Southeast Geometry Seminar

The Southeast Geometry Seminar (SGS) is a new semiannual series of one day events sponsored jointly by:
Georgia Institute of Technology
University of Tennessee, Knoxville
University of Alabama at Birmingham

The organizers are: John McCuan (GIT), Alex Freire (UTK), Gilbert Weinstein (UAB), and Sumio Yamada (UAB).

SGS I: Tuesday, April 30, 2002
University of Alabama at Birmingham

Morning Session: Hill University Center, Room 412

8:00 AM - Coffee and refreshments.
8:30 AM - 9:20 AM
Changyou Wang (University of Kentucky)
Blow Up Analysis of The Heat Flow of Harmonic Maps

Abstract: In this talk, we will discuss the limiting behaviors for a sequence of weakly convergent smooth or suitable weak solutions to the heat equation of harmonic maps. We will describe the obstruction for the strong convergence, smooth convergence, and analyze the defect measure by providing a dimensional stratification decompostion, and the relationship between the limiting flow for the couple of mapping and defect varifolds. In the case that the limiting map flow posses refined properties, we are able to show the defect varifold moves according to the motion of varifold in the sense of Brakke.
 
9:30 AM - 10:20 AM
Margaret Symington (Georgia Institute of Technology)
Almost Toric Symplectic Four-Manifolds

Abstract: Among symplectic manifolds, toric manifolds (those equipped with a Hamiltonian action of a torus of half the dimension) have the remarkable property that the manifold, symplectic form and torus aciton are all encoded in a polytope of half the dimension of the manifold. I will explain a generalization of the moment map on a toric symplectic four-manifolds (whose image is a polygon) to a restricted class of (singular) Lagrangian fibrations. For these "almost toric" manifolds the two-dimensional image of the fibration map almost completely determines the symplectic manifold. With a little extra data that can be added to the two-dimensional diagram, the symplectic manifold is completely determined up to diffeomorphism. I will show how, with such two-dimensional data, one can:
  • determine what four-manifolds admit an almost toric structure;
    and
  • see how to perform certain surgeries symplectically, including a generalized rational blowdown.
10:30 AM - 11:20 AM
Robert Hardt (Rice University)
Size Minimization and Approximating Problems

Abstract: A k dimensional rectifiable current is given by an oriented k dimensional rectifiable set M together with a positive integer-valued density function D . The mass of the current is then simply the integral of D over M (with respect to k dimensional Hausdorff measure). In 1960 Federer and Fleming proved the existence of a rectifiable current of least mass for a given boundary. For q in [0,1] , the q-mass of the current is the integral of Dq over M . The case q = 0 corresponds to size , introduced by Almgren as a way of using currents to model soap films. We will discuss the existence of and partial regularity of a rectifiable current of least q-mass for a given boundary. For that purpose we define scans which are certain functions arising as limits of slices of rectifiable currents and use a new compactness theorem for metric space valued BV functions.
 
11:30 AM - 1:30 PM
Lunch

 

Afternoon Session: Education Building, Room 152
1:30 PM - 2:20 PM
Sumio Yamada (University of Alabama at Birmingham)
On Harmonic Maps into Teichmuller Spaces

Abstract: Given a topological surface, Teichmuller space is defined to be the space of all the hyperbolic metric structures on the surface. In this talk I will consider the geometry of Teichmuller spaces induced by the so-called Weil-Petersson distance function, and discuss its relevance to the geodesic length functionals on the Riemann surfaces.
 
2:30 PM - 3:20 PM
Martial Agueh (Georgia Institute of Technology)
Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory.

Abstract: We show that the nonlinear degenerate parabolic equations:

admits a solution. The method used is variational. It requires less uniform convexity assumption than what is known in the literature (see Alt-Luckhauss). This class of problems includes the Fokker-Planck equation, the Porous-medium equation and the parabolic p-Laplacian equation.
 
3:30 PM - 4:20 PM
Bo Guan (University of Tennessee, Knoxville)
Hypersurfaces of constant curvature with boundary

Abstract: Given a disjoint collection G = {G1, ...,Gm} of closed smooth embedded (n-1) dimensional submanifolds of Rn+1, a fundamental question is to decide whether there exist (immersed) hypersurfaces M of constant Gauss-Kronecker curvature in Rn+1 with boundary M = G. In this talk we will discuss the recent development in the study of this problem and its relation with Monge-Ampere equations.