Iowa State University Mathematics Colloquium Spring 2005

Schedule of Talks

 Location: 290 Carver Hall

Time: 4:10-5p.m.

 

Special colloquia in Summer

·        May 13, Friday   2:10-3p.m. Professor Qiang Du, Pennsylvania State University, Modeling and simulations of vesicle membrane deformation  

·        July 6, Wednesday, 2:10-3p.m.  Ignacio Fernández Rúa, University of Cantabria, Spain.  Finite semifields

 

• Visitor Information

Abstracts:

Jan. 18, Tuesday, 4:10p.m. Professor Michael Doob, University of Manitoba

Title: Mathematics on the web: where are we heading?
Abstract:
We all want to put our mathematics on the web as easily as we do text and to get an attractive legible result. Good news: things do seem to be moving in the right direction. Bad news: it will still take some time to make it all easy and transparent to use. In this talk we'll review some of the current methods being used to post mathematics to the web, observe some of their shortcomings and will look into the near future at some of the developing techniques.
 

Abstracts:

Feb. 1, Tuesday, 4:10p.m. Professor Bokhee Im,  Chonnam National University

Title: Introduction to Comtrans Algebras
Abstract:
A comtrans algebra E over a unital commutative ring R is an R-module E equipped with two trilinear ternary operations, a commutator [x,y,z] and a translator < x,y,z> such that the commutator satisfies the left alternative identity [x,x,y] = 0, the translator satisfies the Jacobi identity and together the commutator and translator satisfy the comtrans identity [x,y,x] = < x,y,x >. In 1988, comtrans algebras were introduced in answer to a problem from differential geometry, asking for the algebraic structure in the tangent bundle corresponding to the coordinate n-ary loop of an (n+1)-web. The role played by comtrans algebras is analogous to the role played by the Lie algebra of a Lie group. The standard Lie algebra multiplication is the binary commutator [x,y]=xy-yx of a bilinear and associative operation xy. Similarly, the standard ternary comtrans algebra operations are the ternary commutator [x,y,z] = xyz - yxz and translator < x,y,z > = xyz - yzx of a trilinear operation xyz. In this talk, we will review old results and give some new prospects for the ongoing and future research related to comtrans algebras. 
 

Abstracts:

Feb. 8, Tuesday, 4:10p.m. Professor Fernando Souza, University of Iowa

Title: Diagrammatic morphisms and some applications
Abstract:
Graphical calculi abound in several branches of mathematics, computer science, and theoretical physics, providing a powerful tool and a concise language. They often consist of graphs immersed in the plane or in 3-space with vertices labeled by morphisms of a convenient category. These labeled graphs form categories with additional structures, which abstract familiar algebraic notions like tensor products, duality, traces, and irreducible representations, for example. These diagrammatic morphisms have been crucial for the development of subjects like quantum topology, quantum algebra, some approaches to semantics of programming languages, and some quantum field theories, among others. In this colloquium, we survey some major approaches to diagrammatic morphisms, and discuss their relationship with the well-studied case of freely generated categories with additional structure. Examples will be drawn from different areas. In spite of specific details for some of the examples, the only overall requisite is multilinear algebra.
 

Tuesday 15 Feb, 4:10p.m. Professor Mikhail Klibanov, North Carolina

Title: Inverse Problems and Carleman Estimates: Theory and Numerics

 

Abstract: 

Currently, the method of Carleman estimates is the only tool enabling on to prove global uniqueness theorems and to establish stability estimates for multidimensional coefficient inverse problems with single measurement data. This method was introduced in the field in 1981 by A.L. Bukhgeim and M.V. Klibanov (simultaneously and independently) and is widely explored now in the inverse problems community. Recently, M.V. Klibanov and A. Timonov have applied this method for constructing of globally convergent algorithms for a wide class of inverse problems. It is worthwhile to mention that except of a very few, current algorithms for coefficient inverse problems are locally convergent, i.e., their convergence is guaranteed only if the starting vector is located in a close proximity of the solution, which means that the solution is basically known in advance, except of a small perturbation. The talk will be focused on questions of uniqueness, stability and globally convergent algorithms for coefficient inverse problems.

