## tba

## Schinzel's Hypothesis with probability 1 and rational points on varieties in families

We prove that Schinzel's Hypothesis (H) holds for 100% of polynomials of any fixed degree. I will discuss the proof of this result and also explain how to deduce from this the Hasse principle for certain surfaces.

## Differential graded Lie algebras and stability problems in geometry

Given a Lie algebroid structure on a vector bundle $A$ and a leaf $L$ of this Lie algebroid, a natural question is whether all nearby Lie algebroid structures on A have a leaf which is diffeomorphic to $L$. This question was answered by M. Crainic and R. Fernandes. In this talk, I will show that this question is an instance of a general question about differential graded Lie algebras: Given a differential graded Lie algebra $\mathfrak{g}$, a differential graded Lie subalgebra $\mathfrak{h}$, and a Maurer-Cartan element $Q$ of the subalgebra, are all Maurer-Cartan elements of $\mathfrak{g}$ near $Q$ gauge equivalent to an element of $\mathfrak{h}$? In the case that $\mathfrak{h}$ has finite codimension in $\mathfrak{g}$, I will give a sufficient criterium for a positive answer to the question. As a consequence, I will mention some stability results for zero-dimensional leaves of various geometric structures that can be obtained from this.

## Euler characteristics and 4-manifolds

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

We consider the Euler characteristics of closed orientable 2n-manifolds with a given fundamental group G, and highly-connected universal cover. We strengthen the 4-dimensional Hausmann-Weinberger estimates and extend to higher dimensions. As an application we obtain new restrictions for non-abelian finite groups arising as fundamental groups of rational homology 4-spheres.

## Low-dimensional topology seminar

Contact: Aru Ray

## Group actions on manifolds, rigidity and local rigidity

[MPI-Oberseminar currently in-person only]

One classical approach to study manifolds and groups is studying their interactions. The basic question whether a certain group can act on a certain manifold was looked at, and is the main theme of Zimmer's program. Depending on the context of the manifolds, the class of groups, and the types of actions, many results have been found over the last several decades. The talk will give an overview from old results, even before Zimmer proposed his program, to recent developments. We then focus on some recent local rigidity phenomena discovered in the settings of actions with low regularity.

## Euler sums and the cyclotomic Grothendieck-Teichmüller group

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

Euler sums are real numbers defined by iterated integrals on the projective line minus $0,\infty,1,-1$. In this talk, we introduce a family of linear relations among Euler sums which exhausts all motivic linear relations. This gives an explicit description of the level two motivic Galois group. We also show that the level two motivic Galois group coincides with the cyclotomic Grothendieck-Teichmüller group introduce by Benjamin Enriquez. Some part of this talk is based on a joint work with Nobuo Sato.

## Complexity of h-cobordisms

In-person only.

Two n-manifolds X_0,X_1 are h-cobordant if exists an n+1 dimensional manifold whose boundary is the union of X_0 (with opposite orientation) and X_1, such that the inclusion maps are homotopy equivalences. Diffeomorphic manifolds are clearly h-cobordant and Smale’s celebrated h-cobordism theorem states that the converse holds when n>= 5 and X_i is simply connected. On the other hand, Donaldson showed that this is not true for n=4. Still, it is possible to define a notion of complexity of an h-cobordism to measure the failure of it from being trivial. This notion has relations to embedded surfaces in the X_i and to their automorphism group. I will talk about some old and recent result about complexity of h-cobordisms of dimension (4+1).

## Conic fibrations over elliptic curves

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

A theorem of Serre states that almost all plane conics over Q have no rational point. We prove an analogue of this for a family of conics parametrised by certain elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve. Another way to think about this result is: we show that 0% of points on elliptic curves have a denominator which is a sum of two squares.

## Wondering what Kurt Heegner had in mind.

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

Reporting on the current status of our book project about Heegner with Samuel J. Patterson, we will briefly go through biographical elements, before zooming in on the overall mathematical program that Kurt Heegner was apparently following from the 1930s to the 1950s.

## From quantum toroidal algebras to wreath Macdonald operators

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

In this talk, I will explain how quantum toroidal algebras lead to a novel family of commuting difference operators whose joint eigenfunctions are the wreath Macdonald polynomials. The latter polynomials arise from the study of symplectic quotient singularities involving the wreath products of symmetric groups with an arbitrary finite cyclic group. When the cyclic group is trivial, one recovers the usual (type A) Macdonald difference operators and polynomials, whose theory has been extensively developed. In contrast, very little is known about wreath Macdonald polynomials in general. Our 'wreath Macdonald operators' provide a new, more direct characterization of wreath Macdonald polynomials for arbitrary finite cyclic groups. This is joint work with Mark Shimozono and Joshua Wen.

## WKB-analysis of nonabelian Hodge theory

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

We discuss the P=W (perversity equals weight) conjecture in non-abelian Hodge theory (recently proven by Maulik--Shen and Hausel--Mellit--Minets--Schiffmann), and our Geometric approach to attacking it that could be turned to an actual proof in several particular cases. The idea is to use recent advances on the large-scale asymptotic behaviour of solutions to Hitchin's equations due to T.Mochizuki and R.Mazzeo et al. Time permitting, relationship to the Gaiotto--Moore--Neitzke conjecture and the Hitchin WKB-problem will also be mentioned.

