University of Georgia
Mathematics Department Colloquium 2005-2006
Time and place: Thursday at 3:30p.m., Room 304 Boyd Graduate
Studies, (unless otherwise specified).
September 2005
September 15, 2005
Speaker: Mike Wolf, Rice University
Title of talk: Minimal Desingularizations
of Planes in Space
Abstract: We prove that there is only one
way to 'desingularize' the intersection of two planes in
space to and obtain a periodic minimal surface as a result.
After explaining the statement and its context, with an
update on recent progress and challenges in minimal surface
theory, we give an overview of the proof of the result.
The argument is mostly an exercise in, and an introduction
to, the basics of Riemann surface deformation theory: we
translate the geometry of the minimal surface in space into
a statement about an elementary moduli space of planar domains,
and then study how those domains vary and degenerate. |
October 2005
October 14, 2005 - Please
note this is a Friday.
Speaker: Yuesheng Xu, Dept. of Math., Syracuse
University
Title of talk: Convergence Theorems
for Multi-Parameter Regularization Methods
for Solving Ill-Posed Operator Equations
Abstract: We consider in this talk
solving ill-posed operator equations. Based on a decomposition
for the solution space, we propose a multi-parameter regularization
for solving the equations. We provide convergence theorems
for the regularization methods to be convergent
October 20, 2005
Speaker: Alexander Kleshchev, University
of Oregon
Title: Polynomial representations of
GL(n)
Abstract: We trace the 120 year old history
of polynomial representations of GL(n), which has its roots
in classical invariant theory, through the work of Deruyts,
Schur, Weyl, Gelfand-Tsetlin, Lusztig and others. In the
end we will sketch a categorification of polynomial GL(n)-modules
obtained recently by Brundan and the speaker.
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November 2005
November 10, 2005
Speaker: Sergei Levendorski, Dept. of Economics,
University of Texas, Austin
Title of talk: Financial Mathematics
In the talk, it will be explained how finance have
served as a source of new mathematical objects and new variations
of mathematical methods; why many problems in mathematical
finance remain unsolved, and why new methods will be needed
as long as financial markets exist; why mathematical methods
applied to finance lead to explosion of financial markets,
and why, sometimes, the influence of the mathematical methods
on finance lead to very serious crises; that, contrary to
the wide-spread perception, mathematical finance
cannot be reduced to a subfield of Statistics and the theory
of Stochastic Processes in general and Stochastic Differential
Equations and Optimal Stopping theory in particular; there
are important applications for Complex Analysis, Integrals
Transforms, Partial Differential Equations and Pseudo-Differential
equations, Lie groups, Dynamic Systems, Lattice models and
related fields in Algebra, nothing to say about Numerical
Analysis.
Several basic types of problems will be discussed in some
detail. |
January 2006
January 10, 2006 (Please
note this is a Tuesday.)
3:30pm, Room 302
Speaker: Pete Clark, McGill
Title: Acquisition of rational points
on algebraic varieties
Abstract: A fundamental problem in arithmetic
geometry is to understand the set of Q-rational points
on an algebraic variety. It would seem that if there are
no rational points at all, we have a perfect understanding.
But in fact there is interesting geometry in "pointless
varieties." Especially, given such a variety V, over
which field extensions K/Q does V acquire K-rational points?
In this talk we concentrate on the case of algebraic curves,
presenting -- via results and conjectures -- the beginnings
of a general theory.
January 12, 2006
3:30pm, Room 302
Speaker: Xiang, Tang, UC Davis
Title of talk: Foliations, Hopf algebras
and modular forms
Abstract: Recently Hopf algebras have
played a role in several areas of mathematics and physics.
The fact that the same Hopf algebra was useful in the
study of foliations, renormalization in quantum field
theory, and number theory has led to interesting discoveries.
Inspired by the Rankin-Cohen brackets on modular forms,
Connes and Moscovici constructed a universal deformation
formula for the Hopf algebra associated to a codimension
one foliation. In this talk, we will explain how to use
differential geometry to understand their deformation
and various structures involved. In particular, we will
show that the Rankin-Cohen deformation is closely related
to the Weyl-Moyal product.
