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.

__Speaker:__** **** Marcus Khuri, Stony Brook University
**

** 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.

**Friday****,
Saturday
March
22,
23,
**

__Speaker:__** ****
Shanks workshop: "Kahler geometry on the edge"****
**

__Schedule of talks__**:
http://www.math.vanderbilt.edu/%7Esuvaini/Workshop-2013/****
**

**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