 

Feb. 22 Tuesday, 4:10p.m. Professor Yuxi Zheng, Pennsylvania State University

Title: Shock reflection in two space dimensions  for the pressure-gradient and Euler systems of conservation laws

 Abstract: 

E. Mach revealed a major mathematical problem in 1878 in his famous physical experiment in which a shock wave is sent to hit a wedge, known as ``Mach reflection experiment'' or a ``plane shock hitting a wedge.'' The problem became very prominent during the atomic bomb age or supersonic flight age, when von Neumann devoted a tremendous amount of time and energy to the problem -- only to reveal more problems  (rather than answers) for the problem, and several von Neumann paradoxes were since known. The problem, determining the patterns of reflections, remains a high-status problem but little progress has been made. In the past few years, however, we have seen promising new work coming up, which includes Gary Lieberman and co-workers' work on the unsteady transonic small disturbance equations. In my colloquium talk, I will present the current status of the problem. In my seminar talk, I will give details of my recent results on a model called the pressure-gradient system and its implication on the full Euler system.

 

Mar. 1st, Tuesday, 4:10p.m. Professor Sriram Pemmaraju, Department of Computer Science The University of Iowa

Title: Approximation Algorithms for Max-Coloring
Abstract:
The minimum vertex coloring problem seeks to find the fewest colors to assign to the vertices of a given graph such that no two adjacent vertices have the same color. It is well known that this problem is hard to solve, even approximately, in general. However, for certain special classes of graphs, for example, perfect graphs, the minimum vertex coloring problem can be solved efficiently.

This talk will focus on a class of ``constrained coloring'' problems that arise from applications in scheduling. These problems are in general harder than the minimum vertex coloring problems. We investigate the question of whether these problems can be solved efficiently on classes of graphs on which minimum vertex coloring is easy. We will present approximation algorithms and hardness of approximation results for a certain constrained coloring problem.

 

Mar. 8 Tuesday, 4:10p.m. Professor Chun Liu, Pennsylvania State University

Title: Shock reflection in two space dimensions for the pressure gradient and Euler systems of conservation laws

Abstract:
I will discuss the uniform energetic variational approach in the study of free interface motions in fluids. We will employ a diffusive interface approach that emphasize the special coupling between the transport of the material variable and the induced stress. The examples include surface tension, Marangoni effect, elastic membranes and the effects of the special elastic properties of the materials.

 

March 22, Tuesday, 4:10p.m. Professor Peter  Polykov University of Wyoming
Title: On a boundary value problem in 2D subsonic aeroelasticity
Abstract:
We consider the linearized equation of the subsonic inviscid compressible 2-dimensional flow with three boundary conditions: flow tangency condition, which corresponds to the physical property of the continuity of the flow on the boundary of the wing, Kutta-Joukowski condition of the acceleration potential being zero outside of the wing, and the velocity potential tending to zero at infinity. We reduce the problem to the solution of a linear integral equation and prove the existence of a special solution of this equation under a special condition on the normal velocity of the wing. The proof is based on the results of Carleman on linear integral equations and results of Tricomi on the finite Hilbert transform.  

Mar. 31 Tursday, 4:10p.m. Professor Shi Jin, University of Wisconsin-Madison

Title: Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials

Abstract:
When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of zero time step due to the CFL constraint, the inconsistency to the constant Hamiltonian, and more seriously, the uniqueness of weak solutions. We propose a class of Hamiltonian-preserving schemes that are able to overcome these numerical and mathematical deficiencies. The key idea is to build into the numerical flux the behavior of a classical particle at a potential barrier. This defines a unique weak solution. We establish the stability theory of these new schemes, and analyze their numerical accuracy. Numerical experiments are carried out to verify the theoretical results. This method can also be applied to the level set methods for the computations of multivalued physical observables in the semiclassical limit of the linear Schrodinger equation with a discontinuous potential, high frequency wave propabagation with a discontinuous local wave speed, among other applications.

