**Extremal**** Length Geometry
Seminar**

Organized by Fred
Gardiner (frederick.gardiner@gmail.com) with Yunping Jiang and Linda Keen

Department of
Mathematics

CUNY Graduate Center

Fall of 2014
Schedule

Wednesday, 3:30pm-4:55pm (__Math
Thesis Room, Rm. 4214-03)__:

September 3, Qualifying
Exam (room occupied)

September 10, Dylan Thurston (Indiana University, Bloomington),

Title: From Rubber Bands to Rational Maps

Abstract: we develop parallel theories of elastic graphs and conformal
surfaces with boundary, that let us on the one hand tell
when one rubber band network is looser than another, and on the other hand tell
when one conformal surface embeds in another. As an application, we give a new
characterization of hyperbolic critically finite rational maps among branched
self-coverings of the sphere. Our characterization takes the form of a
necessary and sufficient positive criterion: a branched covering is equivalent
to a hyperbolic rational map if and only if it has a 1-dimensional object
satisfying a particular ``self-embedding'' property. Portions of this project
are joint work with Jeremy Kahn and Kevin Pilgrim.

September
17, Fred Gardiner, On Teichmuller spaces of uniformly
asymptotic conformal circle endomorphisms and the quasidisk cocyle I.

October 1, Fred
Gardiner, On Teichmuller spaces of uniformly
asymptotic conformal circle endomorphisms and the quasidisk cocyle II.

October 8, Quanlei Fang, Explicit eigenvalue estimates for
transfer operators acting on spaces of holomorphic functions I

October 15, Quanlei Fang, Explicit eigenvalue estimates for
transfer operators acting on spaces of holomorphic functions II

October 22, Fred
Gardiner, On Teichmuller spaces of uniformly asymptotic conformal circle endomorphisms and the quasidisk cocyle III.

October 29, Fred
Gardiner, On Teichmuller spaces of uniformly
asymptotic conformal circle endomorphisms and the quasidisk cocyle IV.

Title: Some
applications of conformal field theory to Teichmuller
theory

Abstract: Consider the collection of compact Riemann surfaces of genus g
with n punctures, endowed with n non-overlapping

Conformal maps of the disk into neighbourhoods
of each puncture. The moduli space of these "rigged" surfaces,
up to

conformal equivalence, was considered by
Friedan-Shenker and Vafa.
It plays a central role in conformal field

theory. David Radnell
and I showed that this moduli space can be identified (up to a discrete group
action) with the

quasiconformal Teichmuller space of surfaces of genus g with n boundary
curves. In this talk I will describe some

applications of this correspondence to Teichmuller theory. Namely, it can be used to prove
that the quasiconformal

Teichmuller space of bordered surfaces fibers over the Teichmuller
space of compact surfaces.

This in turn can be used to construct a Teichmuller
space of surfaces of genus g with n boundary curves,

modelled on L^2
Beltrami differentials. This Teichmuller space
is in some sense the largest space

on which the Weil- Petersson
metric converges. Joint work with D. Radnell and W. Staubach.

November
12, Fred Gardiner, On Teichmuller
spaces of uniformly asymptotic conformal circle endomorphisms
and the quasidisk cocyle V.

November
19, Mariela Carvacho
(Chile), Title: Some invariants on the classification of group actions

on compact Riemann surfaces.

Abstract: Given a closed orientable
topological surface

$X_g$ of genus $g \geq
2$ and $H$ an orientation

preserving self-homeomorphism of $X_{g}$

there exists (Nielsen realization) a complex

structure on $X_{g}$ such that $H$ extends

to a conformal action on $X_{g}$. It is known

that such a structure (in general) is not
unique.

It
depends on the number of equivalence

classes of actions of $H$ on $X_{g}$. In this
talk

the idea is to give a description of the problem

of equivalent actions and to show some recent

results about this subject.

November
26, Yunping Jiang, Infinmum of
metric entropy for Anosov diffeomorphisms

and expanding circle endomorphisms
preserving Lebegue measure.

December 3,
Fred Gardiner, On Teichmuller
spaces of uniformly asymptotic conformal circle endomorphisms
and the quasidisk cocyle
VI.

December 10, Fred Gardiner, On Teichmuller spaces of uniformly asymptotic conformal circle endomorphisms and the quasidisk cocyle