** Loc****ation: Zoom**

### Program

Date/Time | Title | Speaker |
---|---|---|

17.02.2020, 10.15 CEST |
Exotic Spheres | Jean-Claude Hausmann, Université de Genève |

24.02.2020, 10.15 CEST |
An effective Proof of the Cartan Formula | Anibal Maximiliano Medina-Mardones, EPFL |

03.03.2020, 10.15 CEST |
Genuine Equivariant Factorization Homology | Asaf Horev, Stockholms Universitet |

21.04.2020, 10.15 CEST | Morava K-theory of Algebraic K-theory and Topological Periodic Cyclic Homology | Gabriel Angelini-Knoll, Freie Universität Berlin |

28.04.2020, 10.15 CEST | The Exit Path ∞-Category of the Reductive Borel-Serre Compactification | Mikala Ørsnes Jansen, Københavns Universitet |

05.05.2020, 10.15 CEST |
Twisted Dyer-Lashof Operations |
Calista Bernard, Københavns Universitet/Stanford University |

12.05.2020, 10.15 CEST | The Long Exact Sequence of Higher Groups | Egbert Rijke, Univerza v Ljubljani |

26.05.2020, 10.15 CEST | Koszul Duality of Algebras for Operads | Gijs Heuts, Universiteit Utrecht |

08.06.2020, 17:15 CEST | The Coalgebra of Chains and the Fundamental Group | Manuel Rivera, Purdue University |

16.06.2020, 10.15 CEST | Orthogonal Calculus and R^n-Cubical Homotopy Theory | Niall Taggart, Queen’s University Belfast |

23.06.2020, 10.15 CEST | Nonconilpotent Koszul Duality | Ai Guan, Lancaster University |

30.06.2020, 17.00 CEST | Equivariant Fundamental Classes in RO(C_2)-Graded Cohomology | Christy Hazel, University of Oregon |

07.07.2020, 10.15 CEST | Balmer Spectrum of Derived Mackey Functors | Irakli Patchkoria, University of Aberdeen |

14.07.2020, 17.00 CEST | Two Model Structures for Double Categories | Maru Sarazola, Cornell University |

15.07.2020, 15.00 CEST | A Double (∞,1)-Categorical Nerve for Double Categories | Lyne Moser, École Polytechnique Fédérale de Lausanne |

21.07.2020, 17.00 CEST | Rigidity of the K(1)-Local Stable Homotopy Category | Jocelyne Ishak, Vanderbilt University |

28.07.2020, 17.00 CEST | A classification of Q_0-local A(1)-modules | Katharine Adamyk, University of Colorado Boulder |

### Abstracts

**Jean-Claude Hausmann,Exotic Spheres:**

In 1956, John Milnor astonished the mathematical community by discovering exotic spheres, i.e. smooth manifolds homeomorphic but not diffeomorphic to a standard sphere. Less than seven year later, exotic spheres in high dimensions were fully classified up to diffeomorphism by the work of Kervaire-Milnor, which gave birth to surgery theory. These important achievements have been possible thanks to great advances in differential topology, as well as in homotopy theory. This talk will survey those effervescent years in topology.

**Anibal Maximiliano Medina-Mardones,An effective Proof of the Cartan Formula:**

The Cartan formula encodes the relationship between the cup product and the action of the Steenrod algebra in F_p-cohomology. In this work, we present an effective proof of the Cartan formula at the cochain level when the field is F_2. More explicitly, for an arbitrary pair of cocycles and any non-negative integer, we construct a natural coboundary that descends to the associated instance of the Cartan formula. Our construction of Cartan coboundaries works for general algebras over the Barratt-Eccles operad, in particular, for the singular cochains of spaces, a case that has been used in condensed mater physics and for which we have developed open source software.

**Asaf Horev,Genuine Equivariant Factorization Homology:**

Factorization homology is an invariant of framed manifolds and E_n algebras, used for constructing quantum field theories. For one dimensional manifolds, this invariant recover topological Hochschild homology. This construction has an equivariant extension, where the manifold, the algebra and the result admit an action of a fixed finite group G. For one dimensional G-manifolds these invariants recover the ‘real’ and ‘twisted’ versions of topological Hochschild homology. In this talk we’ll review factorization homology, present its equivariant extension, and discuss E_V algebras. If time permits, we’ll discuss joint work with Inbar Klang and Foling Zou, including Equivariant non abelian Poincare duality, and some calculations of real and twisted topological Hochschild homology of Thom spectra.

**Gabe Angelini Knoll,****Morava K-theory of Algebraic K-theory and Topological Periodic Cyclic Homology:**

The chromatic red-shift program of Ausoni-Rognes suggests that algebraic K-theory shifts chromatic height by one. In my talk, I will describe a computational approach to this program where chromatic height is measured by vanishing of Morava K-theory. In particular, we see that the vanishing range of Morava K-theory of topological periodic cyclic homology of a certain family of Thom spectra y(n) increases by one. We also prove that algebraic K-theory preserves vanishing of Morava K-theory for y(n), a result recently proven in parallel by Land-Meier-Tamme by entirely different methods. Our theorem relies on a technical result about when commuting Morava K-theory with a sequential limit is possible, which I will discuss. As second application of this technical result, we prove a higher chromatic height analogue of Mitchell’s theorem for truncated Brown-Peterson spectra associated to a prime p and an integer n, which remains conditional for large primes p and integers n. This is based on joint work with J.D. Quigley and joint work with A. Salch.

