Seminars and Colloquia by Series

The four-vertex-property and topology of surfaces with constant curvature

Series
Geometry Topology Seminar
Time
Monday, October 27, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mohammad GhomiSchool of Mathematics, Georgia Tech
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.

A new topological bound for energy of fluid flows

Series
Geometry Topology Seminar
Time
Friday, October 24, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Rafal KomendarczykUniversity of Pennsylvania
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.

Ribbon graphs and knots

Series
Geometry Topology Seminar
Time
Monday, October 20, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Iain MoffattUniversity of Southern Alabama
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.

Knots in contact 3-manifolds

Series
Geometry Topology Seminar
Time
Friday, October 10, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Vera VertesiSchool of Mathematics, Georgia Tech
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).

3-manifolds and lagrangian subspaces of character varieties

Series
Geometry Topology Seminar
Time
Monday, October 6, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
A. SikoraSUNY Buffalo
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.

Large N duality and integrable systems

Series
Geometry Topology Seminar
Time
Friday, October 3, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Tony PantevDept of Mathematics, University of Penn
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.

Stable equivalence of manifolds

Series
Geometry Topology Seminar
Time
Monday, September 29, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Igor BelegradekSchool of Mathematics, Georgia Tech
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.

Spectral invariants, the energy-capacity inequality, and the non-squeezing theorem

Series
Geometry Topology Seminar
Time
Monday, September 22, 2008 - 16:00 for 1 hour (actually 50 minutes)
Location
Room 322, Boyd Graduate Studies UGA
Speaker
Michael UsherDepartment of Mathematics, University of Georgia
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.

The HOMFLY polynomial, the trilogarithm and zeta(3)

Series
Geometry Topology Seminar
Time
Monday, September 22, 2008 - 14:30 for 2 hours
Location
Room 322, Boyd Graduate Studies UGA
Speaker
Stavros GaroufalidisSchool of Mathematics, Georgia Tech
I will discuss a relation between the HOMFLY polynomial of a knot, its extension for a closed 3-manifold, a special function, the trilogarithm, and zeta(3).  Technically, this means that we consider perturbative U(N) Chern-Simons theory around the trivial flat connection, for all N, in an ambient 3-manifold. This is rigorous, and joint with Marcos Marino and Thang Le.

Filling invariants for groups

Series
Geometry Topology Seminar
Time
Monday, September 15, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Pallavi DaniEmory University and LSU
The Dehn function of a finitely presented group measures the difficulty in filling loops in the presentation complex of the group. Higher dimensional Dehn functions are a natural generalization: the n-dimensional Dehn function of a group captures the difficulty of filling n-spheres with (n+1)-balls in suitably defined complexes associated with the group. A fundamental question in the area is that of determining which functions arise as Dehn functions. I will give an overview of known results and describe recent progress in the 2-dimensional case. This is joint work with Josh Barnard and Noel Brady.

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