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Series: Geometry Topology Seminar

In a joint work with D.Tamarkin we study analytic continuability of solutions of theLaplace-transformed Schroedinger equation by methods of Kashiwara-Schapira style microlocal theoryof sheaves.

Series: Geometry Topology Seminar

In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the H^k(P\Sigma_n, \Q) tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n, that is, in some precise sense, the description of the decomposition of the cohomology group into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

In the talk, I plan to give a definition of loose Legendrian knots inside contact manifolds of dimension 5 or greater. The definition is significantly different from the 3 dimensional case, in particular loose knots exist in local charts. I'll discuss an h-principle for such knots. This implies their classification, a bijective correspondence with their formal (algebraic topology) invariants. I'll also discuss applications of this result, comparisons with 3D contact toplogy, and some open questions.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

We will use a new concordance invariant, epsilon, associated to the knot Floer complex, to define a smooth concordance homomorphism. Applications include a new infinite family of smoothly independent topologically slice knots, bounds on the concordance genus, and information about tau of satellites. We will also discuss various algebraic properties of this construction.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

We will discuss aspects of Chern-Simons theory, quantization and algebraic curves that appear in moduli spaces problems.

Series: Geometry Topology Seminar

Note that this talk is on the UGA campus.

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.

Series: Geometry Topology Seminar

Note that this talk is on the UGA campus.

To every homeomorphism of a surface, we can attach a positive real number, the entropy. We are interested in the question of what these homeomorphisms look like when the entropy is positive, but small. We give several perspectives on this problem, considering it from the complex analytic, surface topological, 3-manifold theoretical, and numerical points of view. This is joint work with Benson Farb and Chris Leininger.