## Workshop on "$\infty$-categories and their applications", August 17 - 21, 2020

The quest for solid foundations for $\infty$-categories started several decades ago, but from the mid-2000s there have been spectacular developments which excitingly accelerated the pace of progress. This workshop will focus on these objects, both in their foundations and in the applications to different fields of mathematics. There will be minicourses on $\infty$-categories, and also on $\infty$-operads, on algebraic K-theory and on parametrized homotopy theory, all three of which have benefited substantially from the $\infty$-categorical viewpoint. There will be talks by invited speakers working on the field, as well as shorter contributed talks by younger participants.

We think this workshop will prove interesting for researchers working in the area, for mathematicians in related fields, for seasoned homotopy theorists who want to learn the new $\infty$-categorical approaches to the field, and more generally for everybody interested in new developments

with $\infty$-categories.

#### Speakers for minicourses

Bergner, Julie* (University of Virginia)

Haugseng, Rune (Norwegian University of Science and Technology)

Nikolaus, Thomas (WWU Münster)

Barwick, Clark (University of Edinburgh)

* to be confirmed

#### Speakers

Hackney, Philip (University of Louisiana at Lafayette)

Ozornova, Viktoriya (Ruhr-Universität Bochum)

Spitzweck, Markus (Universität Osnabrück)

Riehl, Emily* (John Hopkins University)

Gepner, David* (Purdue University)

Gutierrez, Javier (Universitat de Barcelona)

Scherotzke, Sarah* (University of Luxembourg)

Harpaz, Yonatan* (Université de Paris 13)

* to be confirmed

#### Registration

To register please use the registration webform. The registration will not be closed before August 2, 2020. There is no registration fee.

#### Financial support

Limited financial support is available. If you need support, please add a request on your registration webform with a brief justification. Please note that the workshop is relatively local in scope, so that we might be able to cover your accommodation but probably not your travel expenses. The Deadline is **May 31, 2020**.

#### Contributes Talk

If you would like to give a contributed talk, please indicate so on our registration webform by **May 31, 2020**.

#### Hotel reservations

If you would like us to make a hotel reservation for you, please state this on the registration webform. We can only honor requests for hotel reservations made by **July 15, 2020**. If you would like to make a hotel reservation by yourself you will find a list of hotels here.

#### Practical information

For practical information such as how to get to the institute and a list of hotels see here.

#### Contact

If you have any questions, please feel free to contact catsandapps@mpim-bonn.mpg.de.

## Some faces of the Poincaré homology sphere

## IMPRS seminar on various topics: Knot theory

## Higher Geometric Structures along the Lower Rhine XIV, March 12 - 13, 2020

This is the eleventh of a series of short workshops jointly organized by the Geometry/Topology groups in Bonn, Nijmegen, and Utrecht, all situated along the Lower Rhine. The focus lies on the development and application of new structures in geometry and topology such as Lie groupoids, differentiable stacks, Lie algebroids, generalized complex geometry, topological quantum field theories, higher categories, homotopy algebraic structures, higher operads, derived categories, and related topics.

Webpages of the previous meetings: I (Bonn), II (Nijmegen), III (Utrecht), IV (Bonn), V (Nijmegen), VI (Utrecht), VII (Leuven), VIII (Bonn), IX (Nijmegen), X (Utrecht), XI (Bonn), XII (Nijmegen), **XIII(Utrecht) **

#### Speakers

tba.

#### Program

Here are the links to the program and the abstracts as soon as they become available.

#### Registration

To register please use the registration webform. There is no registration fee. For any further inquiries please send an email to LowerRhineXIV@mpim-bonn.mpg.de.

#### Financial support

Limited financial support is available. If you need support, please add a request on your registration webform with a brief justification. Please note that the workshop is relatively local in scope, so that we might be able to cover your accommodation but probably not your travel expenses. The Deadline is **February 16, 2020**.

#### Conference dinner

The conference dinner will be on Thursday evening at 18:00h at the institute.

#### Practical information

For practical information such as how to get to the institute and a list of hotels see here.

