## 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.

## 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)

## Margulis Lemma and RCD spaces

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

## tba

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

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

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

## tba

Contact: Nicolas Addington

## tba

Contact: Nicolas Addington

## tba

Contact: Nicolas Addington

## On opers and complex lagrangians

Contact: Christian Kaiser

In 2014 Gaiotto conjectured a correspondence between a holomorphic lagrangian (Hitchin section) in the Dolbeault moduli space of Higgs bundles on a curve and the holomorphic lagrangian of opers in the de Rham moduli space of holomorphic connections. This conjecture was established in 2016 for holomorphic opers with an arbitrary complex simple Lie group; the construction is known today as the conformal limit.

## Algebraic structures in symplectic geometry I

Contact: Christian Kaiser

## Algebraic structures in symplectic geometry II

Contact: Christian Kaiser

## On Chowla's non-vanishing conjecture over function fields

Chowla's conjecture postulates that the L-function of a quadratic character over Q should never vanish at s=1/2. At present we know that this holds for a positive proportion of characters, thanks to Soundararajan. We also know that over function fields Fq(T) the naive analogue of Chowla's conjecture is false: Li has constructed infinitely many quadratic characters with L-function vanishing at 1/2. The refined form of Chowla's conjecture postulates that the non-vanishing should hold with probability 1.

## The Kodaira classification of the moduli of hyperelliptic curves

We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces are much worst from the point of view of singularities than the moduli space of pointed stable curves M_{g,n}, but much better from the point of view of birational complexity. We provide a full Kadaira classification with the surprising feature that for n large fixed, the Kodaira dimension decreases as the genus grows. In other words, the space becomes "simpler" as the genus grows. Further, we study the polyhedrality of the effective cone.

## Course on slice knots and knot concordance

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

## Dirichlet eigenvalues of regular polygons

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

## tba

## Course on slice knots and knot concordance

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

## Strongly quasipositive knots are concordant to infinitely many strongly quasipositive knots

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

## Differentiation of higher groupoids in tangent categories

Hybrid.

Contact: Christian Kaiser

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