Holomorphic Dynamics

This course provides an introduction to the theory of dynamical systems generated by the iteration of holomorphic maps. Beginning with rational maps on the Riemann sphere, we study the fundamental dichotomy between the Fatou and Julia sets and develop the basic tools used to analyze their structure. Topics include normal families, Montel’s theorem, periodic points and their classification, critical orbits, and the role of the postcritical set. We will explore key examples such as quadratic polynomials and the Mandelbrot set, emphasizing the interplay between complex analysis, geometry, and dynamical systems. Depending on time and interests, additional topics may include polynomial-like mappings, quasiconformal techniques, and selected connections with Teichmüller theory and moduli spaces. The course is intended for graduate students with a background in complex analysis.

7th April ~ 25th June, 2026

Weekday	Time	Venue	Online	ID	Password
Tuesday,Thursday	10:40 - 12:15	A14-101	Zoom 17	442 374 5045	BIMSA

PART 1: Riemann Surfaces
PART 2: Julia sets
PART 3: Local and global fixed point theory
PART 4: Fatou sets
PART 5: Caratheodory theory
PART 6: Polynomial maps
PART 7: Introduction to holomorphic dynamics in several variables

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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.