Note: There is no lecture on April 14, May 3, May 5, May 17, May24, May31, 2022. The last lecture will be delivered  on June 2.

Description

Professor Anton Zorich:

I plan to give a panorama of geometry and dynamics of moduli spaces and discuss certain applications of this subject. The course would be informal: I plan to omit most of the proofs in order to concentrate on conceptual ideas and especially on ties between various facets of the discussed fields. This would allow to describe almost everything from scratch: I do not assume any particular knowledge of dynamics, geometry or combinatorics.

I plan to start with flat surfaces, similar to those which arise from the study of billiards in rational polygons. This would lead us to the moduli space of Abelian differentials and to the natural action of the group GL(2,R) on this space. I plan to present the Magic Wand Theorem of Eskin-Mirzakhani-Mohammadi and the complement to it due to Filip. I will introduce the notion of Lyapunov exponents and Eskin-Kontsevich-Zorich formula for the sum of Lyapunov exponents of the Teichmuller flow. As an illustration of all this technique we will compute the diffusion rate of the Ehrenfest wind-tree billiard and of its relatives.

We will proceed with the fascinating world of hyperbolic surfaces. I plan to give an idea of how Maryam Mirzakhani has counted simple closed geodesic multicurves on closed hyperbolic surfaces. We will discuss in this context Witten-Kontsevich correlators, mention Witten's conjecture, Fenhel-Nielsen coordinates and symplectic structure of the moduli space of hyperbolic surfaces. If time allows, we will discuss train-tracks.

Finally, in the last part of the course I plan to bridge flat and hyperbolic worlds. Using the count of metric ribbon graphs by Kontsevich and Norbury, we will compute, following Delecroix-Goujard-Zorich, Masur-Veech volume of the moduli space of quadratic differentials through count of square-tiled surfaces. We will conclude with recent spectacular results of Amol Aggarwal on large genus asymptotics of Masur-Veech volumes and of Witten-Kontsevich correlators. As an application I will describe (following Delecroix-Goujard-Zorich) random geodesic multicurves on surfaces of large genera and random square-tiled surfaces of large genus. If time allows, we will count meanders on a surface of any genus.

Profile

Anton Zorich is Distinguished Professor of Mathematics at Université Paris Diderot - Paris 7. He is a former member of the Institut Universitaire de France. His research lies on the border between dynamical systems, geometry and topology. He often performs computer experiments which sometimes lead to conjectures proved years or decades later. He usually works in collaboration; often with Alex Eskin and Maxim Kontsevich. He was an invited speaker at the International Congress of Mathematicians in Madrid in 2006.

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Course Slides:

1_Panorama_From_billiards_to_flat_surfaces_presentation.pdf

2_Panorama_Magic_Wand_presentation.pdf

3_Panorama_count_of_saddle_connections.pdf

4_Panorama_Right_angled_billiards_and_quadratic_differentials.pdf

5_Panorama_square_tiled_surfaces.pdf

6_Panorama_Equidistribution_Noncorrelation.pdf

7_Panorama_Meanders.pdf

8_Panorama_Mirzakhani_count.pdf

9_Panorama_Train_tracks_Measured_laminations.pdf

10_Panorama_Large_genus.pdf

11_Panorama_Large_genus_2.pdf

12_Panorama_Calabi_Theorem_and_Connected_Components_of_Strata.pdf

Course videos:

Lecture1: https://u-paris.zoom.us/rec/share/BI66s8djNZxWMxo_FUn64WXVqhefBrhcIa2iabs4J1My0thuYBhO19SQ_FD8c4A.KFrvnD8mlE8UF5q2

Lecture2: https://u-paris.zoom.us/rec/share/slLvp1Y-mhcZJObe2GhpHbeRSrB4lWZ00K8BEy6SJLJ_a5OxI77h1c5l-GZ0sEYf.y5utf1Jd7DIzKAfK

Lecture3: https://u-paris.zoom.us/rec/share/z4fEEKiKbdjv79PyQzxr3S_lJPRC7fuQm1vGosT5s294kvww8VTpOrb8cBEdJad-.IBHqM07wE6tDylqw?startTime=1649748048000

Lecture4: https://u-paris.zoom.us/rec/share/4BwkAtqfYUOhpo8Vtob2kh5XZ1vEPF7tQJhVBNbVCYX9eg2dsQ_tz1uYPhiaDW_h.bD5KjevylMVr78TG?startTime=1650352823000

Lecture5: https://u-paris.zoom.us/rec/share/JE0CwQLaeq_co19BX1XfdRzsTaIZOdB09QUAlSxzXdo2WTLxrV1NMBkYztemylfl.O1vOUnsDsCW8azEh?startTime=1650525680000

Lecture6: https://u-paris.zoom.us/rec/share/ZN9WUh_6NMRZfi9LKhMOoHzfZztUMUET_zGBTSeJygeTMrJ2rpd6R0U8uuKi2Y8O.m5AtwItyUE-uzK-l?startTime=1650957621000

Lecture9:https://u-paris.zoom.us/rec/share/Q6rjoNzccFEEDj4oVrZMirw2CxyOvO7qFegEXFGJCW41YP-cMpsuUHxIJysh0IXm.9z_FroPd4E8Q3Mhn?startTime=1652340268000

Lecture10:https://u-paris.zoom.us/rec/share/HG40S80jDRFedL26L2pmgvaLK6bYRDfOhITe3zmvoEZHQKcAN2jrbQm4UcBSA24C.o9eF2N4Jy0o0yY8k?startTime=1652944936000

Lecture11:https://u-paris.zoom.us/rec/share/ePTCBdcZn7Ge5aTkZ1m9ovGkYJ5tXmWnx_F-EIJ-qbygtBq6W86zLlvLyugnTwZi.zqa8IxukqFtCaTdR?startTime=1653549895000

Lecture12:https://u-paris.zoom.us/rec/share/5sPZ_tu8sdsDYqC18KfCZ2MH337a-xUvEfgh_VeEu1GmnisFuC6tFL6pPJ3hU6uB.XpcZibRshvRfHh0W?startTime=1654154480000