Southeast Geometry Conference 2008
Titles and Abstracts
- Yuanan
Diao (University of North Carolina at Charlotte)
- Title:
Jones Polynomial of Knots from Repeated
Tangle Replacement
Operations.
- Abstract: I will show that the Jones polynomial of a link
diagram obtained through repeated tangle replacement operations
can be computed by a sequence of suitable variable substitutions
in simpler polynomials. In the case that all the tangles
involved in the construction of the link diagram have at most $k$
crossings (where $k$ is a constant independent of the total number
$n$ of crossings in the link diagram), the computation time needed to
calculate the Jones polynomial of the link diagram is bounded above by
$O(n^k)$. In particular, the Jones polynomial of any Conway algebraic
link diagram with $n$ crossings can be computed in $O(n^2)$ time. A
consequence
of this result is that the Jones polynomial of any Montesinos link
(this would include any two bridge knot or link) of $n$ crossings
can be computed in $O(n^2)$ time.
- Mohammad
Ghomi (Georgia Institute of Technology)
- Title: Topology of Riemannian submanifolds with
prescribed boundary.
- Abstract: In this talk we show that a closed
submanifold of codimension 2 in
Euclidean space bounds at most finitely many topological types of
complete hypersurfaces with nonnegative curvature. This settles a
question of Guan and Spruck. Further we discuss analogous results for
arbitrary Riemannian submanifolds. On the other hand, we show that
these finiteness theorems may not hold if the codimension is too high
or the boundary is not sufficiently regular. The proofs employ, among
other methods, the Gromov-Perelman theory of Alexandrov spaces with
curvature bounded below, and a relative version of Nash's isometric
embedding theorem. These results include joint works with Stephanie
Alexander, Robert Greene, and Jeremy Wong.
- Tom Ivey (College of Charleston )
- Title: Some Results on Austere 4-folds.
- Abstract:
Austere submanifolds in Euclidean space are for which the eigenvalues
of the second fundamental form, in any normal direction, are symmetrically
arranged around zero. The class of austere submanifolds was
introduced by Harvey and Lawson in 1982, as a way of generating
special Lagrangian submanifolds via the conormal bundle. Austere
3-folds were classified by Bryant, and austere submanifolds with
rank 2 Gauss map were classified by Dajczer and Florit.
In this talk, I will discuss joint work with Marianty Ionel
on classifying austere submanifolds of dimension 4.
These include complex surfaces in C^n, but we show that
there are a wealth of new examples with the same type of
compex-linear second fundamental form. We also obtain
finiteness theorems and sharp lower bounds
on the dimension of the linear subspace in which the submanifold lies,
both depending on the type of the second fundamental form.
- Alex
Kasman (College of Charleston)
- Title: Grassmannians are simpler than we thought.
- Abstract:
The Grassmannian is a fundamental object in geometry which
generalizes the notion of projective space by parametrizing k-
dimensional subspaces of an n-dimensional space. As an algebraic
variety, the Grassmannian is the intersection of a collection of
quadratic polynomials: the Pl�cker Relations. A common measure of
the complexity of a quadratic polynomial is the quadric rank, which
completely identifies the corresponding variety up to isomorphism.
Since the quadric rank of Pl�cker relations can be arbitrarily large,
it has long been assumed that Grassmannians in general have large
rank and that it is only in the case of k=2 that the rank reaches its
minimum value of 6. Our main result, that every Grassmannian is in
fact the intersection of a finite number of rank 6 quadrics, is
therefore quite a surprise. (Or, to put it another way, we
demonstrate the existence of an alternative set of polynomials which
also "cut out" the Grassmannian but are actually simpler than the
Pl�cker relations.) The conjecture that this would be so was
inspired by the theory of solitons in mathematical physics, but the
proof is entirely algebro-geometric and (again, surprisingly)
decidedly affine. This is joint work with Takahiro Shiota (Kyoto)
along with Kathryn Pedings and Amy Reiszl, two undergraduate students
at the College of Charleston.
-
John McCuan
(Georgia Institute of Technology)
- Title:
Symmetric constant mean curvature surfaces in the three sphere.
