** Time: Mondays at 10:15**

**Location: MA A1 10**

### Program

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

23.09.2019 | Unstable v-periodic Homotopy Groups through Goodwillie Calculus_{1} |
Jens Kjaer EPFL |

30.09.2019 | Adjoint Functor Theorems for Infinity Categories | Hoang-Kim Nguyen Regensburg |

07.10.2019 | Modules over Algebraic Cobordism | Maria Yakerson Osnabrück |

21.10.2019 | Computing THH via Tensors of Higher Categories | Nima Rasekh EPFL |

28.10.2019 | Homotopy Theory for 2-Categories | Viktoriya Ozornova Bochum |

04.11.2019 | Higher Homotopy Categories, Higher Derivators, and K-Theory | George Raptis Regensburg |

11.11.2019 | Galois Symmetries on Knot Spaces | Pedro Boavida Lisbon |

18.11.2019 | The oo-Categorical Eckmann-Hilton Argument | Lior Yanofsky MPIM |

25.11.2019 | The Loday Construction on Hopf Algebras | Alice Hedenlund Oslo |

02.12.2019 | The Topological Hochschild Homology of Algebraic -theory of finite Fields |
Eva Höning Hamburg |

16.12.2019 | Reconstructing Tensor-Triangulated Categories | Scott Balchin Warwick |

### Abstracts

**Jens Kjaer,
Unstable v_{1}-periodic Homotopy Groups through Goodwillie Calculus:**

It is a classical result that the rational homotopy groups, $\pi_*(X) \otimes \mathbb{Q}$, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on $X$.

This result can also be recovered from applying Goodwillie calculus to rational homotopy theory.

A different simplification of the homotopy theory, is $v_h$-periodic homotopy theory. For $h = 1$ we are able to compute the K-theory based $v_1$-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute $v_1^{-1} \pi_* SU(d)$ in a very different way from the original computation by Davis.

**Hoang-Kim Nguyen,
Adjoint Functor Theorems for Infinity Categories:**

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this talk, I will explain infinity categorical generalizations of Freyd’s classical adjoint functor theorem. I will also discuss conditions when a functor between infinity categories, whose induced functor on homotopy categories admits an adjoint, admits an adjoint at the infinity categorical level.

**Maria Yakerson,
Modules over Algebraic Cobordism:**

When k is a field with resolution of singularities, it is known that Voevodsky’s category of motives DM(k) is equivalent to the category of modules over the motivic cohomology spectrum HZ. This means that a structure of an HZ-module on a motivic spectrum is equivalent to a structure of transfers in the sense of Voevodsky. In this talk, we will discuss an analogous result for modules over the algebraic cobordism spectrum MGL. Concretely, a structure of an MGL-module is equivalent to a structure of coherent transfers along finite syntomic maps, over arbitrary base scheme. Time permitting, we will see a generalization of this result to modules over other motivic Thom spectra, such as the algebraic special linear cobordism spectrum MSL. This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.

**Nima Rasekh**

**Computing THH via Tensors of Higher Categories:**

In this talk we will show how we can use the fact that presentable higher categories are enriched over spaces to give a simple way to compute THH of various spectra and in particular Thom spectra. This is joint work with Bruno Stonek and Gabriel Valenzuela.

**Viktoriya Ozornova
Homotopy Theory for 2-Categories:
**Grothendieck and Quillen introduced a notion of homotopy equivalences for categories using a by-now-standard tool called “nerve” of a category. This idea leads to various models of categories-up-to-homotopy. In the joint ongoing projects with Julie Bergner and Martina Rovelli, we study variants of the Roberts-Street-nerve for 2-categories and notions of homotopy equivalences arising from this nerve, with an eye towards 2-categories-up-to-homotopy

**Georgios Raptis**

**Higher Homotopy Categories, Higher Derivators, and K-theory:**

I will discuss the construction of the homotopy n-category and its properties, especially in connection with the existence of higher weak colimits. Inspired by this construction, I will introduce higher categorical generalizations of the notion of a (pre)derivator which take values in n-categories for any fixed n. These may be regarded as a sequence of approximations that bridge the gap between $\infty$-categories and derivators. Then I will define $K$-theory for higher homotopy categories and for higher derivators, and present some results on the comparison with Waldhausen $K$-theory.

