# Workshop on the Geometry and Topology Of Low - Dimensional Maniolds

### Dept. of Mathematics, University of Pune and Bhaskaracharya Pratishthana, Pune

#### Venue : Department of Mathematics, University of Pune, Pune.

 In the past twenty/thirty years, the geometry and topology of low dimensional manifolds have seen major developments. These developments also have direct inputs from theoretical physics, for example from quantum field theories and string theory. The Ricci flow, introduced by Hamilton, and its use by Perelman for the proof of the Poincare Conjecture has been one of the major advances in this area in recent times. A generalization of the Poincare Conjecture is the Geometrization Conjecture of Thurston, which proposed special Riemannian geometries for all compact 3-manifolds. It is also claimed to be proved by the Ricci Flow Techniques. A special case of Haken manifolds was already proved independently by very different techniques coming from Teichmuller Theory. There will be special lectures on topics in Riemann surfaces, quasiconformal mappings, Kleinian groups and Teichmullar theory.

#### Prof.s S. T. Yau and Hamilton will be during Jan 2-Jan 5, 2008. Prof. Penner will be during Jan 10-Jan 14, 2008.

The workshop will discuss some of these developments and their role in the study of invariants of manifolds. The workshop is particularly suitable for young research workers and advanced post graduate/ Ph.D. students and university and college teachers interested in these new developments. It is being organized as a satellite workshop associated with the 100th year of the Indian Mathematical Society and its annual conference to be held in the University of Pune from December 27 -30, 2007.
The Workshop is partially funded by the National Science Foundation of the United States.

#### Speakers: The principal speakers will include:

Part-I

• Prof. C. S. Aravinda (TIFR, Bangalore),
• Prof. R. Hamilton (Univ. of California), Jan. 3-4, 2008
• Prof. Ravi Kulkarni (IIT, Bombay and BP), Jan 2-16, 2008
• Prof. A. Mangasuli (BP), Jan 2-16, 2008
• Prof. Kishore Marathe (CUNY), Jan 2-16, 2008
• Prof. Santhanam (IIT, Kanpur), Jan 2-6, 2008
• Prof. Vemuri (CMI)
• Prof. S. T. Yau (Harvard Univ.), Jan 3-4, 2008

Part-II

• Prof. Sudeb Mitra (CUNY), Jan 2-16, 2008
• Prof. R. Penner (Aarhus), Jan 10-16, 2008
• Prof. D. Saric (CUNY), Jan 9-16, 2008
• Prof. Ara Basmajian (CUNY), Jan 9-16, 2008
• Prof. Indranil Biswas (TIFR),
• Prof. Perry Suskind (Connecticut), Jan 9-16, 2008
• Prof. E. Taylor (Weylayan U), Jan 9-16, 2008
• Prof. Petra B. Taylor (Weylayan U), Jan 9-16, 2008

##### Updated list of speakers and topics

How to apply: Applications giving the following details should be sent to the address given below:
1. Name 2. Age 3. M/F 4. Institution 5. Address for correspondence, e-mail, phone no.
6. Educational qualifications / Research interests 7. Any additional information.

Research students should send a letter of recommendation from their supervisor / Head.
The applications should be sent, before October 20, 2007 to:

The Organizing Committee,
Workshop ( January 2008),
Department of Mathematics,
University of Pune,
Pune 411 007.
Phone No. 020-2560-1272,
or by e-mail to hbhate at math.unipune.ernet.in

Additional information will be available at the departmental website: math.unipune.ernet.in or www.bprim.org. Interested participants should check for updates.

Local hospitality will be extended to all participants. Limited travel support may be available to a few participants, subject to availability of funds.

Organising Committee:

 Indranil Biswas,(Convener) TIFR, Mumbai Sudeb Mitra (Convener, US) City Univ. of New York K. B. Marathe (Convener) City Univ. of New York H. Bhate (Local coodinator) University of Pune B. N. Waphare ( HoD, Maths.) University of Pune Ravi Kulkarni (Convener) Bhaskaracharya Pratishthana and IIT Bombay S. A. Katre University of Pune V. S. Gejji University of Pune

Prof.s Ravi Kulkarni (IIT, Bombay, and Bhaskaracharya Pratishthan) and Sathanam (IIT, Kanpur)) will give some preliminary lectures on Curvature, Connections.

