Georgia Institute of Technology College of Sciences

For a genus g surface with s > 0 punctures and 2g+s > 2, decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over the usual Teichmuller space, where the fiber corresponds to families of horocycles peripheral to each puncture. As proved by R. Penner, DTeich admits a mapping class group-invariant cell decomposition, which then descends to a cell decomposition of Riemann's moduli space. In this talk we introduce a new cellular bordification of DTeich which is also MCG-invariant, namely the space of filtered screens. After an appropriate quotient, we obtain a cell decomposition for a new compactification of moduli space, which is shown to be homotopy equivalent to the Deligne-Mumford compactification. This work is joint with R. Penner.

In 1997 Hausmann and Knutson discovered a remarkable correspondence between complex Grassmannians and closed polygons which yields a natural symmetric Riemannian metric on the space of polygons. In this talk I will describe how these symmetries can be exploited to make interesting calculations in the probability theory of the space of polygons, including simple and explicit formulae for the expected values of chord lengths. I will also give a simple and fast algorithm for sampling random polygons--which serve as a statistical model for polymers--directly from this probability distribution.

A contact manifold with boundary naturally gives rise to a sutured manifold, as defined by Gabai. Honda, Kazez and Matic have used this relationship to define an invariant of contact manifolds with boundary in sutured Floer homology, a Heegaard-Floer-type invariant of sutured manifolds developed by Juhasz. More recently, Kronheimer and Mrowka have defined an invariant of sutured manifolds in the setting of monopole Floer homology. In this talk, I'll describe work-in-progress to define an invariant of contact manifolds with boundary in their sutured monopole theory. If time permits, I'll talk about analogues of Juhasz' sutured cobordism maps and the Honda-Kazez-Matic gluing maps in the monopole setting. Likely applications of this work include an obstruction to the existence of Lagrangian cobordisms between Legendrian knots in S^3. Other potential applications include the construction of a bordered monopole theory, following an outline of Zarev. This is joint work with Steven Sivek.