PATCH Day on September 28

Our speakers on September 28th are:

 Joan Birman, who will be talking on:

Title: A polynomial invariant of pseudo-Anosov maps.


Pseudo-Anosov mapping classes on surfaces have a rich structure, uncovered by William Thurston in the 1980’s. We will discuss the 1995 Bestvina–Handel algorithmic proof of Thurston’s theorem, and in particular the “transition matrix” their algorithm computes. We study the Bestvina–Handel proof carefully, and show tha the dilatation of the mapping class is the largest real root of a particular polynomial divisor $P(X)$ of the characteristic polynomial C(x)=|xI-T|.  While C(x) is not, in general, an invariant of the mapping class, we prove that P(x) is. The polynomial P(x) contains the minimal polynomial M(x) of the mapping class as a divisor, but does not, in general, coincide with M(x).

In this talk we will review the background and describe the mathematics which underlies the new invariant. This represents joint work with Peter Brinkmann and Keiko Kawamuro.

We also have:

Shea Vela-Vick

Who will be speaking on:

Title: Recovering HFK^- from sutured Floer homology

Abstract: I plan to discuss a method for reconstructing the knot Floer homology “minus” theory of a knot K as a direct limit of a collection of sutured Floer homology groups associated to K, each equipped with a naturally defined U–action. Generalizing the techniques used in this construction, one can define useful invariants of contact structures on open 3–manifolds with F^2 \times [0, \infty) ends and appropriate boundary data at infinity. In this talk, I’ll focus on the construction when the surface F is a 2–torus and discuss applications if time permits.


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