January 17, 2006 (Please note this is a Tuesday.)
3:30pm, Room 302
Speaker: Evgueni Tevelev, Univ. of Texas
at Austin
Title of talk: Equations of the moduli
space of stable rational curves
Abstract: At the most basic level, algebraic
geometry studies (systems of) polynomial equations and
the geometry of their solutions. Nowadays algebraic varieties
are usually defined in an abstract functorial way and
their equations (if one can find them!) provide an important
information about their geometry, deformations, degenerations,
etc. I will explain when equations are considered nice
(Green-Lazarsfeld properties and Koszul algebras). I'll
describe joint work with Sean Keel where we find equations
in the Lie operad of the moduli space of stable rational
curves .
January 19, 2006
3:30pm, Room 302
Speaker: Wee Lian Gan, MIT
Title of talk: Symplectic reflection
algebras and quantum Hamiltonian reduction
Abstract: Symplectic reflection algebras
of wreath-product type give noncommutative deformations
of the symmetric products of Kleinian singularities.
The representation theory of these algebras is expected
to be closely related to the geometry of Hilbert schemes
of points on minimal resolutions of the Kleinian singularities.
I will give an overview of some recent developments.
January 24, 2006
3:30pm, Room 302
Speaker: Ambrus Pal (IHES, France)
Title of talk: K_2 of elliptic surfaces
and the rigid analytic regulator
Abstract: Milnor K-groups of algebraic varieties play
a significant role in algebra, geometry, number theory
and even in mathematical logic. In spite of some spectacular
results, such as the work of Voevodsky on the Bloch-Kato
conjecture, some fundamental finiteness conjectures remain
open about these objects. In this talk I will explain
how a refined form of the Langlands correspondence over
function fields were used to make progress in this problem.
January 26, 2006
3:30pm, Room 302
Speaker: Alexander Iosevich, Univ. of
Missouri-Columbia
Title of talk: "Analysis, combinatorics
and number theory of distance sets".
Abstract: The Erdos distance conjecture
says that $N$ points in "d"-dimensional Euclidean
space determine at least $CN^{\frac{2}{d}}$ distinct distances.
The continuous analog of this conjecture, introduced by
Falconer says that if the Hausdorff dimension of a set
in Euclian space exceeds $d/2$ than the Lebesgue measure
of the set of distances is positive. We shall discuss
these conjectures and connections between them. We shall
also describe the finite field analog of these problems
where Gauss and Kloosterman sums play a crucial role.
January 30, 2006 (Please note this is a Monday.)
3:30pm, Room 302
Speaker: Paul Balmer, ETH Zurich
Title of talk: Triangular geometry
and applications
Abstract: We shall start
with the concept of triangulated category, reviewing examples
from Algebraic Geometry, Homotopy Theory, Modular Representation
Theory, Motivic Theory and more. We will then introduce
the basic ideas of how to do "geometry of triangulated
categories". Finally, we will see how these techniques
can be useful in some of the above examples.
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February 2006
February 7, 2006
3:30pm, Room 302
Speaker: Xiaoqiang Wang, University of Minnesota
Title of talk: Phase Field Models and
Simulations of Vesicle Bio-Membranes
Abstract: Recently, we began to systematically
model and simulate the shape deformation of vesicle membranes
using a unified energetic variational phase field method based
on the minimization of elastic bending energy with volume
and surface area constraints. Mathematical theory and numerical
algorithms are developed to for the phase field models. Rigorous
convergence theories of the numerical methods are investigated.
Many simulations are carried out in static and dynamic, axis-symmetric
and full 3D, one component and multi-component cases. The
new phase field modeling approach has the advantage of avoiding
tracking the free interfaces, and it can easily handle topological
changes. Meanwhile, a series of formulae for retrieving the
Euler number of the vesicles have been given and discussed
which may be useful for detection and control purposes.
