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Joint Meetings:
I will be attending the 2009 Joint Meetings in Washington, DC. Please contact me if you would like to arrange an interview.

Talks

Audience: Researchers and Graduate Students

2008

1. Extending the normal ruling invariant to Legendrian surfaces
American Institute of Mathematics Research Workshop
Abstract: Applying Morse theory to a generating family of a Legendrian link motivates a geometric pairing of critical points which leads to a combinatorial invariant called a graded normal ruling. This technique also motivates a geometric pairing on a Legendrian surface in R^5 coming from a generating family. What should be the appropriate combinatorial invariant of closed Legendrian surfaces in R^5 coming from this geometric pairing? We will investigate this question in detail. On our way towards a solution, we will describe the classification of local neighborhoods of front projections of Legendrian surfaces and the local Legendrian isotopy moves, thus giving some insight into the Legendrian Reidemeister moves in dimension 5.
2. Building and distinguishing manifolds in dimensions 2 and 3
Wash U Graduate Organized Seminar
Abstract: A common task in mathematics is to determine when two objects are the same and when they are different. For example, the Poincare conjecture, which was recently turned into the Poincare Theorem, asks if every closed, simply-connected 3-dimensional manifold is homeomorphic to the 3-dimensional sphere. The other way to say this is, "If it looks like the 3-sphere and it acts like the 3-sphere, must it be the 3-sphere?"

In this talk, we will consider various ways of constructing 2 and 3 dimensional manifolds and try to find techniques for determining when two are the same and when they are different. In the 2-dimensional case, all closed, orientable manifolds can be distinguished by a handy little gadget called the genus. In the 3-dimensional case, the problem of distinguishing manifolds is much harder.

This talk is specifically geared for first and second year graduate students. It will involve a lot of pictures and a heavy dose of hand waving. I will not assume any previous experience with the following concepts: manifold, orientable, simply-connected, closed, homeomorphic.
3. Generating families, normal rulings and Legendrian knots
Wash U Combinatorics and Topology Seminar
Abstract: Using some fairly elementary function theory and Morse theory, we'll develop a technique for creating Legendrian knots in R^3. From this technique we'll derive a handy Legendrian knot invariant called a normal ruling. As with all of the invariants in this field, our new invariant will have a nice combinatorial description.

Recently, the normal ruling invariant was found to be intimately related to the Chekanov-Eliashberg differential graded algebra invariant. This relationship provides a link between classical Morse theory and low dimensional symplectic field theory. This relationship has led to many useful results.
4. Legendrian Knot Theory via Combinatorics (2 Parts)
Wash U Combinatorics and Topology Seminar
Abstract: Legendrian knots are knots in R^3 that respect a certain 2-dim tangent space distribution called a contact structure. These knots are interesting because the set of Legendrian knots up to isotopy through Legendrian knots is a refinement of the set of knots up to isotopy, i.e. for any knot type K there are Legendrian knots L and L' that are isotopic to K but not isotopic to each other through Legendrian knots. The central question in this theory is what type of invariants distinguish L and L'. Few Legendrian invariants exist and those that do are largely grounded in symplectic topology and Floer homology theory.

However, the question of finding Legendrian invariants can be translated into combinatorics. After a thorough introduction to the basic theory, we'll explore a method of combinatorializing the Legendrian invariant question that involves grid diagrams and elementary grid moves. This combinatorial system came out of an attempt to derive a Legendrian invariant from Ozsvath and Szabo's Heegaard Knot Floer Homology Theory. This system is less than 6 months old!

