An n-braid is a certain kind of arrangement of n pieces of string in three
dimensional space. You can glue two of these braids together to make a new
braid. This turns the set of n-braids into a group. Algebraists like this
because it is a nice group, and topologists like it because it is made of
string. I will give an introduction to this, assuming as little background
as possible.
I will briefly recall the definition of braid groups from last
week, and then talk about ways to map these into groups of matrices.
Infinite-dimensional spaces play an important role in many applications. The
theory of differentiable manifolds modeled on R^n can be generalized to define
manifolds modeled on arbitrary Banach spaces. Doing this gives us a
differential calculus for maps defined on spaces that are often non-linear and
infinite-dimensional. After a quick overview of differential calculus in
Banach space, we will present the basic definitions and a few results for
Banach manifolds. We will also look at some interesting examples of
infinite-dimensional manifolds.
I will define the mapping class group and discuss the classification of the elements of the group. Then I
will state a recent result about pseudo Anosov mapping classes and discuss one way to examine them.
Riemann-Roch theorem is stated and the equivalence of arithmetic
genus (as in Riemann-Roch) and topological genus (as can be determined by
counting holes) is discussed.
One characteristic of soliton equations is an infinite number of conservation laws. These conservation laws give rise to a hierarchy
of partial differential equations. We will see how these differential equations can be realized in a geometric setting using loop
groups and how two separate hierarchies became linked into one. The talk will focus on geometry and not PDEs.
We define and consider the basic properties of foliations on surfaces and quadratic
differentials. Once these objects are introduced, we will examine their
relationship to each other.
Every holomorphic quadratic differential defines an equivalence class of
measured foliations by horizontal trajectories of the differential and the
converse is true as well. We consider the geometry of foliations around
singular points as well as the global behavior of the trajectories.
We will use the path straightening method to find geodesics on
submanifolds of S^2.
We will briefly define Coxeter systems and then show how one can find a "Coxeter Link" given a Coxeter system.
A homotopy n-type is given by a space whose homotopy groups
vanish in degrees bigger than n. They are useful for classification
purposes and to study the homotopy category, say of topological
spaces. I will review how and why they are described in terms of
n-categories, and more generally n-stacks, and describe their
morphisms, focusing on the case of 1-types, where morphisms are well
understood in terms of butterfly diagrams.
The S1 -category of a space X is the smallest number of open sets, each homotopic into
a circle
in X, needed to cover X. We classify all knots whose spaces have S1-cat = 2