## A higher order Levin-Fainleib theorem

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

Landau was the first to obtain an asymptotic formula for the number of integers up to a given number that are sum of two coprime squares, where he used analytic method. Later, this procedure was further developed by Delange and Selberg allowing them to obtain asymptotic for partial sums of arithmetic functions whose Dirichlet series can be written in terms of complex powers of the Riemann zeta-function. In 1967 Levin and Fainleib using differential equation established the logarithmic density of the same set by an elementary argument under more general conditions. We generalise the Levin and Fainleib approach and allow more general hypotheses as well. When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for the mean value of f(n)/n when summed over n <x provided that we have the mean value over primes, namely, f(p)(logp)/p summed over p < Q.

This is a joint work with Olivier Ramare and Rita Sharma.

## Talk 8: Cohomology of Shimura varieties via the Hodge-Tate period map

Recently, a number of results have been obtained on the cohomology of Shimura varieties by studying it via a Leray spectral sequence along the Hodge-Tate period map. I will survey some of these results (some joint with Ana Caraiani, some by others) and stress in particular how they enable a study of the cohomology of the boundary. This relates to a long-standing theme of Harder's work on congruences between Eisenstein series and cusp forms, and has yielded substantial progress on the Langlands program for non-algebraic arithmetic locally symmetric spaces. If time permits, I may close with a note on another theme of this conference, motives.

## Talk 6: Joint equidistribution: arithmetic and L-functions

A classical set of equidistribution problems concerns torus orbits of growing discriminants inside projective units of quaternion algebras. Examples include Heegner points or closed geodesics on the modular surface, integral points on ellipsoids, and supersingular reduction of CM elliptic curves. A new generation of equidistribution problems, motivated by the mixing conjecture of Michel and Venkatesh, considers torus orbits inside pairs of quaternion algebras which in special cases has a rather concrete arithmetic description. I will show how automorphic forms and L-functions shed light on such questions.

## Talk 5: On the local Langlands parametrization for $G(F_q((t)))$

I will report on several recent results on the Langlands parametrization of irreducible representations of groups over local fields of positive characteristic. The parametrization is defined algebraically in the work of V. Lafforgue, Genestier-Lafforgue, and Fargues-Scholze, using variants of the theory of moduli of vector bundles on curves; however, the results I will discuss are primarily obtained by analytic methods. I will specifically report on results on ramification of local parameters (joint with Gan and Sawin), on the generalized Ramanujan conjecture (joint with Ciubotaru), and on joint work in progress with Beuzart-Plessis and Thorne on a strategy to invert the parametrization.

## Talk 2: Modular forms of weight one, motivic cohomology and the Jacquet-Langlands correspondence

I will discuss some joint work (in progress) with Ichino. In a previous paper, we showed that the Jacquet-Langlands correspondence for cohomological Hilbert modular forms preserves rational Hodge structures, as predicted by the Tate conjecture. In this talk, I will discuss a related result in the case of weight one forms. Since weight one forms are not cohomological, the Tate conjecture does not apply and thus it is not at all obvious what the content of such a result should be. I will motivate and explain the statement, which is suggested by another recent development, namely the conjectural connection between motivic cohomology and the cohomology of locally symmetric spaces.

## Talk 4: Finding proper moduli spaces

The Harder-Narasimhan stratification explained how to cut out subspaces of the moduli problem of principal bundles on curves that admit a proper moduli space. This prompted many results of similar nature and by now we know geometric criteria that decide whether an open substack of a moduli problem admits a proper quotient. In many examples these are in the end determined by the datum of a line bundle or a cohomology class in degree 2 of the large moduli problem. In the simple example of quotient stacks obtained from a torus action there is a combinatorial conjecture by Bialynicki-Birula and Sommese indicating that there should be other methods to cut out open subsets admitting proper quotients. I would like to explain how this can be understood by looking at other cohomological degrees.

## Short talk: Restriction problems of Hilbert theta series

We consider the problem of whether the diagonal restrictions of a Hilbert theta series over a totally real number field span the whole space of elliptic modular forms. We transfer this problem to definite quaternionic Hilbert modular forms via theta correspondences and get the surjectivity of the restriction map by applying equidistribution properties of unipotent orbits in SL(2) as well as Jacquet-Langlands correspondences.

## Short talk: Motivic cohomology of Carlitz twists

The torsion in the algebraic K-groups of the ring of integers is related to the denominator of Bernoulli numbers. In this short talk we discuss the function field counterpart of this. While there is no known analogue of a function field valued K-theory, one can still define Ext modules among analogues of Tate twists — called Carlitz twists — in an avatar of the category of mixed motives in this context, namely that of Anderson t-motives. We explain how the torsion of those modules are related to Bernoulli-Carlitz numbers. This is joint work with Andreas Maurischat.

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