**Mikala Ørsnes Jansen,****The Exit Path ∞-Category of the Reductive Borel-Serre Compactification:**

For neat arithmetic groups Γ ≤ SL_{n}(ℤ), the locally symmetric space X associated with Γ provides a nice model for the classifying space BΓ : it is a smooth manifold and thus allows for the study of the discrete group to move into the geometric realm. Unfortunately, X is very rarely compact. To remedy this, Borel and Serre in 1973 constructed a compact manifold with corners into which X embeds as the interior. It is now known as the *Borel-Serre compactification* of X, and it enabled Borel to calculate the ranks of the algebraic K-groups K_{i}(ℤ). For some purposes, however, the Borel-Serre compactification is “too big”. Motivated by an interest in L_{2}-cohomology, Zucker introduced another compactification of X in 1982, later coined the *reductive Borel-Serre compactification*. It is defined as a quotient of the Borel-Serre compactification and is no longer a manifold with corners. It does, however, come equipped with a natural stratification. We set out to understand this stratified space by determining its *exit path **∞-**category*. This is an analogue for stratified spaces of the fundamental ∞-groupoid for topological spaces: it provides information not only about the individual strata but also about how these strata are “glued” together. We show that the reductive Borel-Serre compactification is in some sense a K(π,1) of stratified spaces by showing that its exit path ∞-category is equivalent to the nerve of a 1-category. Moreover, some interesting questions arise when looking back at algebraic K-theory.

This is joint work with Dustin Clausen.

**Calista Bernard,****Twisted Dyer-Lashof Operations:**

In the 70s, Fred Cohen and Peter May gave a description of the mod $p$ homology of a free $E_n$ algebra in terms of certain homology operations, known as Dyer-Lashof operations, and the Browder bracket. These operations capture the failure of the $E_n$ multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen’s work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and explain computational results that show the existence of additional operations in the twisted case.

**Egbert Rijke,****The Long Exact Sequence of Higher Groups:**

An n-group in homotopy type theory is simply defined to be a pointed connected n-type. With this definition, an ordinary 1-group is a pointed connected 1-type. In other words, a 1-group is presented as its classifying space, and we think of higher groups as presented in that way too. The fundamental n-group of a type X at x is then simply the n-truncation of the connected component of X. We formulate a notion of n-exactness, and show that any fiber sequence F -> E -> B of pointed types induces a long n-exact sequence of homotopy n-groups.

**Gijs Heuts,****Koszul Duality of Algebras for Operads:**

Ginzburg-Kapranov and Getzler-Jones exhibited a duality between algebras for an operad O and coalgebras (with divided powers) for a “Koszul dual” cooperad BO, taking the form of an adjoint pair of functors between these categories. Instances of this duality include that between Lie algebras and cocommutative coalgebras, as in Quillen’s work on rational homotopy theory, and bar-cobar duality for associative (co)algebras, as in the work of Moore. I will review this formalism and discuss the following basic question: on what subcategories of O-algebras and BO-coalgebras does this duality adjunction restrict to an equivalence? I will discuss an answer to this question and explain the relation to a conjecture of Francis and Gaitsgory.

**Manuel Rivera,****The Coalgebra of Chains and the Fundamental Group:**

I will explain the sense in which the natural algebraic structure of the singular chains on a path-connected space determines its fundamental group. This is a conceptual observation which has several important consequences, one of them being the following extension of a classical theorem of Whitehead: a continuous map between path-connected pointed spaces is a weak homotopy equivalence if and only if the induced map between the differential graded coalgebras of singular chains is a cobar-quasi-isomorphism (i.e. a quasi-isomorphism after applying the cobar functor). Another consequence is the following statement over a field F of arbitrary characteristic: two path-connected pointed spaces X and Y are connected by a zig-zag of maps inducing isomorphisms on fundamental groups and on homology with coefficients in any local system if and only if the simplicial cocommutative coalgebras of chains FX and FY are cobar-quasi-isomorphic.

The main ingredients needed to formulate and prove these results are: 1) an extension of a classical theorem of Adams which relates the cobar construction to the based loop space of any path-connected space 2) the symmetry of the diagonal map and its chain approximations, and 3) a theorem of P. Goerss relating Bousfield localization and the simplicial coalgebra of chains.

**Niall Taggart,Orthogonal Calculus and R^n-Cubical Homotopy Theory:**

Goodwillie calculus has at its core, cubical homotopy theory, for example, a functor in Goodwillie calculus is linear if it sends homotopy pushouts to homotopy pullbacks. In this talk, I will describe the connection between the orthogonal calculus of Weiss, and R^n-cubical homotopy theory, which is the analogue of cubical homotopy theory, with the set of n elements replaced by the Euclidean space R^n.

