Abstracts of upcoming talks at the MPIM. For an overview see also the calendar.

## Recent developments in Quantum Topology -- Cancelled --

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

## Universal quadratic forms over number fields

I will talk about several recent results on universal quadratic forms over rings of integers of totally real number fields (i.e., totally positive quadratic forms that represent all totally positive integers). Over real quadratic fields, one can obtain a fairly precise information concerning the smallest rank of a universal form in terms of the associated continued fraction; in particular, the rank can be arbitrarily large.

## Free subgroups of Kleinian groups

We show that any cocompact Kleinian group $\Gamma$ has an exhaustive filtration by normal subgroups $\Gamma_i$ such that any subgroup of $\Gamma_i$ generated by $k_i$ elements is free, where $k_i \ge [\Gamma:\Gamma_i]^C$ and $C = C(\Gamma) > 0$. Together with this result we prove that $\log k_i \ge C_1 \mathrm{sys}_1(M_i)$, where $\mathrm{sys}_1(M_i)$ denotes the systole of $M_i$, thus providing a large set of new examples for a conjecture of Gromov. In the second theorem $C_1> 0$ is an absolute constant.

## Absolute algebraic geometry using gamma sets, after A.Connes and C.Consani

This is an expository talk on Connes-Consani's theory of geometry over field with one element. It is based on Segal's theory of gamma sets (see arXiv:1502.05585 [math.AG]). I will explain the background, the basic constructions and some interesting examples.

## The icosahedron, the Rogers-Ramanujan identities, and beyond

The first part of this lecture series is meant to illustrate the dictum that all suÿciently beautiful mathematical objects are connected. The two objects we choose to illustrate this are the icosahedron, the most subtle of the Platonic solids, and the Rogers-Ramanujan identities, often considered the most beautiful formulas in all of mathematics.

## Weierstrass mock modular forms and certain VOAs

Using techniques from the theory of mock modular forms and harmonic Maaß forms, especially Weierstrass mock modular forms,

we establish several dimension formulas for certain holomorphic, strongly rational vertex operator algebras, complementing previous

work by van Ekeren, Möller, and Scheithauer. As an application, we show that certain special values of the completed Weierstrass zeta

function are rational. This talk is based on joint work with Lea Beneish.

## Multiple Zeta Values and Their Extended Family

A multiple zeta value (MZV) is a real number specified by a string of positive integers.

The sum of the string is the weight, and its length is the depth. MZVs of depth one are

just values of the Riemann zeta function at positive integers; MZVs of depth two were

already studied by Euler. MZVs of arbitrary depth became a hot topic in the early 1990s

with their simultaneous appearance in perturbative quantum field theory and knot

theory. Since then the field has continued to develop rapidly. I will describe various

## Foliated Open Books -- or, How I Learned to Stop Worrying About How to Glue Open Books, I

In this talk, I'll discuss recent joint work with Vera Vertesi to develop a new version of an open book decomposition for contact three-manifolds. Our foliated open books join a zoo of existing types of open books, so I'll focus on the senses in which ours is a natural construction which provides a user-friendly approach to cutting and gluing.

## Foliated Open Books -- or, How I Learned to Stop Worrying About How to Glue Open Books, II

In this talk, I'll discuss recent joint work with Vera Vertesi to develop a new version of an open book decomposition for contact three-manifolds. Our foliated open books join a zoo of existing types of open books, so I'll focus on the senses in which ours is a natural construction which provides a user-friendly approach to cutting and gluing.

## On the GCD of shifted polynomial powers, iterations and their relatives

Let $a,b$ be multiplicatively independent positive integers and $\varepsilon>0$. Bugeaud, Corvaja and Zannier (2003)

proved that $\gcd(a^n-1,b^n-1)\le \exp(\varepsilon n)$ for sufficiently large $n$. Ailon and Rudnick (2004) considered

the function field analogue and proved a much stronger result. In this talk we present several extensions of the result of

Ailon and Rudnick. We also look at some gcd problems for linear recurrence sequences, posing some open questions,

and if time allows on compositional iterates of univariate polynomials.

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