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.

## On the natural stratification of the space of differentiable functions and the space of Morse functions

Hybrid. Contact: Christian Kaiser (kaiser @ mpim-bonn.mpg.de)

## The Stark conjecture over the field of rational numbers: A new approach

Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)

We present a new proof for the algebraicity of the cyclotomic Stark units. These units are defined in a transcendental way, via the derivative at s=0 of a partial zeta function. The proof links these units to the special values of the Riemann zeta function at positive even integers. The rationality of these special values (modulo a power of pi) implies the algebraicity of the cyclotomic Stark units.

## The Nielsen realization for non-spin 4-manifolds

Contact: Aru Ray (aruray @ mpim-bonn.mpg.de)

## 4-genus bounds from the 10/8+4 theorem

Contact: Aru Ray (aruray @ mpim-bonn.mpg.de)

Donald and Vafaee described a way to use Furuta’s 10/8 theorem to obstruct sliceness in the 4-ball and Linh Truong used a refinement, the 10/8 +4 theorem of Hopkins-Lin-Shi and Hu, to strengthen this sliceness obstruction. We will show how to expand on this technique to obtain lower bounds for four-ball genus and present some calculations for satellite knots. This is joint work in progress with Sashka Kjuchukova and Linh Truong.

## Margulis Lemma and RCD spaces

Contact: Christian Kaiser (kaiser @ mpim-bonn.mpg.de)

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Contact: Aru Ray (aruray @ mpim-bonn.mpg.de), Steven Sivek (sivek @ mpim-bonn.mpg.de)

## Conley index theory and condensed sets

The Conley index is a spatial refinement of the Morse index. Informally speaking, it is a ‘space’ that describes the local dynamics around an isolated invariant subset of a topological dynamical system. In this talk, I will explain a new formulation of Conley index theory, which I think is simpler and more flexible than the traditional formulation. One important point is that the Conley index should be defined as a based equivariant condensed set/anima, not as a mere homotopy type of topological spaces.

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