** Loc****ation: Zoom/Real World**

### Program

Date | Time | Place | Title | Speaker |
---|---|---|---|---|

01.03.2022 | 10:15 CET | MA A3 30 | Generalized Character Varieties and Quantization via Factorization Homology | Corina Keller, Université de Montpellier |

08.03.2022 | 10:15 CET | MA A3 30/Zoom | Associativity of Cosmash Products in Algebra | Corentin Vienne, UC Louvain |

15.03.2022 | 10:15 CET | MA A3 30 | Structured Geometric Sheaves in Higher Category Theory | Raffael Stenzel, Masaryk University |

22.03.2022 | 10:15 CET | MA A3 30 | How Algebraic is a Stable Model Category? | Constanze Roitzheim, University of Kent |

29.03.2022 | 17:00 CEST | Zoom | Ramification in Homotopy Theory | John Berman, University of Massachusetts Amherst |

05.04.2022 | 10:15 CEST | MA A3 30 | Comparison of Nerve Functors for Strict n-Categories | Dimitri Ara, Aix-Marseille Université |

12.04.2022 | 10:15 CEST | MA A3 30 | Sheaves as Topological Descriptors of Datasets | Nicolas Berkouk, EPFL |

26.04.2022 | 10:15 CEST | MA A3 30 | Costabilisation of vₙ-Periodic Homotopy Types | Yuqing Shi, Universiteit Utrecht |

03.05.2022 | 10:15 CEST | MA A3 30 | The Barr-Beck Theorem in Symplectic Geometry | Nate Bottman, Max-Planck-Institut |

10.05.2022 | 10:15 CEST | MA A3 30 | Postnikov Systems for Higher Categories | Joost Nuiten, Université Paul Sabatier |

17.05.2022 | 10:15 CEST | MA A3 30 | Homotopy coherent Hopf Algebras | Aras Ergus, EPFL |

19.05.2022 | 14:15 CEST | MA B2 485 | Smooth Structures and Embedding Calculus | Ben Knudsen, Northeastern University |

24.05.2022 | 10:15 CEST | MA A3 30 | A Hopf Algebra Model for Dwyer’s Tame Homotopy Theory | Haoqing Wu, EPFL |

31.05.2022 | 10:15 CEST | MA A3 30 | Homological Stability for Asymptotic Monopole Moduli Spaces | Martin Palmer, IMAR |

07.07.2022 | 09:00 CEST | MA B1 524 | An Introduction to the Geometry of Jet Schemes and Arc Spaces | Ilaria Rossinelli, EPFL |

### Abstracts

**Corina Keller,****Generalized Character Varieties and Quantization via Factorization Homology:**Factorization homology is a local-to-global invariant which “integrates” disk algebras in symmetric monoidal higher categories over manifolds. In this talk I will focus on a particular instance of factorization homology on surfaces where the input algebraic data is a braided monoidal category. If one takes the representation category of a quantum group as an input, it was shown by Ben-Zvi, Brochier and Jordan (BZBJ) that categorical factorization homology quantizes the category of quasi-coherent sheaves on the moduli space of G-local systems. I will discuss two applications of the factorization homology approach for quantizing (generalized) character varieties. First, I will explain how to compute categorical factorization homology on surfaces with principal D-bundles decorations, for D a finite group. The main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We will see that in this case factorization homology gives rise to a quantization of Out(G)-twisted character varieties (This is based on joint work with Lukas Müller). In a second part we will consider surfaces that are decorated with marked points. It was shown by BZBJ that the algebraic data governing marked points are braided module categories and I will discuss an example related to the theory of dynamical quantum groups.

**Corentin Vienne,****Associativity of Cosmash Products in Algebra:**

A. Carboni and G. Janelidze extend the definition of smash product from pointed topological spaces to pointed objects in a suitable category. Moreover, they study a condition they call smash associativity. In this talk, we interest ourselves in the dual notion: the associativity of cosmash products. An interesting fact about this categorical notion is that it characterizes the variety of commutative and associative K-algebras for an infinite field K. The promise of this talk is to convince you of the validity of the latter statement.

In order to present things in an understandable way, we will first introduce the notion of a binary cosmash product. We see how it naturally leads to a suitable definition of binary commutators by looking at some classical examples. We then try to extend these notions the ternary case, and even to the n-ary case for some natural number n. From here, what cosmash associativity means can be explained essentially without effort. In the end, we discuss the main result and, if time allows it, the techniques used to prove it.

Joint work with Ülo Reimaa and Tim Van der Linden.