**Pedro Boavida de Brito**

**Galois Symmetries on Knot Spaces:**

I’ll explain how the absolute Galois group of the rationals acts on a space which is closely related to the space of all knots. The path components of this space form a finitely generated abelian group which is, conjecturally, a universal receptacle for finite-type knot invariants. The added Galois symmetry allows us to extract new information about its homotopy and homology beyond characteristic zero. This is joint work with Geoffroy Horel.

**Lior Yanovski**

**The oo-Categorical Eckmann-Hilton Argument:**

The classical “Eckmann-Hilton argument” states that given a set with two unital binary operations that satisfy the interchange law, the two operations must coincide and moreover, this operation is associative and commutative. If we assume that both binary operations where associative to begin with, this result says that two interchanging commutative monoid structures on a set must coincide and be commutative. In this form, the Eckmann-Hilton argument has a higher homotopical generalization in terms of the “additivity theorem”. Namely, the Boardman-Vogt tensor product of the operads E_n and E_m is E_(n+m). In joint work with Tomer Schlank we give a (different) generalization of the non-associative Eckmann-Hilton argument in terms of a lower bound on the connectivity of the spaces of n-ary operations of the Boardman-Vogt tensor product of *any *two reduced oo-operads P and Q in terms of the connectivity of P and Q. In this talk, I will give a quick introduction to oo-operads and the Boardman-Vogt tensor product, state the main results and, if time permits, sketch the proof.

**Alice Hedenlund**

**The Loday Construction on Hopf Algebras:**

Topological Hochschild homology can be viewed as a special case of the more general Loday construction. This is known to not be a stable invariant using a counterexample by Dundas-Tenti employing the stably splitting of a torus into a wedge of spheres. However, while stability for the Loday construction does not hold in general, extra structure on the input ring spectrum can guarantee stability nonetheless. For example, Berest-Ramadoss-Yeung proved that stability holds for Hopf algebras by relating the Loday construction to representation homology. In this talk I will explain a direct categorical proof of this fact, which avoids representation homology, using the framework of infinity categories. The result is part of a “Women in Topology” project on the stability of the Loday construction together with S. Klanderman, A. Lindenstrauss, B. Richter, and F. Zou.

**Eva Höning**

**The Topological Hochschild Homology of Algebraic K-theory of finite Fields**

In this talk I will explain how to generalize a spectral sequence of Brun for the computation of topological Hochschild homology. As a first example we use the generalized Brun spectral sequence to give a simple computation of the mod $p$ and $v_1$ topological Hochschild homology of connective complex $K$-theory which has first been calculated by Ausoni using the Bökstedt spectral sequence. We then consider the generalized Brun spectral sequence for the topological Hochschild homology of algebraic $K$-theory of finite fields.

**Scott Balchin**

**Reconstructing Tensor-Triangulated Categories**

Tensor-triangulated categories naturally occur in various settings, for example, as the derived category of a commutative ring, the category of (possibly equivariant) cohomology theories, or the stable module category of a Frobenius ring. To any such category, one can assign the so-called Balmer spectrum, which is a categorification of the Zariski spectrum of a ring. I will report on joint work with J.P.C. Greenlees which provides a machinery to build up the unit object of the category from smaller building blocks. These building blocks are formed by taking localised completions of the unit at primes of the Balmer spectrum. Once one has this reconstruction result for the unit object, various models can be considered which fracture the category in question. Not only does this result give an insight into the structure of the category, but it usually provides a convenient setting to do calculations in.