Geometric structures on surfaces
In this course, we give a panorama of geometric structures a topological surface can be endowed with. We will discuss combinatorial decompositions, natural dynamical systems and moduli spaces of these structures. We will focus on geometric structures compatible with a structure of Riemann surface (translation structures, dilation structures, cone spherical metrics...).
Lecturer
Date
28th March ~ 13th June, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Tuesday | 14:20 - 16:05 | A3-2a-202 | ZOOM 08 | 787 662 9899 | BIMSA |
Prerequisite
Undergraduate general topology and complex analysis
Syllabus
Part I: Simplicial structures and Polyhedra
Part II: Basic theory of Riemann surfaces
Part III: Introduction to Translation surfaces
Part IV: Geometry of meromorphic differentials
Part V: A flat illustration of Morse Theory on surfaces
Part VI: Walls-and-chamber structures on strata of meromorphic differentials, Veech groups
Part VII: Isoresidual fibration and Mittag-Leffler theorem
Part VIII: Quadratic differentials, half-translation surfaces and cone spherical metrics with special monodromy
Part IX: Geometry of the moduli spaces of cone spherical metrics
If there is enough time, we will also discuss dilation and complex affine structures.
Part II: Basic theory of Riemann surfaces
Part III: Introduction to Translation surfaces
Part IV: Geometry of meromorphic differentials
Part V: A flat illustration of Morse Theory on surfaces
Part VI: Walls-and-chamber structures on strata of meromorphic differentials, Veech groups
Part VII: Isoresidual fibration and Mittag-Leffler theorem
Part VIII: Quadratic differentials, half-translation surfaces and cone spherical metrics with special monodromy
Part IX: Geometry of the moduli spaces of cone spherical metrics
If there is enough time, we will also discuss dilation and complex affine structures.
Reference
Donaldson: Riemann Surfaces
Zorich: Flat Surfaces
Zorich: Flat Surfaces
Audience
Undergraduate
, Graduate
Video Public
No
Notes Public
No
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 at the Weizmann Institute of Science and joined BIMSA as an Assistant Professor in 2022. His research focuses on geometric structures on surfaces, with applications to moduli spaces and dynamical systems. He contributed to the study of various flavours of geometric structures, including translation surfaces, polyhedral metrics, cone spherical metrics and complex affine structures. His approach typically involves a mix of complex analysis, geometric constructions, and combinatorial reasoning.
His key results include the proof of the existence of closed geodesics in dilation surfaces, the complete characterization of configurations of local invariants realized by a differential on a Riemann surface and the establishment of Grünbaum's asymptotic classification for simplicial line arrangements with few double points.
His recent research interests include the topological interpretation of quantum invariants of knots, the counting of BPS states in quantum field theory and holomorphic dynamics in higher dimensions.
His key results include the proof of the existence of closed geodesics in dilation surfaces, the complete characterization of configurations of local invariants realized by a differential on a Riemann surface and the establishment of Grünbaum's asymptotic classification for simplicial line arrangements with few double points.
His recent research interests include the topological interpretation of quantum invariants of knots, the counting of BPS states in quantum field theory and holomorphic dynamics in higher dimensions.