Extremal Length Geometry Seminar

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

Department of Mathematics

Fall of 2015 Schedule

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

### September 11, Organization and Fred Gardiner, Caratheodory's metric and Kobayashi's for certain directions in Teichmueller space I

September 18, Dynamical System: Ideas and Application, 11am-1pm, 2pm-4pm, Room 9204 (Pedro Dal Bo (Brown Univ.) and Enrique Pujals (IMPA and CUNY-GC))

September 25, (Class follows Tuesday schedule)

### October 2, Fred Gardiner, Caratheodory's metric and Kobayashi's for certain directions in Teichmueller space II

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,