**Raffael Stenzel,****Structured Geometric Sheaves in Higher Category Theory:**

Much like ordinary topos theory is the theory of sheaves on a category equipped with a topology, higher topos theory can be understood as the theory of homotopy-coherent sheaves on a higher category equipped with a “structured” topology. In essence, the latter notion replaces coherent families of covering sieves with coherent families of fibered covering structures.

In this talk we make use of that additional freedom in the definition of higher sites to move away from the classical sheaf condition over topological spaces — and over geometric categories more generally — and introduce a stronger limit-preserving property. We therefore define and study a new class of higher toposes: the structured geometric sheaf theories on suitably equipped higher categories. We will see that the two notions of structured and unstructured (i.e. ordinary) geometric sheaves differ only by a subtle cotopological fragment. Yet it turns out that this fragment is crucial in various aspects. As a case in point, we will show that every higher topos is the theory of structured geometric sheaves over itself, while the same is generally not true for the ordinary geometric sheaves over itself.

**Constanze Roitzheim,****How Algebraic is a Stable Model Category?:**

There are many different notions of “being algebraic” used in stable homotopy theory. The relationships between those turn out to be unexpectedly subtle. We will explain the different ways in which a category of interest can be algebraic, explore the different implications between them and illustrate those with plenty of examples.(This is joint work with Jocelyne Ishak and Jordan Williamson.)

**John Berman,****Ramification in Homotopy Theory:**

I will discuss a new homotopy theoretic generalization of the idea of ramification, in the sense of number theory. Understanding the ramification of an extension of number fields is useful for calculations. By comparison, the new homotopical version leads to a greatly simplified calculation of Topological Hochschild Homology (THH) of a ring of integers in a number field. On the other hand, the new definition allows us to talk about ramification of extensions of ring spectra like ku/ko. Therefore, there are hopeful applications to computing THH and K-theory in chromatic homotopy theory settings. I will survey some of these ideas, assuming some background with spectra but not with algebraic number theory.

**Dimitri Ara,****Comparison of Nerve Functors for Strict n-Categories:**

The nerve functor from categories to simplicial sets allows to associate to any small category a homotopy type giving rise to a homotopy theory of small categories studied by Quillen, Thomason and Grothendieck. To generalize this picture to higher categories, one needs an analogous nerve functor. But there exists plenty of possibilities! The purpose of this talk will be to explain generalities about comparison of nerve functors and to show that the for strict n-categories the multi-simplicial nerve, the cellular nerve and Street’s nerve are all equivalent.

**Nicolas Berkouk,****Sheaves as Topological Descriptors of Datasets:**

Ever wondered why I keep talking about sheaves while being part of the “applied” side of the group ? In this talk, I will explain how (derived) sheaf theory (after Kashiwara-Schapira) can be used as a theoretical foundation for persistence theory (the algebraic theory associated to most Topological Data Analysis constructions). If time permits, I will also explain how taking this point of view allows to define new computable topological descriptors of datasets using…derived Grothendieck operations!

**Yuqing Shi,****Costabilisation of vₙ-Periodic Homotopy Types:**

One can consider the stabilisation of a symmetric monoidal ∞-category as the ∞-category of objects that admit an infinite delooping. For example, the ∞-category of spectra is the stabilisation of the ∞-category of homotopy types. Costabilisation is the opposite notion of stabilisation, where we are interested in objects that admits infinite desuspensions. It is easy to see that the costablisation of the ∞-category of homotopy types is trivial. The ∞-category of vₙ-periodic homotopy types is a localisation of the ∞-category of homotopy types which is the analogue of rational localisation in higher chromatic height. In this work we showed that the costabilisation of vₙ-periodic homotopy types is the ∞-category of T(n)-local spectra. As a consequence, we obtain the universal property of the Bousfield–Kuhn functor. This is a joint work with Gijs Heuts.

**Nate Bottman,****The Barr-Beck Theorem in Symplectic Geometry:**

The Barr–Beck theorem gives conditions under which an adjunction F -| G is monadic. Monadicity, in turn, means that the category on the right can be computed in terms of the data of F and its endomorphism GF. I will present joint work-in-progress with Abouzaid, in which we consider this theorem in the case of the functors between Fuk(M1) and Fuk(M2) associated to a Lagrangian correspondence L12 and its transpose. These functors are often adjoint, and under the hypothesis that a certain map to symplectic cohomology hits the unit, the hypotheses of Barr-Beck are satisfied. This can be interpreted as an extension of Abouzaid’s generation criterion, and we hope that it will be a useful tool in the computation of Fukaya categories.

