Introduction to Alexandrov geometry and Degenerations of geometric objects
The goal of this course is to present two theories about degenerations of spaces in geometry. In the first part, we introduce Alexandrov Geometry. In differential geometry, being positively or negatively curved is defined in terms of inequalities satisfied by the coefficients of the curvature tensor. However, riemannian manifolds can degenerate to objects that do not keep a manifold structure. Alexandrov geometry is about giving a meaning to curvature inequalities in the general settings of metric spaces.
In the second part, we will discuss translation structures as examples of flat metrics with conical singularities and present the various compactification theories for their moduli spaces. If time allows, we will also discuss degenerations of complex affine structures with the recent proof of the existence of closed geodesics in dilation surfaces.
In the second part, we will discuss translation structures as examples of flat metrics with conical singularities and present the various compactification theories for their moduli spaces. If time allows, we will also discuss degenerations of complex affine structures with the recent proof of the existence of closed geodesics in dilation surfaces.
Lecturer
Date
8th October ~ 25th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Wednesday | 10:40 - 12:15 | Online | ZOOM 14 | 712 322 9571 | BIMSA |
Prerequisite
Basic Differential Geometry and Complex Analysis
Syllabus
Part 1: Alexandrov Geometry
1) Gromov-Hausdorff convergence of metric spaces
2) CAT(0) spaces
3) Dimension theory for CAT and CBB spaces
4) Gradient Flow
Part2 : Geometric structures on surfaces
1) Introduction to the flat geometry of translation surfaces
2) A flat geometric picture: The WYSWYG compactification of strata of meromorphic differentials
3) An algebro-geometric picture: The Multi-scale compactification of strata of meromorphic differentials
4) Degenerations of complex affine structures: proving the existence of closed geodesics in dilation surfaces
1) Gromov-Hausdorff convergence of metric spaces
2) CAT(0) spaces
3) Dimension theory for CAT and CBB spaces
4) Gradient Flow
Part2 : Geometric structures on surfaces
1) Introduction to the flat geometry of translation surfaces
2) A flat geometric picture: The WYSWYG compactification of strata of meromorphic differentials
3) An algebro-geometric picture: The Multi-scale compactification of strata of meromorphic differentials
4) Degenerations of complex affine structures: proving the existence of closed geodesics in dilation surfaces
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Guillaume Tahar obtained his Ph.D from Université Paris Diderot, under the supervision of Anton Zorich. He was a senior postdoctoral fellow in Weizmann Institute of Science and joined BIMSA as an assistant professor in 2022. He contributed to the study of moduli spaces of various flavours of geometric structures on surfaces. His results include proving the existence of closed geodesics in dilation surfaces and the complete characterization of configurations of local invariants realized by a differential on a Riemann surface. His recent research interests include linear differential operators, simplicial arrangements of lines and quantum invariants of knots.