Seminars
Home
Schools
Mathematics
Seminars
- FIELD
- Math:Geometry
- DATE
-
Mar 25 (Tue), 2025
- TIME
- 10:30 ~ 12:00
- PLACE
- 1503
- SPEAKER
- Rogozinnikov, Eugen
- HOST
- Xu, David
- INSTITUTE
- 수학부
- TITLE
- Parametrizing spaces of positive representations
- ABSTRACT
- Higher Techmüller theory deals with spaces of representations of the fundamental group of a surface into a reductive Lie group $G$, modulo the conjugation, especially with the connected components (called higher Teichmüller spaces) that consist entirely of injective representations with discrete image.
In the last two decades in works of Fock, Goncharov, Burger, Iozzi, Guichard, Wienhard, and others researchers, it was discovered that the most interesting higher Teichmüller spaces are emerging from the groups $G$ having a positive structure, i.e. certain submonoid $G_+$ with no invertible non-unit elements. Some of these submonoids have been known since 1930’s as totally positive matrices and then generalized by Lustzig for split real Lie groups. However it left out a large class of non-split reductive Lie groups such as $SO(p,q)$. O. Guichard and A. Wienhard filled this gap in 2018 by introducing the Theta-positivity, which also includes submonoids $SO(p,q)_+$ sitting in the unipotent group of $SO(p,q)$ and $Sp(2n,R)_+$ which is the set of upper uni-triangular block 2x2-matrices with a symmetric positive definite matrix in the upper right corner.
In my talk, I introduce the Theta-positivity for Lie groups and explain how the spaces of positive representations of the fundamental group of a punctured surface into a Lie group with a positive structure can be parametrized, and how we can describe the topology of these spaces using this parametrization. This is a joint work with O. Guichard and A. Wienhard.
- FILE
-