Groups, Geometry and Topology Seminar

Organizer: Gye-Seon Lee (Seoul National University)

Schedule

Shinpei Baba (Osaka University)

Title: ???
Abstract: ???

Gianluca Faraco (University of Milano-Bicocca)

Title: Flat Geometry in Broader Contexts: A Journey Through Translation Surfaces
Abstract: In this mini-course, we will explore translation surfaces - flat geometric structures that have attracted significant attention from a dynamical perspective. Shifting the focus, these lectures will examine translation surfaces also through topological and geometric lenses, with the main goal of understanding the moduli spaces they inhabit, known as strata. We will begin with a general introduction to translation surfaces from both topological and complex-analytic viewpoints. I will then introduce the notion of periods and explain how translation surfaces can be represented through their period data. The final part of the course will focus on the so-called isoperiodic foliations, presenting recent developments in the field along with some open problems that remain at the frontier of current research.

Clarence Kineider (Max Planck Institute for Mathematics in the Sciences)

Title: Fock-Goncharov coordinates and spectral networks
Abstract: In 2006 V. Fock and A. Goncharov introduced moduli spaces generalizing the (decorated) Teichmüller space for punctured surfaces. These moduli spaces are built upon the character varieties for higher-rank split real Lie groups (though we will focus in this lecture on PGL_n/SL_n). The theory they developed revolves around building so-called "cluster coordinates" on these moduli spaces that satisfy very strong algebraic and combinatorial properties. One of the outcome of their work is the construction of the first examples of what is now called "higher Teichm ller spaces". In the first part of this mini-course, I will give an introduction to these moduli spaces and their cluster coordinates. In the second part, I will draw the link between the theory of Fock-Goncharov and another type of object called "spectral networks", introduced by D. Gaiotto, G. Moore and A. Neitzke, bringing a more geometric/topological aspect to the story.

Eugen Rogozinnikov (Max Planck Institute for Mathematics in the Sciences)

Time  /  Place:
3 June 2024 (Monday) 17:00 - 18:15  /  129-301
5 June 2024 (Wednesday) 17:00 - 18:15  /  129-309
11 June 2024 (Tuesday) 17:00 - 18:15  /  129-301
12 June 2024 (Wednesday) 17:00 - 18:15  /  129-301

Title: Introduction to spectral network
Abstract: Spectral networks were introduced in a seminal article by Davide Gaiotto, Gregory W. Moore and Andrew Neitzke published in 2013. These are networks of trajectories on surfaces that naturally arise in the study of various four-dimensional quantum field theories. From a purely geometric point of view, they yield a new map between flat connections over a Riemann surface and flat abelian connections on a spectral covering of the surface. At the same time, these networks of trajectories provide local coordinate systems on the moduli space of flat connections that are valuable in the study of higher Teichmüller spaces.
In the first part of this mini-course, I will review key concepts from geometric group theory, including hyperbolic groups and boundaries at infinity of hyperbolic groups and spaces. Following this, I will discuss the theory of vector bundles and the Riemann-Hilbert correspondence. In the second part, I will define spectral networks explicitly for surfaces with punctures. I will also present and discuss their most prominent applications in geometry: non-abelianization and abelianization, which connect higher Teichmüller spaces of a base surface to abelian character varieties of its ramified cover, the spectral curve.

Arnaud Maret (Sorbonne Université)

Time: 7 May 2024 (Tuesday) 15:30 - 18:00  &  9 May 2024 (Thursday) 15:30 - 18:00
Place: 27 (Building) 220 (Room)

Title: Surface group representations - it's not all about Teichmüller!
Abstract: This mini-course is an introduction to surface group representations which are not discrete and faithful. We'll mostly look at representations into PSL(2,R). I'll start with some generalities about surface group representations and character varieties (Goldman symplectic structure, Toledo number, etc). I'll then explain what is known and conjectured about these non-(discrete and faithful) representations. I'll insist on two aspects: mapping class group dynamics and geometrization. We'll finally study a very specific family of such representations: the so-called Deroin-Tholozan representations. They are in some sense the best model of non-(discrete and faithful) representations that we know. Along the way, we'll encounter triangle groups, some elementary algebraic geometry, group cohomology, and much more!