Prof. Vemuri (CMI) will lecture on Maximum Principles.

Prof. Mangasuli (BP) will lecture on Heat equations on Riemannian manifolds.

#### List of speakers and topics:

1) Ara Basmajian
Title: "Isometries of Hyperbolic space as commutators".

2) Petra Bonfert-Taylor
Title: "Quasiconformal homogeneity of hyperbolic domains and their boundaries".

3) Edward Taylor
Title: "Quasiconformal symmetry and ammenability".

4) Dragomir Saric [Jan., 9-16, 2008]
Title: "The Teichmuller distance between finite index subgroups of $PSL_2(\mathbb{Z})$".

5) Perry Suskind
Title: "The Geometry at Infinity of a Hyperbolic Riemann Surface of Infinite Type."

Abstract: (This is joint work with Andrew Haas.) We study geodesics on a planar Riemann surface of infinite type having a single infinite end. Of particular interest is the class of geodesics that go out the infinite end in a most efficient manner. We investigate properties of these geodesics and relate them to the structure of the boundary of a Dirichlet polygon for a Fuchsian group representing the surface.

6) Sudeb Mitra
Title: "Some extensions of holomorphic motions."

7) Ravi S. Kulkarni

Time - Table for Part-II

 9th, Wed. Jan. 2008 10th, Thurs. 11th, Fri. 12th, Sat. 13th, Sun. 9.30 to 10.45 Petra B. Taylor Robert Penner 9.30 to 10.30 Robert Penner H. Mukherjee 11.15 to 12.30 Ed Taylor Perry Susskind 10.45 to 11.45 Ara Basmajian Sudev Mitra 12.00 to 1.00 D. Saric K. Gangopadhyay 1.00 to 2.30 Registration(2-2.30) Lunch 2.30 to 3.45 Ravi Kulkarni Ara Basmajian Ed Taylor 2.30 to 3.30 Dragomir Saric K. Marathe 4.15 to 5.30 A. P. Singh Ravi Kulkarni Petra B. Taylor 4.00-5.00 P. Susskind

 Dragomir Saric: The Teichmular distance between finite index subgroups of PSL_2(Z) Krishnendu Gangopadhyay: Z-Classes of Isometries of Pseudo-Riemannian geometries of constant curvature Sudeb Mitra: 1. Extensions of holomorphic motions2. Quasiconformal motions Robert Penner: TBA Indranil Biswas: TBA Ravi Kulkarni: 1. Introduction to Teichmuller Theory2. Subgroups of the modular group A. P. Singh: Spiralling Baker domains Petra Bonfert Taylor: 1. Quasiconformal Groups2. Quasiconformal Homogeneity Ed Taylor: 1. Quasiconformal Homogeneity2. Quasiconformal Groups Ara Basmajian: Isometries of Hyperbolic space as commutators Perry Susskind: The geometry at infinity of a hyperbolic Riemann surface of infinite type H. Mukherjee TBA
Time - Table for Part-I

 2nd Jan.2008 3rd 4th 5th 7th Jan,2008 8th Jan,2008 9.30 to 10.45 K. Marathe Yau K. Marathe G. Santhanam W. Freyn Arvinda 11.15 to 1.00 G. Santhanam G. Santhanam G. Santhanam Arvinda Varadrajan A. Mangasuli 1.00 to 2.30 Lunch 2.30 to 4.00 Yau K. Marathe Arvinda Hamilton Arvinda A. Mangasuli 3.45 to 5.00 K. Marathe Varadrajan Problem Session, Santhanam, Marathe Mangasuli Mangasuli

Accommodation

Participants of the workshop are requested to go to SET GUEST House near the MAIN GUEST House of the University. There is a possibility of change of place of accommodation for a few days later.