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). |
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SGS I: Tuesday, April 30, 2002
University of Alabama at Birmingham
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Morning Session:
Hill University Center, Room 412
8:00 AM - Coffee and refreshments.
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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.
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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.
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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.
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11:30 AM - 1:30 PM
Lunch
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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.
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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.
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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. |
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