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The Southeast Geometry
Seminar (SGS) is a 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 III:
Tuesday, May 6, 2003
University of Alabama at Birmingham
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Morning Session: Campbell
Hall, 435
8:00 AM - Coffee and refreshments, CH 451
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8:30 AM - 9:20 AM
Mohammad
Ghomi (University of South Carolina)
Locally Convex Hypersurfaces and
Monge-Ampere Equations
Abstract:
We will give a survey of some recent results concerning the
geometry and topology of positively curved hypersurfaces with boundary
in Euclidean space. These include a recipe for constructing closed
hypersurfaces of positive curvature with prescribed submanifolds and
tangent planes, an optimal regularity theorem for the boundary of convex
hulls of submanifolds, and the speaker's joint work with S. Alexander on
a convex hull property for positively curved surfaces which is dual to
that of the negatively curved case, and a convergence theorem for
locally convex surfaces. The latter has been used recently by Guan and
Spruck in their work on existence of surfaces with constant positive
curvature and prescribed boundary.
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9:30 AM - 10:20 AM
Joseph Fu (University
of Georgia)
Absolute curvature integrals and
the metric stability of immersed surfaces
Abstract:
We will explain how bounds on the area of the unit normal
bundle N(S) of a surface S immersed in R3 control (to some extent) the
intrinsic geometry of S. For example, the intrinsic diameter of S is
dominated by a multiple of area N(S), and with suitable topological
control a sequence of surfaces with bounded unit normal areas converges
to any Hausdorff limit even as length spaces .
Since area N(S) is roughly proportional to the sum of area S and the
L1 norms of the second fundamental form and the Gauss curvature
K of
S, a bound on area N(S) is equivalent to a bound on these curvature
integrals. Our guiding principle is that these results are a kind of
converse to the Theorema Egregium: put very roughly, not only is K
computable from the intrinsic metric, but also K - or rather the Gauss
map, whose graph is N(S) - is essential in taming the metric. This
principle may very well not hold water, but whether or not it does
seems to turn on some open questions.
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10:30 AM - 11:20 AM
William Minicozzi (Johns Hopkins
University)
Embedded Minimal Disks
Abstract:
The two classical examples
of embedded minimal disks are the plane and the helicoid - the latter is
a double spiral staircase. I will talk about joint work with Toby
Colding which shows that these are the only possible local pictures and
then uses this local structure to prove a global compactness theorem for
embedded minimal disks. |
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11:30 AM - 1:30 PM
Lunch
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1:30 PM - 2:20 PM
Conrad
Plaut (University
of Tennessee)
Finitely Presented
Groups Arising from Schreier Groups
Abstract:
What we call Schreier groups come from a natural construction discovered
in the 1920's by O. Schreier--so natural, in fact, that it was independently
rediscovered by Malcev in the 1940's and by Tits in the 1960's. Like most
natural constructions, Schreier groups have proved useful. Schreier and Tits
used them for covering group theory; Valera Berestovskii and I recently
followed the same path, and found (among other things) a very general covering
group theory for topological groups. Schreier's construction builds a new
group using only information contained in some symmetric open neighborhood of
the identity in a topological group. For covering group theory one uses a
connected open set or a decreasing collection of open sets (in which case
connectedness is not an issue). This talk will consist of a summary of the
covering group theory followed by some preliminary results concerning Schreier
groups obtained from open sets with finitely many connected components, and
the finitely presented groups that arise. In particular, we will discuss how
the geometric interaction of the components of the open set determines the
relations in the finitely presented group (part of the latter is work of a
student, David Phillipi.)
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2:30 PM - 3:20
PM
Marcel Griesemer (University
of Alabama at Birmingham)
On a Variational Principle for
Eigenvalues of Dirac Operators
Abstract:
The classical Courant-Hilbert minimax principle gives a variational characterization of the
eigenvalues at one end of the spectrum of self-adjoint semi-bounded operators.
It is not applicable to non-semibounded operators, such as Dirac operators.
I shall report on a generalization of this minimax principle that does not have this limitation.
This new variational principle has lead to interesting new results on the spectrum of Dirac operators
(describing a massive quantum particle) on Euclidean space.
To exploit it in the analysis of Dirac operators on compact manifolds is an open challenge.
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3:30 PM - 4:20 PM
Xiangsheng
Xu (Mississippi State University)
Regularity Theorems for a Class of Degenerate
Elliptic Systems
Abstract:
The thermistor problem is a mathematical
model, describing the interaction between temperature
and electrical current in a conductor where electrical
and thermal properties are highly
temperature-dependent. The stationary problem consists
of a system of two partial differential equations with
quadratic nonlinearities and diffusive coupling.
It is well-known that one can obtain the boundedness of the temperature
by suitably imposing the boundary conditions.
We are interested in the local regularity of solutions to the system.
We show that certain regularity properties still hold even
in the case where the temperature blows up in a region with non-empty interior.
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