral chains fit into a derived equivalence has proved resistant to all attacks. In this talk I will outline how we recently resolved, in the affirmative, the integral chains problem. Our approach exploits a mixture of (co)simplicial techniques together with ideas from Hess’ work on homotopical descent and the framework exploited in Ching-Harper’s recent resolution of the 0-connected Francis-Gaitsgory con-jecture. If time permits, I will also describe how we recently resolved that iterated suspension satisfies homotopical descent on objects and morphisms. This is joint work with J.E. Harper.

**Knudsen: ** Using ideas borrowed from the theory of chiral algebras, we supply Lie algebras with enveloping algebras over the operad of little n-dimensional disks. By globalizing these algebras over a manifold using the theory of factorization homology, we arrive at a new approach to the study of configuration spaces.

**Ergus: ** In this talk, we will first discuss various descriptions of the gravity operad which is obtained from the moduli spaces of n-marked Riemannian surfaces of genus 0. Following Getzler, we will then show that it is the Koszul dual of the hypercommutative operad which is given by the homology of the Deligne–Mumford–Knudsen compactifications of these moduli spaces.

**Chacholski: ** In this talk I will describe how one can attempt to measure differences between the moduli of complexes of dimension n and the moduli of complexes of dimension m.

**Wu: ** This talk consists of three parts. I will begin by recalling the bar and cobar constructions of operads. Then I will extend these constructions to operadic infinitesimal bimodules. This gives us a quasi-free resolution of an infinitesimal bimodule. In the second part, I will state the definition of the deformation complex of an operad map and its dg Lie algebra structure. Then I will give the definition of the deformation complex of an infinitesimal bimodule map and show it is a representation of dg Lie algebra. Finally, I will introduce the knowledge of graph complex and use it to show the deformation complex of e_n operads is quasi-isomorphic to the deformation complex of e_n infinitesimal bimodules up to a degree shift. This is an algebraic analogue of a theorem by Dwyer-Hess.

**Biedermann: ** (joint with M. Anel, E. Finster, A. Joyal) We explain our generalized Blakers-Massey theorem. For this we introduce the notion of modality: a unique factorization system whose left class is closed under base change. The example of n-connected/n-truncated maps leads to the classical Blakers-Massey. In the context of Goodwillie calculus we find another example: factoring a natural transformation into a P_n-equivalence followed by an n-excisive map. This leads to a Blakers-Massey Theorem for the Goodwillie tower. This gives a quick and independent proof of the fact that homogeneous functors deloop.

**Sinha: ** The homology of symmetric groups has been well-known since Nakaoka’s seminal work in 1962 and then Cohen-Lada-May’s reformulation and extension in 1974. But the cup co-product structure of the latter relies on Adem relations, and so has been of limited value in applications, for example providing no insight as to the number of ring generators or existence of nilpotent elements. For that reason, mathematicians such as Madsen, Milgram, Magannis, Adem and ultimately Feshbach devoted considerable energy to understanding cup product structure in more detail, building on its beautiful connection to invariant theory.

What was missing, from our perspective, was an induction or transfer product which along with known structures forms a Hopf ring, first discovered by Strickland and Turner in 1997. Using this, Giusti, Salvatore and myself have a “one-line” description of the mod-two cohomology of symmetric groups published in 2012, and Giusti and I have just finished a corresponding presentation for alternating groups, which is substantially more technically challenging.

In this talk, I will give a hands-on introduction to our calculation for symmetric groups and then talk about the broader picture before discussing alternating groups.

**Sinha: ** The calculus of functors was first invented to study pseudo-isotopy in differential topology. The formal structure first led to the calculus of homotopy functors, but then was adapted by Weiss to address questions about embeddings and other isotopy functors. In settings where it fully applies, for example to the rational homology of spaces of embeddings of a codimension three manifold, it has been remarkably successful. But one case where it does not fully apply is for classical knot theory.

In this setting, all knowledge points to a conjecture that the Goodwillie-Weiss tower serves as a universal finite-type knot invariant over the integers. I will give the background to make this conjecture meaningful (and significant), and then share recent progress and potential next steps in a longstanding program to address this conjecture.

**Scherer: ** This is joint work with Ramon Flores. Emmanuel Dror Farjoun asked whether the composition of a localization and a cellularization functor is always idempotent. Ramon Flores gave a negative answer for such functors in the category of spaces. We provide now a group theoretical counter-example based on the intriguing computation by Ol’shanskii of the Schur multiplier of Burnside groups at large primes. I will start with definitions and examples of localization and cellularization before introducing the main characters in this story.

**Kauffman: ** Knotoids are open-ended knot diagrams whose endpoints can be in different regions of the diagram. Two knotoids are said to be isotopic if there is a sequence of Reidemeister moves that connects one diagram to the other without moving arcs across endpoints. The definition is due to Turaev. We will discuss three dimensional interpretations of knotoids in terms of projections of open-ended embeddings of intervals into three dimensional space, and we shall discuss a number of invariants of knotoids based on concepts from virtual knot theory. Knotoids are a new branch of classical knot theory and they promise to provide a way to measure the “knottiness” of open interval embeddings in three space. This talk is joint work with Neslihan Gugumcu.

**Walter: ** I will outline recent advances generalizing, simplifying, and applying the configuration pairing of graphs and trees originally developed to compute the homology of configuration spaces. Earlier work applies this framework to investigate knot invariants as well as rational homotopy theory. I will present an new application directly to Lie algebras and Lie coalgebras, where this framework provides an interesting tool for making computations and an alternate view of some basic, classical Lie algebra constructions.

In the theory of finite type knot invariants, we filter knots and their invariants into towers of Lie algebras; reducing the study of knots to the study of Lie algebras. This work is morally dual — filtering Lie algebras to reduce them to towers of “simplified knots.”

**Sanders: **I will begin with some historical background on the spectrum of the cohomology ring and its significance in modular representation theory. I will then discuss Paul Balmer’s approach to Quillen stratification, facilitated by his notion of the spectrum of a tensor triangulated category. Time permitting, I will then discuss an application of these methods to the problem of classifying the thick tensor-ideals of the stable category of the mod 2 Steenrod algebra