Panorama of Geometric structures on surfaces
In this course, we introduce the general notion of (G,X)-structures on manifolds. After providing the basic results on the topology of surfaces (including the topological classification theorem), we will give a general panorama of (G,X)-structures in dimension two. We will focus on two classes of geometric structures:
- metrics with constant curvature (possibly with singularities);
- geometric structures compatible with a structure of Riemann surface.
We will discuss fruitful interactions with combinatorics, dynamical systems and the general theory of moduli spaces.
- metrics with constant curvature (possibly with singularities);
- geometric structures compatible with a structure of Riemann surface.
We will discuss fruitful interactions with combinatorics, dynamical systems and the general theory of moduli spaces.
Lecturer
Date
11th October ~ 29th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 14:20 - 16:05 | A3-2a-201 | ZOOM 06 | 537 192 5549 | BIMSA |
Prerequisite
Undergraduate general topology and complex analysis
Syllabus
PART 1: Axiomatic topology of surfaces (Radò theorem, Invariance of domain, Brouwer fixed point theorem)
PART 2: Atlases and (G,X)-structures
PART 3: Combinatorial topology of bidimensional complexes
PART 4: Metrics with constant curvature
PART 5: Introduction to Riemann Surfaces
PART 6: Translation surfaces
PART 7: Complex projective and affine structures on surfaces
PART 2: Atlases and (G,X)-structures
PART 3: Combinatorial topology of bidimensional complexes
PART 4: Metrics with constant curvature
PART 5: Introduction to Riemann Surfaces
PART 6: Translation surfaces
PART 7: Complex projective and affine structures on surfaces
Video Public
Yes
Notes Public
Yes
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.