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The Southeast Geometry
Seminar (SGS) is a semiannual series of one day events sponsored jointly by:
- University of Alabama at Birmingham
- Georgia Institute of Technology
- Emory University
The organizers are: John McCuan (GIT), Vladimir Oliker (Emory), and
Gilbert Weinstein (UAB). |
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SGS
VI:
Wednesday, December 8, 2004
University of Alabama at Birmingham
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Morning Session:
Campbell Hall, Room 435
9:00 AM - Coffee and refreshments, CH 435
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9:30 AM - 10:20 AM
Jyotshana Prajapat (UAB)
On a New Characterisation of the Sphere
Abstract: We establish a relationship between stationary
isothermic surfaces and
uniformly dense
domains. A stationary isothermic surface is a level surface of
temperature which does not evolve with time. A domain
Ω
in the N-dimensional Euclidean space RN
is said to be uniformly dense in a surface
Γ
in RN
of codimension 1 if for every small r > 0 the volume of the
intersection of Ω
with a ball of radius r and center x does not depend on
x for x in Γ.
We
prove that the boundary of every uniformly dense domain which is bounded
(or whose complement is bounded) must be a sphere. We then examine a
uniformly dense domain with unbounded boundary
∂Ω and we show that
the principal curvatures of
∂Ω satisfy certain identities.
The case
in which the surface
Γ
coincides with
∂Ω is
particularly interesting. In fact, we show that if the boundary of a
uniformly dense domain is connected then (i) if N = 2 it must be either
a circle or a straight line and (ii) if N = 3 it must be either a
sphere, a spherical cylinder or a minimal surface. We conclude with a
discussion on uniformly dense domains whose boundary is a minimal
surface.
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10:30 AM - 11:20 AM
Plenary Speaker:
Igor
Rodnianski (Princeton University)
Recent developments in the Cauchy problem in General Relativity
Abstract:
The talk
will describe recent advances concerning local and global properties of
solutions of the Einstein equations. These will include: a new approach
to the problem of stability of Minkowski space for the Einstein-vacuum
and Einstein-scalar field equations, local continuation results and the
L2-curvature
conjecture, and the Price law in the collapse of a self-gravitating
scalar field.
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11:30 AM - 1:30 PM
Lunch
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Afternoon Session:
Campbell Hall,
Room 435
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1:30 PM - 2:20 PM
Nándor Simányi
(UAB)
The Boltzmann-Sinai Ergodic Hypothesis in Two Dimensions
(Without Exceptional Models)
Abstract:
We
consider the system of N (N
≥ 2)) elastically colliding hard
balls of masses m1,..., mN
and radius r in the flat unit torus Td
(d
≥ 2). In the case
d=2
we prove (the full hyperbolicity and) the ergodicity of such systems for
every selection (m1,..., mN;
r) of the external geometric parameters, without exceptional values.
In higher dimensions, for hard ball systems in Td
(d ≥ 3),
we prove that every such system (is fully hyperbolic and) has open
ergodic components. |
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2:30 PM - 3:20
PM
John McCuan
(GIT)
Symmetric CMC
surfaces in the three-sphere
Abstract: I will describe a Delaunay-type classification theorem
for surfaces in the three-sphere. |
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3:30 PM - 4:20 PM
Vladimir
Oliker
(Emory)
A solution of A.D. Aleksandrov's problem via Monge-Kantorovich Theory
Abstract: We
give a variational solution of the A.D. Aleksandrov problem of existence
of a noncompact complete convex hypersurface with prescribed integral
Gauss curvature. The required functional is motivated by the
Monge-Kantorovich theory of optimal mass transfer. |
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4:30 PM -
5:20
PM
Jason Parsley
(University of Georgia)
Helicity of vector fields on S3
Abstract: The helicity of a vector field measures the extent to
which its flowlines wrap and coil around one another. Helicity is
analogous to the writhing number of a curve, and is closely related to
the linking number of two curves. On the three-sphere, we define
helicity using an integral formula and show this is in accordance with
the definition in Euclidean space. For a vector field V defined
on a subdomain of the three-sphere, upper bounds on the helicity of V
are established. We detail applications of helicity to geometric knot
theory, plasma physics, and energy minimization problems for vector
fields. |
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