The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:

**The National Science Foundation****The University of Alabama at Birmingham****The Georga Institute of Technology****Emory University****University of Tennessee Knoxville**

The organizers are: Vladimir Oliker (Emory), Mohammad Ghomi and John McCuan (Georgia Tech), Fernando Schwartz (UTK), Junfang Li (UAB), and Gilbert Weinstein (previoulsy with UAB).

Thanks to an NSF grant, we have funds to support participants, particularly students and recent Ph.D. recipients. We encourage women and minorities to apply. To apply please write to us.

see http://www.mathcs.emory.edu/contact.php, Letter ``P'' indicates parking for visitors. Save the receipt!

Mathematics & Science Center,

Emory Campus, 400 Dowman Dr.,

2-nd floor, Room W201,

9:30AM -
** Alex Freire (Univ. of Tennessee, Knoxville)
**

Mass, capacity and inverse mean curvature flow

Mass, capacity and inverse mean curvature flow

**Abstract:** I'll describe a sharp inequality relating mass and capacity in
all dimensions, and applications to geometric inequalities for mean-convex
hypersurfaces. The proofs involve the inverse mean curvature flow in
euclidean space.
This is joint work with Fernando Schwartz.

10:30AM - ** Matthew Gursky (University of Notre Dame)**

Critical metrics on connected sums of Einstein four-manifolds

Critical metrics on connected sums of Einstein four-manifolds

**Abstract:** We develop a gluing procedure designed to obtain
canonical metrics on connected sums of Einstein four-manifolds. These metrics are
critical points of quadratic Riemannian functionals.
The main application is an existence result, using two well-known Einstein manifolds
as building blocks: the Fubini-Study metric on CP2, and the product metric
on S2 x S2.. Using these metrics in various gluing configurations,
critical metrics are found on connected sums.

11:30AM - ** William Minicozzi II (MIT)**

Singularities in Mean Curvature Flow

Singularities in Mean Curvature Flow

**Abstract: ** Mean curvature flow is a nonlinear geometric flow where a hypersurface evolves to decrease its area as efficiently as possible. Singularities are unavoidable and the point is to understand the possible singularities and the behavior of the flow near a singularity. I will talk about recent joint work with Toby Colding and with Toby Colding and Tom Ilmanen where we prove strong rigidity and uniqueness theorems for generic singularities in all dimensions.

2:30PM - ** Christopher Croke (University of Pennsylvania)**

Scattering and Lens Rigidity with trapped geodesics

Scattering and Lens Rigidity with trapped geodesics

**Abstract:**We will consider compact Riemannian manifolds M with boundary N. We let IN be the unit
vectors to M whose base point is on N and point inwards towards M. Similarly we define OUT. The
scattering data (loosely speaking) of a Riemannian manifold with boundary is a map from IN to OUT which
assigns to each unit vector V of IN a the unit vector W in OUT. W will be the tangent vector to the
geodesic determined by V when that geodesic first hits the boundary N again. This may not be defined
for all V since the geodesic might be trapped (i.e. never hits the boundary again). A manifold is said to
be scattering rigid if for any Riemannian manifold Q with boundary isometric to N and with the same
scattering data, then the boundary isometry extends to an isometry of M and Q. The lens data includes
not only the scattering data but also the lengths of the geodesics. In this talk we will discuss some recent
results on the scattering (and lens) rigidity problems. One thing we will discuss is recent work of my
graduate student Haomin Weh on the relation between scattering data and lens data for surfaces.
There are a number of manifolds that are known to be lens rigid and there are examples that are not
scattering or lens rigid. All of the known examples of non-rigidity have trapped geodesics in them. In
this talk we will see that the flat solid torus is scattering rigid. This is the first global scattering (or lens)
rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors
tangent to trapped geodesics in any such Q have measure 0 in the unit tangent bundle of Q. We will also
consider scattering rigidity of a number of two dimensional manifolds (joint work with Pilar Herreros)
which have trapped geodesics

3:30PM - ** Gaoyong Zhang (NYU-Poly)**

The logarithmic Minkowski problem

The logarithmic Minkowski problem

**Abstract:**The logarithmic Minkowski problem asks for necessary and sufficient
conditions such that a nonnegative finite Borel measure on the unit
sphere is the cone-volume measure of a convex body in the Euclidean
space. This problem and the classical Minkowski problem for surface
area measures are two important cases of the $L_p$ Minkowski problem
in the $L_p$ Brunn-Minkowski theory. The solution to the logarithmic
Minkowski problem for the unit balls of finite dimensional Banach
spaces (origin-symmetric convex bodies) is discussed. Its relation
to geometric inequalities stronger than the Brunn-Minkowski inequality
is explained.

invited to an early dinner at a restaurant near the Emory Conference Center Hotel.