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 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.

Several basic types of problems will be discussed in some detail.

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.

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.

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.