2007

1. The Jones polynomial as an Euler characteristic of a knot homology invariant
Wash U Graduate Student Seminar
Abstract: The standard homology we know and love gives an invariant of topological spaces up to homotopy equivalence. The Euler characteristic of that homology (i.e. the alternating sum of the ranks of the homology groups with coefficients in a field) is also an invariant up to homotopy equivalence. Recently, Mikhail Khovanov ingeniously created a homology theory for knots up to ambient isotopy in R^3 whose graded Euler characteristic is the Jones polynomial of the knot. What he ends up with is a new knot invariant which is at least as strong as the Jones polynomial invariant. This opened the door to several other knot homologies and led to many graduate students in topology earning degrees. We'll see how the Khovanov homology came about and walk through the example of the trefoil. There will be lots of pictures!
2. An Introduction to Khovanov Homology and Applications to Contact Topology (3 Parts)
Wash U Topology Seminar
These three lectures were a lengthy introduction to Khovanov homology. The talks were largely based on the paper, "Khovanov's Homology for Tangles and Cobordisms" by Dror Bar-Natan. As an application of the theory of Khovanov homology, we explored the connection between Khovanov homology and the maximal Thurston-Bennequin number of a knot.
3. Open books, fibered links and the Murasugi sum
University of Texas at Austin Junior Topology Seminar
Abstract: The Murasugi sum is a technique of "gluing" two open books together to form a new open book. We'll investigate what information about the underlying manifolds is preserved by this operation and how the binding of the open books might change. This technique plays an integral part in the correspondence between open books and contact structures.

A link L in S^3 is a fibered link if S^3 \ L has a fibration over S^1 that is well behaved near L. The fibers of the fibration are Seifert surfaces of L. This is equivalent to saying that L is the binding of an open book in S^3. We will see how to build new fibered links by using the Murasugi sum. We'll also see the limitations of this technique.
4. Moser's method, Gray's Theorem and local control of contact structures on 3-manifolds
Wash U Major Oral Exam
Abstract: A contact structure on a 3-manifold M is a 2-plane distribution on M that is completely non-integrable, i.e. there is no embedded 2-manifold F in M whose tangent space agrees with the distribution. In other words, a contact structure is very twisted, even locally. We will begin with an introduction to contact structures on 3-manifolds and move into a discussion of one of the most useful tools in this area: Moser's method. Gray's Theorem is an excellent example of the power of Moser's method. If time permits, I will give some indication as to how this method can be used to prove theorems about local neighborhoods of points and nicely embedded knots.
5. A Comprehensive Introduction to Contact Topology on 3-manifolds (5 Parts)
Wash U Topology Seminar
During the winter break of the 2006-2007 academic year, I gave a series of five talks on the subject of contact topology on 3-manifolds. The talks were aimed at research mathematicians who were not already familiar with the subjects. Topics included; classifying plane fields, local neighborhood theorems, convex surface theory, classifications of tight and overtwisted structures on specific manifolds, open book decompositions, and Legendrian and transverse knot theory.

2006

1. Classifying the mapping group of a genus g surface by Dehn Twists
University of Texas at Austin Junior Topology Seminar
Abstract: We'll work through Lickorish's 1961 paper which proves that (1) every orientation preserving homeomorphism of a closed connected 2-mfld is isotopic to a product of Dehn twists and (2) every closed connected orientable 3-mfld can be obtained from S^3 by doing +-1 Dehn surgery on a link. Lickorish's ideas are creative and easy to understand. And there are plenty of pictures to help us along the way.
2. Thin position of knots and essential surfaces
Wash U Minor Oral Exam
Abstract: In 1983, Gabai introduced the notion of knot width and subsequently defined a knot invariant called thin position. This invariant played an important role in Thompson's algorithm for determining when a closed orientable 3-manifold is, in fact, S^3. It was also essential to Gordon and Luecke's theorem that a knot is completely determined by the homeomorphism class of its complement in S^3.

We will introduce thin position and give some sense as to why this invariant is natural and nice, but still difficult to apply. We will describe a class of knots whose thin position is known. Finally, we will prove a Lemma of Gabai's that asserts the existence of a 2-sphere in S^3 whose intersections with a surface bounding a knot are essential.

2004

1. The Banach-Tarski Paradox
Graduate Teaching Seminar, Wash U
Audience: The Graduate Teaching Seminar course at Washington University is a semester long course designed to introduce first year graduate students to various teaching methods and the practicalities and logistics of teaching. Throughout the term, graduate students gain teaching experience by presenting topics to the class and receiving comments and suggestions. The course culminates in each student leading an entire class period.