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The Southeast Geometry
Seminar (SGS) is a semiannual series of one day events sponsored jointly by:
- Emory University
- Georgia Institute of Technology
- University of Tennessee, Knoxville
- University of Alabama at Birmingham
The organizers are: Vladimir Oliker (Emory), John McCuan (GIT), Alex Freire (UTK), Gilbert Weinstein (UAB), and
Sumio Yamada (UAB). |
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SGS V:
Friday, February 20, 2004
Georgia Institute of Technology
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All lectures will take place in the College of Management, Technology Square, Room 101
9:00 AM - Coffee and refreshments
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9:30 AM - 10:20 AM
Malcolm Adams (University of Georgia)
The Spectrum of Metrized Graphs
Abstract
In this talk I will report on some work of R. Rumely and M. Baker,
number theorists at the University of Georgia, concerning the
spectral theory of metrized graphs. For the puposes of this talk, we
will consider a finite graph with a length assigned to each edge. We
will discuss definitions of the Laplacian on such a graph as well as
computations of Green's functions and the spectrum. Although some
results exist about isospectral graphs and spectral asymptotics, this
seems like a rich enough set of examples to warrant further
exploration.
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11:30 AM - 1:20 AM
Lunch
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Afternoon Session:
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1:30 PM - 2:20 PM
Margaret Symington (Georgia Tech)
Lefschetz fibers in integrable systems
Abstract:
An integrable system induces a singular Lagrangian fibration on
a symplectic manifold. We assume the fibers are compact. Then the
Lagrangian condition forces generic fibers to be tori. If the symplectic
manifold has dimension four then a typical isolated singular fiber of top
dimension is a Lefschetz fiber -- a sphere with one transverse
self-intersection. Such fibers appear with positive self-intersection in
physical integrable systems (such as the spherical pendulum) and in almost
toric fibrations (e.g. of the K3 surface). After discussing some features
of the neighborhood of such a fiber I will prove that a Lefschetz fiber
with negative self-intersection cannot occur.
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2:30 PM - 3:20
PM
Plenary Lecture: Henry C. Wente
(University of Toledo)
Elastica, Pendant Drops and Exotic Containers
Abstract
Elastica are planar curves, first studied by Euler, whose
curvature is a linear function of position. They are extremals of the
squared curvature functional subject to appropriate constraints. When the
curvature is proportional to the vertical coordinate, one obtains the
profile curve of the one-dimensional pendant or sessile liquid drop.
Extended horizontally, one generates surfaces giving solutions to the
pendant or sessile drop equation in Euclidean 3-space. We discuss the
properties of these curves and use them to construct new exotic containers.
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3:30 PM - 4:20 PM
Stelson Lecture:
Gerhard Huisken
(Albert-Einstein-Institut für Gravitationsphysik of the
Max-Planck Society)
A Priori Estimates and Surgery Constructions
for Mean Curvature Flow of Necks
Abstract:
The lecture explains how the geometry of necks in a hypersurface can be
controlled by a priori estimates for the curvature and then gives an
explicit surgery construction. Finally it is shown how the surgery can be
used to extend mean curvature flow beyond singularities for hypersurfaces
with the sum of the two lowest principal curvature positive everywhere.
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4:30 PM - 5:20 PM
Open Problem Session
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Past SGS
is the area
radius of the
outermost horizon and Q is the total charge. This yields a
counter-example to a natural extension of the Penrose Inequality
to charged black holes.