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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.

Series: Geometry Topology Seminar

I will describe a new way to depict any smooth, closed oriented 4-manifold using a surface decorated with circles, along with a set of moves that relate any pair of such depictions.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

I will explain in details starting with the basics, how the
bimodules over some polynomial rings (cohomology of grasmanians)
categorify the irreducible representations of sl(2) or U_q(sl(2).The main goal is to give an introduction to categorification theory. The talk will be accessible to graduate students.

Series: Geometry Topology Seminar

An important structural feature of the kth homology group of SL_n(Z) is that it is independent of n once n is sufficiently large. This property is called "homological stability" for SL_n(Z). Congruence subgroups of SL_n(Z) do not satisfy homological stability; however, I will discuss a theorem that says that they do satisfy a certain equivariant version of homological stability.

Series: Geometry Topology Seminar

Rob Kirby and I have been thinking for a while now about stable maps to 2-manifolds, which we call "Morse 2-functions", to stress the analogy with standard Morse theory, which studies stable maps to 1-manifolds. In this talk I will focus on the extent to which we can extend that analogy to the way in which handle decompositions combinatorialize Morse functions, especially in low dimensions. By drawing the images of attaching maps and some extra data, one describes the total space of a Morse function and the Morse function, up to diffeomorphism. I will discuss how much of that works in the context of Morse 2-functions. This is important because Rob Kirby and I have spent most of our time thinking about stable homotopies between Morse 2-functions, which should be thought of as giving "moves" between Morse 2-functions, but to honestly call them "moves" we need to make sure we have a reasonable way to combinatorialize Morse 2-functions to begin with.

Series: Geometry Topology Seminar

The slice-ribbon conjecture states that a knot in $S^3=partial D^4$ is the boundary of an embedded disc in $D^4$ if and only if it bounds a disc in $S^3$ which has only ribbon singularities. In this seminar we will prove the conjecture for a family of Montesinos knots. The proof is based on Donaldson's diagonalization theorem for definite four manifolds.