**Joost Nuiten,****Postnikov Systems for Higher Categories:**

The Postnikov tower is one of the earliest tools used to investigate the homotopy types of spaces: it allows one to reconstruct a space from Eilenberg–Maclane spaces and hence to study its homotopical properties in terms of cohomology. In this talk, I will describe a version of the Postnikov tower for (oo, n)-categories, inspired by ideas of Lurie. This will follow from a general analysis of Postnikov-type systems in oo-categories, including the “k-invariants” that classify them. For example, given a functorial choice of such a system for the objects of a monoidal category V, one often obtains induced Postnikov systems for algebras in V and for V-enriched categories. Based on joint work with Yonatan Harpaz and Matan Prasma.

**Aras Ergus,****Homotopy coherent Hopf Algebras:**

From the (co)homology of a topological group with coefficients in a field to the (dual) Steenrod algebra, certain algebraic invariants that show up in topology are equipped with a compatible comultiplication that upgrades them to Hopf algebras. One of the main insights of homotopy theory is that algebraic invariants are truncations of objects equipped with homotopy coherent algebraic structures (such as ring spectra), so one might hope that the additional “coalgebraic” structures (and their interactions with algebraic ones) can also be lifted to homotopy coherent ones.

In this talk, we will give an overview of the theory of such homotopy coherent Hopf algebras and (co)modules over them, which in particular provides a unifying point of view on concepts such as (naive) equivariant homotopy theory and Thom spectra.

**Ben Knudsen,****Smooth Structures and Embedding Calculus:**

We ask when embedding calculus can distinguish pairs of smooth manifolds that are homeomorphic but not diffeomorphic. We prove that, in dimension 4, the answer is “almost never.” In contrast, we exhibit an infinite list of high-dimensional exotic spheres detected by embedding calculus. The former result implies that the algebraic topology of knot spaces is insensitive to smooth structure in dimension 4, answering a question of Viro. The latter result gives a partial answer to a question of Francis and hints at the possibility of a new classification of exotic spheres in terms of a stratified obstruction theory applied to compactified configuration spaces.

**Haoqing Wu,****A Hopf Algebra Model for Dwyer’s Tame Homotopy Theory:**

Roughly speaking, a 3-connective space is tame if its homotopy groups become more divisible as its degree increases. As an analogy to Quillen’s rational homotopy theory, Dwyer showed that the homotopy theory of tame spaces is equivalent to the homotopy theory of certain dg Lie algebras over the integer. In this talk, I will first sketch a modern approach to Quillen’s rational homotopy theory. Then I will explain how one can adjust this approach to build an Hopf algebra model for tame spaces. Finally, I will explain the Hopf algebra model is equivalent to Dwyer’s Lie algebra model via a universal enveloping algebra functor.

**Martin Palmer,****Homological Stability for Asymptotic Monopole Moduli Spaces:**

Magnetic monopoles were introduced by Dirac in 1931 to explain the quantisation of electric charges. In his model, they are singular solutions to an extension of Maxwell’s equations allowing non-zero magnetic charges. An alternative model, developed by ‘t Hooft and Polyakov in the 1970s, is given (after a certain simplification) by smooth solutions to a different set of equations, the Bogomolny equations, whose moduli space of solutions has connected components M_k indexed by positive integers k. These have been intensively studied, notably by Segal (stabilisation of their homotopy groups) and Cohen-Cohen-Mann-Milgram (describing their stable homotopy types in terms of braid groups).

A compactification of M_k has recently been proposed by Fritzsch-Kottke-Singer, whose boundary strata we call asymptotic monopole moduli spaces. I will describe ongoing joint work with Ulrike Tillmann in which we study stability patterns in the homology of these spaces.

**Ilaria Rossinelli,****An Introduction to the Geometry of Jet Schemes and Arc Spaces:**

Jet schemes and arc spaces over a variety were introduced by Nash, who was probably the first to notice that geometric properties of arcs are related to ones of the variety. Moreover, they were also discovered to play a fundamental role in the theory of motivic integration started by Kontsevich, which in turn offers powerful tools to work on them and obtain even more properties and results.

This expository talk will be divided into two main parts. We will start defining jet schemes and arc spaces of varieties (eventually in characteristic zero), and discuss their construction as representations of certain functors. Then, we will move onto the study of their geometric properties, focusing in particular on how these provide geometric information and insight about the variety.

If time permits, we will also briefly discuss how jets and arcs are used in motivic integration, and how this powerful machinery leads to even more information.