# Online workshop on geometry, topology, and their applications

## Abstracts for the talks

**Xian Dai ***Ruhr University Bochum***Metric geometry in classical and higher Teichmüller spaces**

In this talk, I will give an overview of metric geometry in the classical Teichmüller space. Some important features of the Weil Petersson metric will be discussed. In the end, I will also say a few words about generalisation of Riemannian and Finsler metrics to Higher Teichmüller spaces.

**Shreyasi Datta ***University of Michigan***Recent progress in S-adic diophantine approximation**

We study the measure theory of non-generic sets (from the Diophantine point of view) in manifolds via orbits of certain ﬂows in an appropriate homogeneous space. We will talk about some new developments in Diophantine approximation in the S-adic set-up, where S is a ﬁnite set of valuations of rationals. The metric Diophantine approximation has seen a tremendous boost in the last two decades, especially after the breakthrough of Kleinbock and Margulis in 1998.

**Subhadip Dey ***Yale University***Combination theorems for Anosov subgroups**

The classical Klein Combination Theorem provides a sufficient condition to construct new Kleinian groups. Subsequently, Maskit gave far-reaching generalizations to the Klein Combination Theorem. A special feature of Maskit's theorems is that they furnish sufficient conditions so that the combined group retains nice geometric features, such as convex-cocompactness or geometric-finiteness.

In recent years, Anosov subgroups have emerged as a natural higher-rank generalization of the convex-cocompact Kleinian groups, exhibiting their robust geometric and dynamical properties. This talk will discuss my recent joint work with Michael Kapovich on the Combination Theorems in the setting of Anosov subgroups.

**Samantha Fairchild ***MPI Mathematics in the Sciences***Saddle connections on translation surfaces**

A translation surface is, informally, a collection of polygons in the plane with parallel sides identified by translation to form a Riemann surface with a singular structure. Understanding the geometry and dynamics of flows on translation surfaces and their moduli spaces led to the development of many new and revolutionary techniques. We will overview some important techniques related to understanding the distribution of saddle connections as well as future directions.

**James Farre ***University of Heidelberg***Long curves and random hyperbolic surfaces**

We will fix some topological data, a pants decomposition, of a closed surface of genus g and build hyperbolic structures by gluing hyperbolic pairs of pants along their boundary. The set of all hyperbolic metrics with a pants decomposition having a given set of lengths defines a (3g-3)-dimensional immersed torus in the (6g-6)-dimensional moduli space of hyperbolic metrics, a twist torus. Mirzakhani conjectured that as the lengths of the pants curves tend to infinity, that the corresponding twist torus equidistributes in the moduli space. In joint work-in-progress with Aaron Calderon, we confirm Mirzakhani`s conjecture. I will give a brief summary of the results.

**Filippo Mazzoli ***University of Virginia***Pleated surfaces and surface group representations inside PSL (n,C)**

Since their introduction by Thurston, pleated surfaces have been extensively deployed to study the geometry of hyperbolic 3-manifolds and representations of the fundamental group of a closed surface of genus larger than 1 (in short, a surface group) inside PSL(2,C). In this talk, we will see how the notion of pleated surface can be suitably generalized to investigate wide classes of surface group representations inside PSL(n,C) for every n larger than 1. We will also discuss applications and questions that arise naturally from their study. This talk is based on an upcoming work with S. Maloni, G. Martone, and T. Zhang.

**Irem Portakal ***TU Munich***Geometric aspects of game theory**

In 1950, Nash published a very influential two-page paper proving the existence of Nash equilibria for any finite game. The proof uses an elegant application of the Kakutani fixed-point theorem from the field of topology. This opened a new horizon not only in game theory, but also in areas such as economics, computer science, evolutionary biology, and social sciences. It has, however, been noted that in some cases the Nash equilibrium fails to predict the most beneficial outcome for all players. To address this, generalizations of Nash equilibria such as correlated and dependency equilibria were introduced. In this talk, I elaborate on how algebraic and convex geometry are indispensable for studying undiscovered facets of these concepts of equilibria in game theory.

**Peter Smillie ***University of Heidelberg***Non-uniqueness of minimal surfaces in locally symmetric spaces**

If G is a split real Lie group of rank 2, for instance SL(3,R), and S is a closed surface of genus at least 2, then Labourie showed that every Hitchin representation of pi_1(S) into G admits a unique equivariant minimal surface. As Labourie pointed out, this lets you parametrise the space of Hitchin representations by the total space of a vector bundle over the Teichmuller space of S. He conjectured that uniqueness should hold more generally, at least for all SL(n,R).

In joint work with Nathaniel Sagman, we show that for any split G of rank at least 3, and for any S, there is a Hitchin representation with two distinct equivariant minimal surfaces, disproving Labourie’s conjecture. I will explain our construction, which starts from minimal surfaces in R^3, and what new questions this raises.

**Christian Urech ***Ecole Polytéchnique Fédérale de Lausanne***Cremona groups over finite fields and Neretin groups**

Cremona groups are the groups of birational transformations of the projective plane, whereas Neretin groups are the groups of almost automorphisms of rooted trees. While the first come from algebraic geometry, the second appear in low-dimensional topology. In this talk, I will introduce both of them and explain how Cremona groups embed as dense subgroups into Neretin groups if we work over a finite field. This point of view allows us to better understand both groups.

**Morgan Weiler ***Cornell University***Low-dimensional symplectic geometry from embedded contact homology computations**

Embedded contact homology (ECH) is a 3D Floer-type homology theory due to Hutchings. Its access to both symplectic and topological information means that it is well-suited to solving problems in low-dimensional symplectic and contact geometry. We will explain recent applications to surface dynamics and 4D symplectic embeddings, which require understanding ECH at the chain level in new topological and geometric settings, including its cobordism maps. In part based on several recent and upcoming joint works with Magill, McDuff, Nelson, and Pires.

## Date and Location

**January 27, 2023**

Max Planck Institute for Mathematics in the Sciences

Videobroadcast

Virtual event - Videobroadcast

## Scientific Organizers

**Anna Wienhard**

MPI for Mathematics in the Sciences

## Administrative Contact

**Katharina Matschke**

MPI for Mathematics in the Sciences