April 5, Tuesday, 4:10p.m. Professor Marta Asaeda, University of Iowa
Title: Quantum invariants for links and khovanov homology
Abstract:
Khovanov homology was first introduced by Khovanov as a categorification of the Jones polynomials. I will talk about its generalization to Skein modules. I will also talk about future perspective, involving knot invariants obtained from operator algebras.
 

April 11, Monday, 4:10p.m. Professor Charles Doran, University of Washington, Seattle

Title: String Theory and Mathematics

Abstract: 

Why are mathematicians so excited about string theory? It is clear why a physicist might be interested in a theory that seeks to unify all the forces of nature, a theory whose stated goal is to obtain a consistent set of equations explaining galaxies and subatomic particles and everything in between. It's not so clear why mathematicians who study geometry, topology, algebra, combinatorics, number theory, etc. should care. The answer lies in the powerful application of ideas from string theory within mathematics. Indeed string theory has pointed out deep connections between whole branches of mathematics that were previously thought unrelated. It is the goal of the speaker to convey the spirit of these developments without the technical details.
 

April 12, Tuesday, 4:10p.m. Professor Jim Keener, University of Utah

Title: A Mechanism for the Onset of Fibrillation following a Heart Attack
Abstract:

Each year in the United States, approximately a quarter of a million people die as the result of a heart attack before reaching a hospital.
In most of these cases, a coronary occlusion led to the sudden onset of fibrillation, a condition, which if not arrested, is fatal.

In this talk, I will use models, mathematical analysis and numerical simulations to describe a possible mechanism for the onset of fibrillation following a coronary occlusion. The mechanism proposed here is substantially different than previously proposed mechanisms for the initiation of reentrant activity (spiral waves, etc.), as it takes into account some of the dynamic processes that are unique to heart attacks.
 

April 22, Friday 3:10-4p.m. Professor E. DiBenedetto,  Vanderbilt University
Title: Homogenized Limits in Phototransduction.
Abstract:
The rod outer segment (ROS) in vertebrates consists of a cylinder including a stack of thin discs carrying photoreceptors. The second messengers Calcium and cGMP diffuse in the cytosol, which is the layered domain formed by the ROS from which the discs are ideally removed.

The discs bear incisures, thus generating a layered geometry with spikes. The diffusion of second messengers in such a domain is discussed by a homogenization limiting process. The main mathematical difficulty is in identifying a suitable notion of convergence and corresponding compactness.

 

April 26, Tuesday, 4:10p.m. Professor Yang Wang,  Georgia Institute of Technology

Title: Hilbert's 3rd Problem, Lattice Tiling and Weyl-Heisenberg Bases

Abstract: 

Hilbert's 3rd Problem asked whether a given polytope can be dissected  into finitely many parts so that these parts can be reassembled into  another given polytope. This problem relates to another interesting problem: Given two lattices in ${\mathbb R}^n$, can we always find a set that tiles ${\mathbb R}^n$ by both lattices? We will discuss these problems in this talk. In particular, the lattice tiling problem has a surprising application in time-frequency analysis regarding the existence of Weyl-Heisenberg orthonormal bases for $L^2({|mathbb R}^n)$.

 

May 13, Friday, 2:10p.m. Professor Qiang Du,  Pennsylvania State University

Title: Modeling and simulations of vesicle membrane deformation
Abstract: In this talk, we report some joint works with colleagues at Penn State on a diffusive interface based energetic variational approach for modeling the vesicle membranes under elastic bending energy and its possible impact on the study of complex bio-membrane systems. The effectiveness of our approach is substantiated via careful analysis and extensive computation. Mathematically, the problem is closely related to the well-known Willmore problem in differential geometry and the Gamma convergence of nonlinear functionals in calculus of variation.  Computationally, various membrane configurations are illustrated, interactions with fluid flow and other objects are examined. The numerical procedures are shown to be insensitive to topological events. In addition, we also discuss the problem of numerically retrieving useful topological information of the membrane from the energetic variational model for both tracking and control purposes which may be of even broader  interests.

 

For more simulation results, please check
http://www.math.psu.edu/qdu/Res/Pic/gallery5.html
Related references can be downloaded from
http://www.math.psu.edu/qdu/Res/year.html
 

 

Contact Information:

boushaba@iastate.edu or linglong@iastate.edu