The Symplectic and Differential Geometry Seminar and the Noncommutative Geometry Seminar have combined into a

                                                Geometry Seminar

Vanderbilt University

   Spring 2013

   Organizers:  Basak Gurel, Gennadi Kasparov, Ioana Suvaina, Zhizhang Xie

   Mondays, 4:10-5:00pm in SC 1432 (unless otherwise noted)

   Related seminars also announced.

Monday, February 4th,

Speaker: Zhizhang Xie, Vanderbilt University

Title: Some secondary geometric invariants

Abstract Secondary invariants are important in geometry and topology. While primary invariants only depend on the topology of the underlining manifolds, secondary invariants also depend on certain auxiliary geometric data (e.g. metrics or connections etc. ) of the underlining manifolds. Some of the well-known secondary invariants are Chern-Simon invariants, eta invariant and rho invariant, where the latter two were introduced by Atiyah, Patodi and Singer.

In this talk, I will discuss some of my recent work and joint work with others on these secondary invariants (and their higher versions). In particular, I shall talk about the higher eta invariant and the higher rho invariant, and their connections to the Baum-Connes conjecture and positive scalar curvature problems.


Monday, February 18th

Speaker: Ioana Suvaina, Vanderbilt University

Title: On the normalized Ricci flow and smooth structures on 4-manifolds

Abstract: There is a strong relation between the existence of non-singular solutions for the normalized Ricci flow and the underlying smooth structure of a 4-manifold. We are going to discuss an obstruction to the existence of non-singular solutions and its applications. The main examples are connected sums of complex projective planes and complex projective planes with reversed orientation. The key ingredients in our methods are the Seiberg-Witten Theory and symplectic topology. This is joint work with M. Ishida and R. Rasdeaconu.

Monday, February 25th

Speaker: Rares Rasdeaconu, Vanderbilt University

Title: ALE Ricci-flat Kahler surfaces and weighted projective spaces

Abstract:  We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are Tian-Yau metrics. The proof relies on a construction of appropriate compactifications of Q-Gorenstein smoothings of quotient singularities as log del Pezzo surfaces. Time permitting,  a geometric description of the compactifications will be provided.  This is a joint work with I. Suvaina.

           Thursday, March 14th,  Colloquium Talk

Speaker:  Marcus Khuri, Stony Brook University

                   Title: Geometric Inequalities and General Relativity

    Friday, March 15th, joint with Subfactor Seminar, 4:10-5:00, in SC 1432

Speaker: Kamran Reihani,  Northern Arizona University

Title: Noncommutative Metrics for Dynamical Systems 

Abstract: Spectral triple is the fundamental object of the metric aspects of Connes' noncommutative geometry. A spectral metric space is a spectral triple (A, H, D) with additional properties guaranteeing that the Connes metric on the state space of A induces the weak*-topology. It is, in fact, the noncommutative analog of a complete metric space. Let (A,H,D) be a spectral metric space and G be a group of automorphisms of A. In this talk I will consider the problem of whether there is a natural spectral triple for the crossed product algebra C*(G,A) that can characterize the metric properties of the dynamical system (G,A). I will discuss a solution to this problem when a single automorphism of A generates G as an equicontinuous family of quasi-isometries. I will also address the converse problem, namely, when a spectral metric space for the crossed product gives rise to one for A. When the action is not equicontinuous (e.g., when the action is uniformly hyperbolic), following the philosophy of Diffeomorphism-Invariant Geometry of Connes and Moscovici, we suggest replacing the dynamical system (G,A) by a dynamical system (G,B), where G acts isometrically. The algebra B is called the metric bundle associated with (G,A). Some candidates for the metric bundle B will be introduced. This talk is based on a joint work with Jean Bellissard and Matilde Marcolli.

Monday,  March 18th, joint with Colloquium, 4:10-5:00 pm, in SC1308

Speaker:  Xiu-Xiong Chen, Stony Brook University

Title: Kaehler Einstein metrics on Fano Manifolds

Abstract: In 1980s, Yau conjectured that the existence of Kaehler Einstein metric on Fano manifold is related to an  algebraic geometric  condition of ``stability''.  The recent work with Donaldson, Sun Song confirmed this conjecture.  In the talk, we will  review history of this problems as well as this subject, and we also will review earlier work of  G. Tian and others on this problems.  We will outline the strategy of proof, which involves deforming through metrics  with cone singularities.  If time permits, we will give more details about various aspects of the proof.

                  This talk is aimed at a  general mathematical audience.

Friday, Saturday March 22, 23,

Speaker:  Shanks workshop: "Kahler geometry on the edge"

Schedule of talks:


Monday, March 25th, 10:00-11:30 am, in SC 1424

Speaker: Carl Tipler, University of Quebec at Montreal

Title:  informal discussion on parabolic structures on ruled surfaces

Monday, April  1st

Speaker: Marcelo Disconzi, Vanderbilt University

Title: Analysis and Geometry in infinite dimensions

Abstract:  In this talk, first I will review how to construct a Riemannian structure on the space of maps between two differentiable manifolds. Then I will discuss how this construction is used to study certain partial differential equations on manifolds, focusing on the Euler equations and, if time allows, on the Einstein equations. If there is still time left, I will introduce some of my recent results and indicate directions of future research.

Monday, April  8st

Speaker: Mustafa Kalafat, Michigan State University

Title: Topology of G_2 manifolds

Abstract: We analyze the topological invariants of some specific Grassmannians, the Lie group G_2, and give some applications. This is a joint work with Selman Akbulut.

Thursday, May  2nd, 

Speaker: Bianca Santoro, CUNY

Title: On complete Kahler Ricci-flat metrics

Abstract: This will  be an informal discussion about what is known about complete Ricci-flat metrics on Kahler manifolds. We will also discuss speculative applications to G_2 - geometry.

Old Seminar Web-Pages: Fall 2009, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012