The 3D codes developed for the equilibrium shape deformations
and the deformations and interactions with fluid fields
allow us to conduct extensive computational studies. Both
known and new equilibrium configurations have been discovered
in our numerical simulations. A detailed examination of
the energetic bifurcation landscape has been carried out.
We have further studied the effect of the spontaneous curvature
and have conducted simulations of vesicle transformations
in fluids. The further development of the phase field approach
for multicomponent vesicles gives us more tools to understand
new and complex phenomena recently being experimentally
studied by biologists.
February 15, 2006
3:30pm, Room 302
Speaker: Olga Plamenevskaya, MIT
Title
of talk: Heegaard Floer theory, knots, and contact
structures
Abstract: Heegaard Floer
theory is one of the most significant recent developments
in low-dimensional topology. Reminiscent of gauge theory,
it provides powerful invariants for 3-manifolds. Although
defined via holomorphic disks, these 3-manifold invariants
have an unexpected connection
to combinatorial knot invariants developed by Khovanov.
I will outline the construction of Heegaard Floer and Khovanov theories, as
well as their relation (due to Ozsvath and Szabo). Then,
I will expand these results to the world of contact topology,
providing a new invariant for transversal knots, and bringing
the correspondence between the two theories to a new level. |
March 2006
April 2006
April 12, 2006 -Please note
this is a Wednesday
Speaker: Frank Zeilfelder (Mannheim, Germany)
Title of talk: Recent Developements
in Multivariate Spline Theory and Its Applications
Abstract: We report on some recent developements
in the field of multivariate splines and its applications,
where the usage of these models turns out to be advantageous.
Multivariate splines are natural generalizations of splines
in one variable, where the piecewise polynomials satisfying
smoothness conditions are associated with partitions (such
as triangulations and tetrahedral partitions) of a given
n-dimensional domain. Having certain applications in mind,
by definition these spaces provide the necessary flexiblity,
but on the other hand, the vast literature shows that these
are very complex mathematical objects. We investigate basic
questions such as local interpolation by the spline spaces,
and discuss some related approximation methods, such as
quasi-interpolation. A common feature of the associated
operators is the locality and stability of the spline constructions,
so that we are able to show that the splines (and its piecewise
derivatives) yield optimal (and nearly-optimal) approximation
order. The algorithmic complexity of the interpolation and
approx-imation methods is linear, and therefore the splines
can be efficiently computed, evaluated, and visualized on
standard PCs. For these purposes, we take advantage of the
piecewise Bernstein-Bezier form of the splines that allows
to apply standard techniques well-known from CAGD (Computer
Aided Geometric Design). We illustrate the effectivity and
efficiency of the new spline methods by showing some applications
involving the (re)construction of terrains and surfaces
of arbitrary topology type, as well as the high quality,
interactive visualization of volume data, which plays a
key role in medical imaging, industrial quality control
and other areas.
April 27, 2006
Speaker: Gavril Farkas (University of Texas,
Austin)
Title of talk: The global geometry
of the moduli space of curves.
Abstract: The moduli space of curves M_g
is the universal parameter space for Riemann surfaces of
given genus. Its study has been initiated by Riemann in
1857 and it has been a long-standing problem to describe
the nature of the moduli space as an algebraic variety.
I will survey the history of the problem starting with Severi's
conjecture from 1915 predicting that M_g is always unirational,
continuing with the work of Harris and Mumford spectacularly
disproving Severi's conjecture and finally discussing a
recent result which settles this problem in one of the most
interesting remaining cases, that of genus 22. |
For abstracts and titles of previous years' colloquia click here:
1998-1999, 1999-2000, 2000-2001, 2001-2002, 2002-2003, 2003-2004, 2004-2005
For weekly seminar schedule click here.
For the Annual Distinguished Cantrell Lectures, click here.
Maps of Campus.
Your comments and suggestions for future
speakers are welcome. Please contact
Valery Alexeev,
valery @ math dot uga
dot edu
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