**Ai Guan,****Nonconilpotent Koszul Duality:**

Koszul duality is a phenomenon appearing in many areas of mathematics, such as rational homotopy theory and deformation theory and representation theory. For differential graded (dg) algebras, it is often formulated as a Quillen equivalence between model categories of augmented dg algebras and conilpotent dg coalgebras, and their corresponding dg modules and comodules. In this talk we consider what happens when the conilpotence condition is removed; the result is an exotic model structure on dg modules that is “of second kind”, i.e. weak equivalences are finer than quasi-isomorphisms. This is joint work with Andrey Lazarev.

**Christy Hazel,****Equivariant Fundamental Classes in RO(C_2)-Graded Cohomology:**

Let C_2 denote the cyclic group of order two. Given a manifold with a C_2-action, we can consider its equivariant Bredon RO(C_2)-graded cohomology. In this talk, we give an overview of RO(C_2)-graded cohomology in constant Z/2 coefficients, and then explain how a version of the Thom isomorphism theorem in this setting can be used to develop a theory of fundamental classes for equivariant submanifolds. We illustrate how these classes can be used to understand the cohomology of any C_2-surface in constant Z/2 coefficients, including the ring structure.

**Irakli Patchkoria,****Balmer Spectrum of Derived Mackey Functors:**

For a finite group G, Kaledin’s category of derived G-Mackey functors can be considered to be an algebraic analog of the G-equivariant stable homotopy category and it is much easier to handle than the derived category of G-Mackey functors. In this talk we will compute the Balmer spectrum of the tensor triangulated category of derived G-Mackey functors for any finite group G. We will start by introducing Kaledin’s category. Then we will give several different descriptions of it in terms of stable infinity categories. Finally, we explain the computation of the Balmer spectrum. This is all joint with B. Sanders and C. Wimmer.

**Maru Sarazola,****Two Model Structures for Double Categories:**

In this talk, I will present recent joint work with Lyne Moser and Paula Verdugo, where we construct two model structures for (the category of) double categories. Unlike the model structures previously present in the literature, both of our model structures contain the homotopy theory of 2-categories, and moreover, they both induce Lack’s model structure on 2- categories, through some version of the horizontal embedding $H: 2Cat \to DblCat$.

After recalling some preliminaries on double categories, 2-categories, and Lack’s model structure on 2Cat, we will introduce our two model structures on DblCat together with their main features, and discuss the pros and cons of each of them.

**Lyne Moser,****A Double (∞,1)-Categorical Nerve for Double Categories:**

A 2-category can be seen as an internal category to categories with discrete category of objects, i.e., a horizontal double category with only trivial vertical morphisms. Some aspects of 2-category theory, such as 2-limits, benefit from a passage to double categories. Going to the ∞-world, we expect to have a similar picture, which would allow one to develop aspects of (∞,2)-category theory, such as (∞,2)-limits, using double (∞,1)-categories.

A double (∞,1)-category was defined by Haugseng as a Segal object in complete Segal spaces, and then an (∞,2)-category in the form of a 2-fold complete Segal space can be interpreted as a “horizontal” double (∞,1)-category. In this talk, I will consider a slightly modified version of these double (∞,1)-categories and give a nerve construction from double categories to double (∞,1)-categories. This nerve is right Quillen and homotopically fully faithful from the category of double categories endowed with a model structure constructed in a joint work with Maru Sarazola and Paula Verdugo. By restricting along a “homotopical” horizontal embedding of 2-categories into double categories, we get a nerve from 2-categories into 2-fold complete Segal spaces, which is also right Quillen and homotopically fully faithful. I will show that these nerves are further compatible in a precise sense with the horizontal embedding of 2-categories into double categories, and this says that the ∞-setting indeed extends the strict setting.

**Jocelyne Ishak,****Rigidity of the K(1)-Local Stable Homotopy Category:**

In some cases, it is sufficient to work in the homotopy category Ho(C) associated to a model category C, but looking at the homotopy level alone does not provide us with higher order structure information. Therefore, we investigate the question of rigidity: If we just had the structure of the homotopy category, how much of the underlying model structure can we recover? This question has been investigated during the last decade, and some examples have been studied, but there are still a lot of open questions regarding this subject. Starting with the stable homotopy category Ho(Sp), that is the homotopy category of spectra, it has been proved to be rigid by S. Schwede. Moreover, the E(1)-local stable homotopy category, for p=2, has been shown to be rigid by C. Roitzheim. In this talk, we investigate a new case of rigidity, which is the localisation of spectra with respect to the Morava K-theory K(1), at p=2.

**Katharine Adamyk,****A Classification of Q_0-local A(1)-Modules:**

This talk will present a classification theorem for modules over A(1), a subalgebra of the mod-2 Steenrod algebra. In order to give the module classification, there will be some background on Margolis homology, an invariant of modules over the Steenrod algebra, and the stable module category. Applications of the classification theorem to lifting A(1)-modules to modules over the Steenrod algebra and applications to the computation of certain localized Adams spectral sequences will also be discussed.