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Series: Geometry Topology Seminar

Lickorish observed a simple way to make two knots in S^3 that produced the same manifold by the same surgery. Many have extended this result with the most dramatic being Osoinach's method (and Teragaito's adaptation) of creating infinitely many distinct knots in S^3 with the same surgery yielding the same manifold. We will turn this line of inquiry around and examine relationships within such families of corresponding knots in the resulting surgered manifold.

Series: Geometry Topology Seminar

In the 1980s Gromov showed that curvature (in the triangle comparison sense) decreases under branched covers. In this expository talk I shall prove Gromov's result, and then discuss its generalization (due to Allcock) that helps show that some moduli spaces arising in algebraic geometry have contractible universal covers. The talk should be accessible to those interested in geometry/topology.

Series: Geometry Topology Seminar

We prove that every metric of constant curvature on a compact 2-manifold M with boundary bdM induces (at least) four vertices, i.e., local extrema of geodesic curvature, on bdM, if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on M which induce only two vertices on bdM. Furthermore, we characterize the sphere as the only closed orientable Riemannian 2-manifold M which has the four-vertex-property, i.e., the boundary of every compact surface immersed in M has 4 vertices.

Series: Geometry Topology Seminar

In many physical situations we are interested in topological lower bounds for L^2-energy of volume preserving vector fields. Such situations include for instance evolution of a magnetic field in ideal magnetohydrodynamics. Classical energy bounds involve topological invariants like helicity which measure linkage of orbits in the flow. In this talk I will present a new lower bound in terms of the third order helicity, which is an invariant measuring a third order linkage of orbits. I will also discuss how the third order helicity can be derived from the Milnor's \mu-bar invariant for 3-component links.

Series: Geometry Topology Seminar

In this talk I will describe some relations between embedded graphs, their polynomials and the Jones polynomial of an associated link. I will explain how relations between graphs, links and their polynomials leads to the definition of the partial dual of a ribbon graph. I will then go on to show that the realizations of the Jones polynomial as the Tutte polynomial of a graph, and as the topological Tutte polynomial of a ribbon graph are related, surprisingly, by the homfly polynomial.

Series: Geometry Topology Seminar

In this talk I will give a purely combinatorial description of Knot Floer Homology for knots in the three-sphere (Manolescu-Ozsvath-Szabo- Thurston). In this homology there is a naturally associated invariant for transverse knots. This invariant gives a combinatorial but still an effective way to distinguish transverse knots (Ng-Ozsvath-Thurston). Moreover it leads to the construction of an infinite family of non-transversely simple knot-types (Vertesi).

Series: Geometry Topology Seminar

W. Goldman proved that the SL(2)-character variety X(F) of a closed surface F is a holonomic symplectic manifold. He also showed that the Sl(2)-characters of every 3-manifold with boundary F form an isotropic subspace of X(F). In fact, for all 3-manifolds whose SL(2)-representations are easy to analyze, these representations form a Lagrangian space. In this talk, we are going to construct explicit examples of 3-manifolds M bounding surfaces of arbitrary genus, whose spaces of SL(2)-characters have dimension as small as possible. We discuss relevance of this problem to quantum and classical low-dimensional topology.

Series: Geometry Topology Seminar

I will describe a framework which relates large N duality to the geometry of degenerating Calabi-Yau spaces and the Hitchin integrable system. I will give a geometric interpretation of the Dijkgraaf-Vafa large N quantization procedure in this context.

Series: Geometry Topology Seminar

This is an expository talk. A classical theorem of Mazur gives a simple criterion for two closed manifolds M, M' to become diffeomorphic after multiplying by the Euclidean n-space, where n large. In the talk I shall prove Mazur's theorem, and then discuss what happens when n is small and M, M' are 3-dimensional lens spaces. The talk shall be accessible to anybody with interest in geometry and topology.

Series: Geometry Topology Seminar

Based on work of Schwarz and Oh, information coming from a filtration in Hamiltonian Floer homology can be used to construct "spectral invariants" for paths of Hamiltonian diffeomorphisms of symplectic manifolds. I will show how these invariants can be used to provide a unified approach to proving various old and new results in symplectic topology, including the non-degeneracy of the Hofer metric and some of its variants; a sharp version of an inequality between the Hofer-Zehnder capacity and the displacement energy; and a generalization of Gromov's non-squeezing theorem.