- Abstract:
I will discuss several notions of symmetry that generalize that
of being invariant under an isometry fixing a geodesic. I will then
discuss a recent classification of all compact cmc surfaces with this
symmetry. The classification extends also to all complete surfaces in
some cases including the basic notion of symmetry mentioned above.
-
Conrad Plaut
(University of Tennessee, Knoxville)
- Title: Homotopy Critical Values and the UU-cover.
- Abstract:
The uniform universal cover (UU-cover) and associated uniform
fundamental group were developed in joint work with Valera Berestovskii as
replacements for the traditional universal cover and fundamental group for
spaces that are not locally nice. We will sketch the construction of the
UU-cover, state the main theorems concerning it, and illustrate with a few
well-known examples of spaces without traditional universal covers: the
Hawaiian earring, the topologist�s sine curve, the Sierpin�ski carpet, and
the infinite dimensional torus. The UU-cover exists for any geodesic
space, including any Gromov-Hausdorff limit of Riemannian manifolds.
Investigation of these limits leads naturally to the notion of homotopy
critical spectrum, which is closely related to the covering spectrum of
Sormani and Wei. We will discuss the known relationships between their
work and ours, and also show that the homotopy critical values determine
an invariant ultrametric on the uniform fundamental group.
-
Jesse Ratzkin
(University of Georgia)
- Title: Rigidty and deformations of constant mean
curvature surfaces.
- Abstract:
A properly embedded constant mean
curvature (CMC) end has a well-defined asymptote,
which means one can assign asymptotic data to
any properly embedded, CMC surface with finite
topology. Does the asymptotic data determine
the CMC surface, at least locally? I will discuss
recent progress in answering this question in
the presence of symmetry and some topological
restrictions, by understanding the linearized
problem.
-
Sema Salur
(University of Rochester)
- Title: Calibrated Geometries.
- Abstract:
Calibrated submanifolds are distinguished classes of minimal
submanifolds and their moduli spaces are expected to play an important role
in geometry, low dimensional topology and theoretical physics. Examples of
these submanifolds are special Lagrangian 3-folds for Calabi-Yau,
associative 3-folds and coassociative 4-folds for G_2, and Cayley 4-folds
for Spin(7) manifolds.
In this talk, I will first give brief introductions to Calabi-Yau and G_2
manifolds, and then a survey of recent research on the deformations of
calibrated submanifolds inside Ricci-flat manifolds.
-
Bianca Santoro
(Duke University)
- Title: Complete kahler metrics with prescribed Ricci curvature
on quasiprojective manifolds.
- Abstract:
In 1990, Tian and Yau \cite{TY1} settled the non-compact version of
Calabi's Conjecture by proving the existence of complete, Ricci-flat
Kahler metrics on smooth, open manifolds that can be compactified by
adding a smooth, ample divisor.
In \cite{S}, we described the asymptotic behavior of the metrics
considered by Tian and Yau.
In this talk, we will discuss that result. in addition to some existence
results, as well as its asymptotic behavior, of complete, Ricci-flat
Kahler metrics on an extended class of open Calabi-Yau manifolds. As a
corollary, we may provide a wider class of examples of complete,
Ricci-flat metrics on open manifolds.
-
Christina Sormani
(Lehman College and CUNY Graduate Center)
- Title: Understanding the Topology of Manifolds with
Nonnegative Ricci Curvature.
- Abstract:
While there have been many advances in the understanding of the topology
of complete noncompact manifolds with nonnegative Ricci curvature, this
area is wide open for further study. In particular, Milnor's famous
conjecture that such a manifold has a finitely generated fundamental group
is still open even in dimension three. In fact it might even be false!
Possible counter examples involving the dyadic solenoid complement might
exist. The speaker will survey a selection of theorems and examples
emphasizing the results with the most geometrically intuitive proofs
including some of her own, and a recent result by her doctoral student,
Michael Munn. A survey of these results was written with Zhongmin Shen
and is available on the arxiv.
-
Matthias Weber
(Indiana University, Bloomington)
- Title: Minimal Surfaces with Little Topology.
- Abstract:
This talk will be a survey about complete,
properly embedded minimal surface in Euclidean space of relatively
small genus. I will contrast recent classification results with
equally recent new examples.