Moduli spaces of algebraic curves and differential forms


Lecturer

Guillaume Tahar (Assistant Professor, BIMSA)
Date

2025-10-14 ~ 2025-12-31
Schedule
Weekday    Time    Venue    Online    ID    Password
Tue,Wed    10:40-12:15    A3-1a-205    Zoom    482 240 1589    BIMSA

Prerequisite

Undergraduate general topology and complex analysis
Introduction

This course provides an introduction to the theory of moduli spaces associated with complex algebraic curves and with various types of differential forms defined on them. We will also discuss different compactifications of these spaces and their applications to enumerative geometry and dynamical systems.
Syllabus

Part 1: Riemann surfaces and their moduli spaces
Part 2: Nodal curves and Deligne-Mumford compactification of the moduli space of algebraic curves
Part 3: Moduli spaces of differentials and their geometric interpretation as translation surfaces
Part 4: Multi-scale compactification of strata of differentials
Part 5: Applications to enumerative geometry and dynamical systems: Masur-Veech volumes of strata and classification of orbit closures in relation with the magic wand theorem of Eskin-Mirazakhani-Mohammadi
Video Public

Yes
Notes Public

Yes
Audience

Undergraduate, Advanced Undergraduate, Graduate, Postdoc, Researcher
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.
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