## tba

## New guests at the MPIM

Wouter Van Limbeek: Symmetry and self-similarity in geometry**Abstract:** In 1893, Hurwitz showed that a Riemann surface of genus $g \geq 2$ admits at most $84(g-1)$ automorphisms; equivalently, any 2-dimensional hyperbolic orbifold $X$ has $Area(X)\geq \pi / 42$. In contrast, such a lower bound on volume fails for the n-dimensional torus $T^n$, which is closely related to the fact that $T^n$ covers itself nontrivially. Which geometries admit bounds as above? Which manifolds cover themselves? In the last decade, more than 100 years after Hurwitz, powerful tools have been developed from the simultaneous study of symmetries of all covers of a given manifold, tying together Lie groups, their lattices, and their appearances in differential geometry. In this talk I will explain some of these recent ideas and how they lead to progress on the above (and other) questions.

## Symplectic bordism: something old and something new, II

I will begin by reviewing the history of the stable homotopy theory approach to understanding (co)bordism groups via the Pontrjagin-Thom isomorphism and analysis of the homotopy type of the associated Thom spectra. This proved remarkably successful and led to the determination of many important classical examples of bordism groups such as unoriented, oriented, unitary, special unitary, spin and spin^c. The two outstanding classical cases which are not completely understood are framed bordism (aka stable homotopy groups of spheres) and symplectic bordism. More recently attention has turned to String bordism and its variants.

## Symplectic bordism: something old and something new

I will begin by reviewing the history of the stable homotopy theory approach to understanding (co)bordism groups via the Pontrjagin-Thom isomorphism and analysis of the homotopy type of the associated Thom spectra. This proved remarkably successful and led to the determination of many important classical examples of bordism groups such as unoriented, oriented, unitary, special unitary, spin and spin^c. The two outstanding classical cases which are not completely understood are framed bordism (aka stable homotopy groups of spheres) and symplectic bordism. More recently attention has turned to String bordism and its variants.

## The Hitchin System

Using the Dolbeault picture of the moduli space of Higgs bundles, we will construct a function from the moduli space to an affine space.This function will be the Hamiltonian of an integrable system called the Hitchin system. This by definition gives a foliation of the moduli space generically by open subsets of abelian varieties. Time permitting, we will see some applications and how this leads to Higgs bundles offering new, and some times unexpected, geometric insight.

## Quantum group symmetry in CFT

I discuss applications of a hidden $U_q({sl}_2)$ symmetry in CFT with central charge $c \leq 1$ (focusing on the generic, semisimple case, withcirrational). This symmetry provides a systematic method for solving Belavin-Polyakov-Zamolodchikov PDE systems, and in partic-ular for explicit calculation of the asymptotics and monodromy properties of the solutions.Using a quantum Schur-Weyl duality, one can understand solution spaces of such PDE systems in a detailed way. The solutions, in turn, are useful both for CFT questions and forrigorous understanding of the connections of 2D CFT with critical models of statistical physics.

## Powers of the Dedekind eta function and the Bessenrodt-Ono inequality

In this talk I present recent results obtained with Markus Neuhauser towards the non-vanishing of the coefficients of the Dedekind eta function in the spirit of G.-C. Rota. This includes Serre’s table, pentagonal numbers, results of Kostant in the context of simple affine Lie algebras and the Lehmer conjecture. In the second part I will talk about partition numbers and the Bessenrodt-Ono inequality.

## Talk coaching for Postdocs

## 1. Singular Hodge theory of matroids; 2. Logarithmic concavity of weight multiplicities for irreducible sln(C)-representations

Talk 1: Title: Singular Hodge theory of matroids

If you take a collection of planes in R

3, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the “Top-Heavy Conjecture”, that Dowling and Wilson conjectured in 1974.On the other hand, given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan–Lusztig polynomial) to it.

The problems of proving the “Top-Heavy Conjecture” and the non-negativity of the coefficients of these Kazhdan–Lusztig polynomials are related, and they are controlled by the Hodge theory of a certain singular projective variety. The “Top-Heavy Conjecture” was proven for hyperplane arrangements by Huh and Wang in 2017, and the non-negativity was proven by Elias, Proudfoot, and Wakefield in 2016. I will discuss work, joint with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang, on these two problems for arbitrary

matroids.

