Workshop on Quasiconformal Mappings and Teichmüller Theory

This will be an introductory workshop intended for PhD students, postdocs and young faculty members.

Dates: 19 - 23 February 2020

Venue: Kerala School of Mathematics, Kozhikode

Organisers: Pranav Haridas and Chitrabhanu Chaudhuri

Speakers

  1. Kingshook Biswas (ISI Kolkata),
  2. Subhojoy Gupta (IISc Bangalore),
  3. Sabyasachi Mukherjee (TIFR Mumbai).
  4. Carsten Lunde Petersen (Roskilde Universitet, Denmark).

Description of Discussion Sessions

Teichmüller's theorem on extremal mappings, Kingshook Biswas (4 x 1.5 hr sessions)

Abstract: We will sketch the proof of Teichmüller's theorem on the characterization of extremal quasiconformal mappings between compact Riemann surfaces as affine maps with respect to flat structures on the surfaces induced by holomorphic quadratic differentials. As a corollary, the Teichmüller space of a surface of genus g is homeomorphic to a cell of dimension 6g - 6. Leading up to the theorem, we will cover basics of quasiconformal maps, the Measurable Riemann Mapping Theorem, Teichmüller spaces and the Teichmuller metric, and holomorphic quadratic differentials and their induced flat structures.


The Bers embedding of Teichmüller spaces, Subhojoy Gupta (4 x 1.5 hr sessions)

Abstract: In these sessions I will introduce Teichmüller spaces from the point of view of deformation theory, and describe how it acquires a complex structure via an embedding into the vector space of holomorphic quadratic differentials. Along the way, I shall talk about the boundary extension of quasiconformal maps, the Simultaneous Uniformization theorem of Bers, and the Schwarzian derivative, amongst other basic topics. The image of the Bers embedding is still mysterious, and I plan to describe some open problems.
References:


Quasiconformal maps in holomorphic dynamics, Sabyasachi Mukherjee (4 x 1.5 hr sessions)

Abstract: The goal of these sessions is to discuss some applications of quasiconformal maps to the theory of dynamics of rational maps on the Riemann sphere. After introducing the basic objects of study in rational dynamics, we will proceed to the proof of Sullivan's no wandering domain theorem, which states that every "stable component" of a rational map is eventually periodic. The proof uses quasiconformal deformations of rational maps in an essential way. We will then describe a quasiconformal surgery technique to count the number of non-repelling cycles of a rational map. If time permits, we will sketch another application of quasiconformal maps that connects the theory of "univalent polynomials" to Teichmüller spaces of reflection groups.

Schedule

9:30 — 11:00 11:30 — 13:00 14:30 — 16:00 16:30 — 18:00
19 Feb Biswas Gupta Mukherjee Tutorial
20 Feb Mukherjee Biswas Gupta Tutorial
21 Feb Mukherjee Biswas Gupta Tutorial
22 Feb Mukherjee Biswas Petersen
23 Feb Gupta Petersen


List of Participants

Name Affiliation
1 Parameswaran A. J. TIFR Mumbai
2 Arpan Kabiraj CMI
3 Divakaran D Azim Premji University
4 Debanjan Nandi TIFR Mumbai
5 Indranath Sengupta IIT Gandhinagar
6 Shilpak Banerjee IIIT Delhi
7 Gianluca Faraco TIFR Mumbai
8 Kuntal Banerjee Presidency University
9 Sachchidanand Prasad IISER Kolkata
10 Subrata Shyam Roy IISER Kolkata
11 Sushil Gorai IISER Kolkata
12 Niladri Sekhar Patra TIFR Mumbai
13 Sandeep S TIFR Mumbai
14 Subhadip Majumder TIFR Mumbai
15 Manodeep Raha TIFR Mumbai
16 Saniya Wagh TIFR Mumbai
17 Biswajit Nag TIFR Mumbai
18 Sayani Bera IACS Kolkata
19 Gobinda Sau IISc Bangalore
20 Pranab Sarkar IISc Bangalore