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.

## "What is...?" seminar

## Homotopy versus Isotopy: Spheres in 5-manifolds, I

## Homotopy versus Isotopy: Spheres in 5-manifolds, II

## Rationality of Fano 3-folds over nonclosed fields

The rationality problem for smooth Fano threefolds over algebraically closed fields is basically solved.

In this talk I will discuss rationality of forms of these Fanos over nonclosed fields of characteristic 0.

I will concentrate on the case where the Picard number equals 1.

The talk is based on joint work with Alexander Kuznetsov (in progress).

## An introduction to quantum computing and quantum error correction (with a connection to Howe duality)

Recently, Gurevitch and Howe have associated a notion of "rank" to representations of the symplectic group, and showed that "highest rank" representations satisfy a form of Howe duality over finite fields (c.f. talk on the 20th). Montealegre-Mora and me then realized that the rank-deficient reps occurring in this context can be characterized in terms of certain quantum error correcting codes. The purpose of this talk is to explain the background of this development, i.e. why physicists care about these objects in the first place.

## Chebyshev’s bias for products of k primes and Greg Martin’s conjecture

For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $\omega(n)=k$ or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes.

## Puzzles about trisections of 4-manifolds

A trisection of a smooth 4-manifold is a very natural kind of decomposition into three elementary pieces which I will describe. Trisections are a natural 4-dimensional analogue of Heegaard splittings of 3-manifolds, a class of decompositions into two pieces that have yielded tremendous insight into 3-dimensional topology, so the philosophy is that trisections should give a way to port 3-dimensional techniques, questions and results to dimension four.

## Rational points on quartic del Pezzo surfaces with a conic bundle structure

There are three possibilities for the quotient of the Brauer group of X modulo constants when X is a del Pezzo surface of degree four over the rational numbers . In this talk we will explain how often each of them occurs when X ranges across a family of quartic del Pezzo surfaces equipped with a conic bundle structure. We will also give an explicit description of the generators of this quotient which allows us to calculate the frequency of such surfaces violating the Hasse principle. This talk is based on a joint work in progress with Cecília Salgado.

## tba

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