# Abstracts for Closing Conference for the Program on Higher Structures in Geometry and Physics

Alternatively have a look at the program.

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

Posted in
Speaker:
Julie Bergner
Zugehörigkeit:
UC Riverside
Datum:
Fre, 2016-03-18 10:00 - 11:00
Location:
MPIM Lecture Hall

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

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Speaker:
Markus Spitzweck
Zugehörigkeit:
Osnabrück
Datum:
Fre, 2016-03-18 11:15 - 12:15
Location:
MPIM Lecture Hall

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

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Speaker:
Jirí Rosický
Zugehörigkeit:
Brno
Datum:
Fre, 2016-03-18 14:00 - 15:00
Location:
MPIM Lecture Hall

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

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Speaker:
Imma Gálvez Carrillo
Zugehörigkeit:
U Politècnica de Catalunya
Datum:
Fre, 2016-03-18 15:30 - 16:30
Location:
MPIM Lecture Hall

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

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Speaker:
Vasily Dolgushev
Zugehörigkeit:
Temple University
Datum:
Sam, 2016-03-19 10:00 - 11:00
Location:
MPIM Lecture Hall

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

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Speaker:
Benjamin Ward
Zugehörigkeit:
Simons Center Stony Brook
Datum:
Sam, 2016-03-19 11:15 - 12:15
Location:
MPIM Lecture Hall

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

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Speaker:
Andrew Tonks
Zugehörigkeit:
University of Leicester
Datum:
Sam, 2016-03-19 13:45 - 14:45
Location:
MPIM Lecture Hall

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 uni es 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 (in nitesimal) bialgebra

## Grothendieck duality via Hochschild homology

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Speaker:
Amnon Neeman
Zugehörigkeit:
Australian National University, Canberra
Datum:
Sam, 2016-03-19 15:00 - 16:00
Location:
MPIM Lecture Hall

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

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Speaker:
Michel van den Bergh
Zugehörigkeit:
Universiteit Hasselt
Datum:
Sam, 2016-03-19 16:30 - 17:30
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Albrecht Klemm
Zugehörigkeit:
University of Bonn
Datum:
Son, 2016-03-20 10:00 - 11:00
Location:
MPIM Lecture Hall

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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