Alternatively have a look at the program.

## Comparing models for $(\infty, n)$-categories

The various models for $(\infty, 1)$-categories each lead to a number of approaches to higher $(\infty, n)$-categories. In this talk we'll consider some of the generalizations of Segal categories and complete Segal spaces, as well as known and conjectured comparisons between them.

## Hermitian multiplicative infinite loop space machines

Multiplicative infinite loop space machines produce out of bimonoidal categories (or

infinity categories) E-infinity ring spectra. This procedure can be viewed as endowing

direct sum K-theory of symmetric monoidal (infinity) categories with a multiplication.

In this talk we will present a similar procedure for hermitian K-theory of infinity categories

with a notion of duality. We will show that any preadditive rigid symmetric monoidal infinity

category gives rise to a direct sum hermitian K-theory E-infinity spectrum. Examples will

## Homotopy locally presentable enriched categories

We will develop a homotopy theory of categories enriched in a monoidal model

category V. In particular, we will deal with homotopy weighted limits and colimits,

and homotopy local presentability. The main result, which was known for

simplicially-enriched categories, links homotopy locally presentable V-categories

with combinatorial model V-categories, in the case where all objects of V are cofibrant.

The talk reports the joint work wit S. Lack.

## Decomposition spaces: theory and applications

Decomposition spaces are simplicial $\infty$-groupoids with a certain exactness condition: they send generic (end--point preserving) and free (distance preserving) pushout squares in the simplicial category $\Delta$ to pullbacks. They encode the information needed for an 'objective' generalisation of the notion of incidence (co)algebra of a poset, and turn out to coincide with the unital 2-Segal spaces of Dyckerhoff and Kapranov.

We establish a general Möbius inversion principle, and construct the universal Möbius decomposition space.

## The Intricate Maze of Graph Complexess

In the paper ``Formal noncommutative symplectic geometry'', Maxim Kontsevich introduced three versions of cochain complexes $\mathcal{GC}_{\text{Com}}$, $\mathcal{GC}_{\text{Lie}}$ and $\mathcal{GC}_{\text{As}}$ ``assembled from'' graphs with some additional structures. The graph complex $\mathcal{GC}_{\text{Com}}$ (resp. $\mathcal{GC}_{\text{Lie}}$, $\mathcal{GC}_{\text{As}}$) is related to the operad $\text{Com}$ (resp. $\text{Lie}$, $\text{As}$) governing commutative (resp. Lie, associative) algebras.

## Feynman Categories

I will discuss how the notion of Feynman categories may be used to consolidate

and generalize familiar constructions and structures which arise when considering

generalizations of operads. These constructions include model structures,

bar/Feynman transforms, and master equations. Time permitting I will then

discuss how functors between Feynman categories intertwine these structures.

## Operadic and simplicial background of some classical Hopf algebras

The Hopf algebras of the title are the cobar contruction on a reduced simplicial set, with

its Hopf algebra structure discovered by Baues, the algebra of rooted forests of Connes

and Kreimer, and the algebra of multi zeta values of Goncharov. In this talk we present

an operadic (and essentially simplicial) construction that encompasses and unies all of

these examples, giving deeper insight into each of them. Indeed, any cooperad which

has a suitably compatible multiplication may be given a canonical (innitesimal) bialgebra

## Grothendieck duality via Hochschild homology

Hochschild cohomology was introduced in a 1945 paper by Hochschild, and Grothendieck

duality dates back to the early 1960s. The fact that the two have some relation with each

other is very new - it came up in papers by Avramov and Iyengar [2008], Avramov, Iyengar,

and Lipman [2010] and Avramov, Iyengar, Lipman and Nayak [2011]. We will review this

history, and the surprising formulas that come out.

We will then discuss more recent progress. The remarkable feature of all this is the role

## An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves

We will explain the construction of a non-Fourier Mukai functor between derived

categories of coherent sheaves on smooth project varieties. This is joint work with

Alice Rizzardo.

## BPS states on elliptic Calabi-Yau, Jacobi-forms and 6d theories

Using the holomorphic anomaly equation we prove that the all genus topological

string theory partition function on elliptic Calabi-Yau can be written in terms of

meromorphic Jacobi-Forms, where the elliptic argument is identified with the

genus counting parameter. We give strong evidence for an universal form of the

denominator with zero at the torsion points and argue that the numerator is a

weak Jacobi form. This gives strong all genus predictions in accordance with

algebraic geometry considerations. We show that if a 6d theory can be decoupled

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