Paderborn, 9-13 September 2024

Emanuele Macrì: Hyper-Kähler manifolds and Lagrangian fibrations
A hyper-Kähler manifold is a complex Kähler manifold that is simply connected, compact, and has a unique holomorphic symplectic form, up to constants. This important class of manifolds has been studied in the past in many contexts, from an arithmetic, algebraic, geometric point of view, and in applications to physics and dynamics.
The theory in dimension two, namely K3 surfaces, is well understood. The aim of this talk is to give an introduction to the theory of hyper-Kähler manifolds in higher dimension, from a point of view of their classification; in particular, about existence of Lagrangian fibrations. We will present some results in dimension four, obtained in collaboration with Olivier Debarre, Daniel Huybrechts and Claire Voisin.

Gunter Malle: Modular representations of finite reductive groups
We will give an overview of current problems and recent results in the representation theory of finite groups of Lie type in non-defining characteristic. These concern the block theory, the shape and precise form of decomposition matrices as well as the application to various conjectures in the field.

Amnon Neeman: Excellent metrics on triangulated categories
The talk will expose recent results, still in progress, which explore the right restrictions to impose on metrics on triangulated categories for them to be particularly well-behaved.
We will begin with a historical survey: Rickard proved, back in 1989, that a pair of coherent rings $R$ and $S$ have the property that $D^b(\text{proj } R)=D^b(\text{proj } S)$ if and only if $D^b(\text{mod } R)=D^b(\text{mod } S)$. Krause asked, in 2018, if there is recipe that constructs $D^b(\text{mod } R)$ out of $D^b(\text{proj } R)$. I will briefly review the work that was done in 2018 - including a paper by myself, introducing good metrics on triangulated categories.
And then we will fast-forward, explaining the recent progress and its motivation.

Alexander Polishchuk: Schwartz spaces associated with the stack of rank $2$ bundles on a curve over a local field
This talk is based on joint works with Alexander Braverman and David Kazhdan. The motivation for this project comes from the Analytic Langlands program, which is an analog of the usual Langlands correspondence involving a curve over a local field. We study the natural Schwartz space of half-densities associated with the stack of rank $2$ bundles on such a curve. The long term goal is to understand the action of Hecke operators on this space. For this it is useful to understand the behavior of elements of this space near stable bundles. Our first result is that a Schwartz half-density on the stack of rank $2$ bundles defines a smooth (locally constant in the non-archimedean case) half-density on the open subscheme of very stable bundles, i.e., bundles without nonzero nilpotent Higgs fields. We then formulate a set of algebro-geometric conjectures (that we can check for low genus) that imply boundedness of these half-densities near stable bundles.

Dipendra Prasad: Homological aspects of the branching problem: GGP for $p$-adics.
Branching laws for (complex) representations of groups describe how a representation of a group $G$ decomposes when restricted to a subgroup $H$ which in the case of compact groups is a direct sum of irreducible representations. When dealing with infinite dimensional representations of a $p$-adic group $G$ restricted to a subgroup $H$, the branching laws, as they are usually studied in the subject — and make up a large body of works in the subject — describe irreducible representations of $H$ which arise as a quotient representation. It is not clear if this allows one to describe how an irreducible representation of $G$ restricted to $H$ looks like. Homological algebra methods suggest thinking not only about $\text{Hom}_H(\pi_1,\pi_2)$, but also $\text{Ext}^i_H(\pi_1,\pi_2)$, and $\text{EP}_H(\pi_1,\pi_2)$. The lecture is an attempt to discuss some of these questions.

Maryna Viazovska: Random lattices with symmetries
What is the densest lattice sphere packing in the $d$-dimensional Euclidean space? In this talk we will investigate this question as dimension $d$ goes to infinity and we will focus on the lower bounds for the best packing density, or in other words on the existence results. I will give a historical overview of the existence of dense lattice packings by H. Minkowski, E. Hlawka, C. L. Siegel, C. A. Rogers, and more recently by S. Vance and A. Venkatesh. For a long time the best asymptotic lower bounds on packing density were known for lattice packings. Recently, a new asymptotic lower bound for sphere packings in high dimensions was proven by M. Campos, M. Jensen, M. Michelin, and J. Sahasrabudhe. This new bound is proven by a combinatorial method and a dense packing guaranteed by their theorem is not necessarily a lattice packing. In the final part of the lecture I will present a recent work done in collaboration with V. Serban and N. Gargava, and Ilaria Viglino on the moments of the number of lattice points in a bounded set for random lattices constructed from a number field.

Eva Viehmann: Affine Deligne-Lusztig varieties beyond the fully Hodge-Newton decomposable case
Affine Deligne-Lusztig varieties are a class of subschemes of affine flag varieties that play an important role in the description of the reduction modulo p of Shimura varieties. However, a complete description of their geometry is only available in very few cases, under the so-called fully Hodge-Newton decomposablility condition. I will explain how to relax this condition, still obtaining a good description of the geometry of the associated ADLV.
This is ongoing work with Xuhua He and Felix Schremmer.

Anna Wienhard: A beautiful world beyond hyperbolic geometry: Anosov representations and higher Teichmüller spaces
I will describe the beautiful world of discrete subgroups of Lie groups of higher rank, taking inspiration from and making connection to hyperbolic geometry.

Efim Zelmanov: Superconformal algebras
In 80s physicists searched for proper superextensions of the celebrated Virasoro algebra. The resulting structures became known as superconformal algebras and attracted a lot of attention. We will discuss representations, cohomologies and further generalisations of superconformal algebras.