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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
Dupree
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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10:30 AM - 11:20 AM
Gilbert
Weinstein (University of Alabama at Birmingham)
On a Penrose inequality with charge
Abstract: We construct a time-symmetric asymptotically flat
initial data set to the Einstein-Maxwell Equations which satisfies
where m is the total mass,
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.
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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