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

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

**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:*

- Teichmüller theory and applications, Volume 1, by John H. Hubbard. Published by Matrix editions, 2006.
- Lectures on Quasiconformal Mappings, 2nd edition, by Lars V. Ahlfors. AMS University Lectures Series, Volume 38. (original 1966, reprinted 2006)
- Univalent Functions and Teichmüller spaces, by Olli Lehto. GTM series, Springer, 1987.
- The Complex Analytic Theory of Teichmüller spaces, by Subhashis Nag. Wiley Interscience, 1988.

**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.

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 |

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 |