**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 2015
Schedule

Friday 10:30am-11:30pm (__Math Thesis Room, Rm. 4214-03)__:

September 18,

September 25, (Class follows Tuesday schedule)

October 9, Fred Gardiner, Caratheodory's metric and Kobayashi's for certain
directions in Teichmueller space III

October 16, Dynamical System: Ideas and Application, 11am-1pm,
2pm-4pm, Room 4102 (Konstantin Mischaikow (Rutgers
Univ.))

October 23, Fred Gardiner, Caratheodory's metric and Kobayashi's for certain
directions in Teichmueller space III

October
30, Mayank Goswami, On
algorithms adapted to theorems of Teichmueller
theory.

ABSTRACT: By the Riemann mapping
theorem, one can bijectively map the interior of an $n$-gon $P$

to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this
mapping

need not necessarily map the vertices of $P$ to those $Q$.
In this case, one wants to find

the "best" mapping between these polygons, i.e.,
one that minimizes the maximum angle distortion

(the dilatation) over all points in $P$. From complex analysis such maps are known to
exist and are unique.

They
are called extremal quasiconformal
maps, or Teichmuller
maps.
Although there are many efficient ways

to compute or approximate conformal maps, there
is currently no such algorithm for extremal quasiconformal maps.

This talk will present the first constructive
method to obtain the extremal quasiconformal
map in the continuous setting.

Our construction is via an iterative procedure that is proven to
converge quickly to the unique extremal map.

To get to within $\epsilon$ of the
dilatation of the extremal map, our method uses $O(1/\epsilon^{4})$
iterations.

Every step of the iteration involves convex optimization and solving differential
equations, and guarantees

a decrease in the dilatation. Our method uses a
reduction of the polygon mapping problem to that of
the punctured

sphere problem, thus solving a more general
problem. Time permitting, I will explore related
applications of computation

in Teichmuller spaces
to tropical geometry and computer graphics and vision.

November
6, Linda Keen, Title: Background and some new results in Fenchel's
theory of the

geometry of right angled hexagons in
hyperbolic three space (joint work with Jane Gilman)

Abstract: Right angled hexagons in H^3 are a generalization

of triangles in R^2. The isometries
are identified with

elements of PSL(2,C). We will discuss this
theory and

some new developments of it which we will apply
in

the second talk

November
13, Linda Keen, Title: Canonical hexagons for two generator subgroups of PSL(2,C)

(Joint work with Jane Gilman)

Abstract: Each presentation of a two generator
subgroup

of PSL(2,C) determines a right angled
hexagon.

We
will show that certain presentations determine

a "canonical" hexagon for the
group.

This has implications for the deep question of

deciding whether the group is free and
discrete.

It also has implications for the character variety

parameterizing these groups.

November 20, Dynamical System: Ideas and Application,
11am-1pm, 2pm-4pm, Room 4102 (Lai-Sang Young (CIMS, NYU))

November 27, (Thanksgivings), no meeting

December 4, Kealey Dias

December 11,