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Talk 2: Logarithmic concavity of weight multiplicities for irreducible sln(C)-representations

Log-concavity is a combinatorial property that can sometimes hint toward an underlying Hodge-theoretic structure. We will show that the sequence of weight multiplicities we encounter is log-concave, as we walk along any root direction in the weight diagram of a finite-dimensional irreducible representation of sln(C). Along the way, we prove both a continuous and discrete kind of log-concavity for Schur polynomials, and we conjecture that many related polynomials in algebraic combinatorics also exhibit log-concavity phenomena. As a consequence of our results, we obtain a special case of Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients. This is joint work with June Huh, Karola Meszaros, and Avery St. Dizier.

## Higgs bundles on Riemann surfaces, II

On a Riemann Surface $\Sigma$, the moduli space of polystable $\mathrm{SL}_n(\mathbb{C}$)-Higgs bundles can be identified with the space of reductive representations $\pi _1 (\Sigma) \to \mathrm{SL}_n(\mathbb{C})$. In this talk, we discuss a proof of this so called non-abelian Hodge correspondence. Our goal is to understand how

to construct a Higgs bundle from a given representation and how this construction relates to the theory of harmonic maps.

## -- Cancelled -- Powers of the Dedekind eta function and the Bessenrodt-Ono inequality

In this talk I present recent results obtained with Markus Neuhauser towards the non-vanishing of the coefficients of the Dedekind eta function in the spirit of G.-C. Rota. This includes Serre’s table, pentagonal numbers, results of Kostant in the context of simple affine Lie algebras and the Lehmer conjecture. In the second part I will talk about partition numbers and the Bessenrodt-Ono inequality.

## Lower Bounds for Discrete Negative Moments of the Riemann zeta Function

I will talk about lower bounds for the discrete negative 2k-th moment of the derivative of the Riemann zeta function for all

fractional k > 0. The bounds are in line with a conjecture of Gonek and Hejhal. This is a joint work with Winston Heap and

Junxian Li.

## Estimates for binary quadratic forms and Apollonian circle packings

Given a positive definite integral binary quadratic form, it is a classical problem in number theory to count the integers that are represented by this form. A modern treatment was given in 2006 by Valentin Blomer and Andrew Granville.

This talk will present a way of extending a theorem of Blomer and Granville to obtain estimates for counting proper representations uniform in the (possibly non-fundamental) discriminant. Subsequently, I will give a sketch of how these estimates were used by Jean Bourgain and Elena Fuchs (2011) in proving the positive density conjecture for Apollonian circle packings.

## Some Feynman diagrams in pure algebra

I will explain how the computation of compositions of maps of a certain natural class, from one polynomial ring into another, naturally leads to a certain composition operation of quadratics and to Feynman diagrams. I will also explain, with very little detail, how this is used in the construction of some very well-behaved poly-time computable knot polynomials.

Joint work with Roland van der Veen.

See also: http://www.math.toronto.edu/~drorbn/Talks/Bonn-200120/

## Algebraic knot theory

This will be a very "light" talk: I will explain why about 13 years ago, in order to have a say on some problems in knot theory, I've set out to find tangle invariants with some nice compositional properties. In my second talk in Bonn I will explain how such invariants were found - though they are yet to be explored and utilized.

See also: http://www.math.toronto.edu/~drorbn/Talks/Bonn-200120/

## Some adjoint L-values and Hilbert modular Eisenstein congruences

I will start from the situation of a cuspidal Hecke eigenform f of real quadratic character, congruent to its complex conjugate modulo a prime P ramified in the coefficient field.

Following Hida, P then divides an appropriately normalised near-central critical value of the adjoint L-function of f. In fact the same is true of the other critical values. The Bloch-Kato conjecture then predicts non-zero elements of P torsion in Selmer groups, which can be constructed using experimentally-observed congruences involving non-parallel weight, level one Hilbert modular forms. Using p-adic deformation in Hida's nearly ordinary family, these appear to be equivalent to congruences between certain parallel-weight (but level p, non-trivial character) Hilbert modular cusp forms and Eisenstein series.

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