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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
IV: Wednesday, December 10, 2003
Emory University
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Morning Session: Mathematics & Science Building, W303
8:00 AM - Coffee and refreshments, TBA
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8:30 AM - 9:20 AM
Shanshuang Yang (Emory University)
Quasisymmetric Parametrizations of Metric Spheres
Abstract: A metric sphere is a metric space which is homeomorphic to a sphere.
An interesting question to ask is under what conditions a metric sphere can be mapped onto
the standard sphere by a quasisymmetric homeomorphism.
In this talk we will present some general results for dimensions one and two,
and discuss how these results can be applied to certain subsets of the Euclidean space that arise naturally in
geometric function theory.
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9:30 AM - 10:20 AM
Sumio
Yamada (University of Alabama at
Birmingham)
Parameterizing Singular Minimal Subvarieties
Abstract: In this talk, we will introduce a new approach where one obtains minimal subvarieties (of length or area) by minimizing a weighted Dirichlet
energy. In minimizing length, one can solve the so-called Steiner minimal network problem, while minimizing area, we present some new insight on
the Plateau problem for singular soap films.
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10:30 AM - 11:20 AM
Herman
Gluck (University of Pennsylvania)
The Gauss Linking Integral on the 3-Sphere and in Hyperbolic 3-space
Abstract: Gauss's integral formula (1833) for the linking number of two disjoint
smooth closed curves serves as a cornerstone of geometric knot theory in Euclidean 3-space,
with rich connections to molecular biology (especially DNA structure) through the
related formula for the writhing number of a single curve, and to fluid dynamics,
electrodynamics and plasma physics through the related formula for the helicity of a vector field.
Gauss was interested in computing the linking number of the earth's orbit with the orbits of certain asteroids,
and although he presented his formula without proof, it is believed that he simply counted up how many times
the vector from the earth to the asteroid covered the celestial sphere...a degree-of-map argument.
Gauss undoubtedly knew another proof: run a current through the first loop,
and calculate the circulation of the resulting magnetic field around the second loop.
By Ampere's Law, this circulation is equal to the total current enclosed by the second loop,
which means the current flowing along the first loop, multiplied by the linking number of the two loops.
Then the Biot-Savart formula for the magnetic field leads directly to Gauss's linking integral.
We report here on the discovery and proof of the corresponding formulas for linking,
writhing and helicity on the 3-sphere and in hyperbolic 3-space.
Gauss's degree-of-map proof does not work on the 3-sphere; instead,
we develop there a rudimentary form of classical electrodynamics,
including a Biot-Savart formula for the magnetic field and a corresponding Ampere's law,
sufficient to lead us to the linking integral.
By contrast, a degree-of-map proof does work in hyperbolic 3-space.
This is joint work with Dennis DeTurck.
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11:30 AM - 1:30 PM
Lunch
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Afternoon Session: Mathematics & Science Building, W303
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1:30 PM - 2:20 PM
Bo Guan (University
of Tennessee)
Spacelike Entire Hypersurfaces of Constant Gauss
Curvature in Minkowski Space
Abstract: We attempt to classify spacelike entire hypersurfaces of constant Gauss curvature in Minkowski
space by their asymptotic tangent cones at infinity. We are also concerned with the question
whether such surfaces are complete. In particular, we will describe such surfaces with
SO(n-1,1) symmetries
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2:30 PM - 3:20
PM
John
McCuan (Georgia Institute of Technology)
On the Shape of Rotating Drops
Abstract: I will describe joint work with R. Hynd which
leads to a fairly complete understanding of the space of all rotationally symmetry surfaces
whose mean curvature is a quadratic function of distance to the axis of rotation (rotating drops).
In particular, we answer several questions posed by R. Gulliver concerning toroidal solutions.
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3:30 PM - 4:20 PM
Tilmann Glimm
(Emory University)
Optical Design of Reflector Systems and the Monge-Kantorovich Mass Transfer Problem
Abstract: We consider the geometric optics problem of constructing a system consisting of two
reflectors which transforms a plane wave front with given intensity into an output plane wave
front with prescribed output intensity.
We show how this problem is deeply connected to the Monge-Kantorovich mass transfer problem (MKP)
with quadratic cost function. This connection yields a new method of solving the two-reflector problem.
Conversely, the connection also gives novel insights into the geometric nature of the dual
formulation of the MKP; for example, the Kantorovich potentials are directly linked to the reflector surfaces.
The techniques extend to a single reflector problem, which can be linked to a
Monge-Kantorovich problem on the sphere. We will further present some numerical computations of the reflectors
based on a linear programming approach to the MKP.
This is joint work with V. Oliker.
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4:30 PM
Open Problems & Pizza
Click here for a write-up of
this session in pdf format. |
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