## Colloquium Schedule Fall 2012

### Colloquium Schedule - Fall 2012

Thursday, December 13, 2012, AB 634, 2:45 - 3:45 pm**Dr. Hong Wang**

Department of Mathematics

University of South Carolina

Fickian diffusion and anomalous diffusion, their fast numerical simulation

Abstract: Diffusion processes arise in nature, sciences, social sciences, and a variety of applications. The fundamental assumption underlying a Fickian diffusion process is that a particle's motion has little or no spatial correlation. The probability of finding a particle somewhere in space can be described by a Gaussian distribution, or equivalently the classical second-order diffusion equation.

Today an increasing number of non-Fickian diffusion processes has been found, ranging from the signaling of biological cells, foraging behavior of animals, to physics, finance, and solute transport in groundwater. In these processes a particle's long walk is not necessarily independent of each other and may have long correlation length, so the processes can have large deviations from the stochastic process of Brownian motion. Recent studies show that fractional diffusion and advection-diffusion equations provide an adequate description of transport processes that exhibit anomalous diffusion, which cannot be modeled properly by classical second-order diffusion equations.

Computationally, an important issue is that the numerical methods for three-dimensional anomalous diffusion processes are prohibitively expensive in terms of computational work and memory requirement. Even the simulation of a three-dimensional linear fractional diffusion equation with a moderate number of grid points can take a state of the art petaflop supercomputer at least hundreds of years to finish.

In this talk we go over classical Fickian diffusion and anomalous diffusion, and their relations with (fractional) calculus. We will also present some fast numerical solution methods for and discuss some mathematical issues on space-fractional diffusion equations.

Thursday, December 6, 2012, DMSC 103, 2:30 - 3:45 pm**Professor Richard Tapia**

Center for Excellence and Equality Education

Rice University

The Isoperimetric Problem Revisited: Extracting a Short Proof of Sufficiency From Euler's 1744 Proof of Necessity

Abstract: We'll watch a presentation of Dr. Richard Tapia from Rice University, given at the Annual Meeting of SIAM 2012. Slides with synchronized audio, talk was delivered in Minneapolis in July 2012.

Thursday, November 8, 2012, DMSC 103, 2:30 - 3:45 pm**Professor David Pask**

School of Mathematics and Applied Statistis

University of Wollongong

Representing a graph on a Hilbert space.

Abstract: In this presentation, I will show how to represent three different types of graphs on a Hilbert space. In the simplest case the edges and vertices are represented by matrix units in a finite matrix algebra. I will then briefly talk about how the properties of the graph translate into properties of the representation of graph.

Thursday, October 25, 2012, DMSC 103, 2:30 - 3:45 pm**Professor Jin Akiyama**

Tokyo University of Science

Math as I like it

Abstract: We'll watch a talk given by Dr. Jin Akiyama from Tokyo University of Science. The presentation was given at International conference "Contemporary Mathematics" in Moscow in June 2009.

Thursday, October 11, 2012, DMSC 103, 2:30 - 3:45 pm**Professor Douglas Arnold**

School of Mathematics, University of Minnesota

Past President's Address: Reflections on SIAM, Publishing and the Opportunities before us.

Presentation given at the Annual Meeting of SIAM 2012.

Slides with synchronized audio, talk was delived in Minneapolis in July 2012.

Thursday, September 27, 2012, DMSC 103, 2:30 - 3:45 pm**Professor Bruce Blackadar**

Department of Mathematics and Statistics, UNR

Homotopies and Liftings

Abstract: Absolute retracts and absolute neighborhood retracts (ANR's) are nice kinds of topological spaces which are very useful in various aspects of topology. One of the most important results about these spaces is the Borsuk Homotopy Extension Theorem.

C*-Algebras were originally algebras of bounded operators on Hilbert spaces, but they are now often thought of as "noncommutative topological spaces." There is good reason for this: commutative C*-algebras are, in a precise sense, really the same thing as (locally compact Hausdorff) topological spaces, and many topological ideas and results can be rephrased nicely in the language of C*-algebras by "turning arrows around" and remain true also for noncommutative C*-algebras. Some powerful machinery has thus come into the theory of operator algebras from topology, and conversely C*-algebras have become an important tool in some aspects of topology, such as the study of dynamical systems.

One can turn the arrows around in the definition of ANR's to obtain a class of C*-algebras called semiprojective C*-algebras, which have proved to be interesting and useful in C*-algebra theory analogous to the uses of ANR's in topology. I recently found a proof of the version of the Borsuk Homotopy Extension Theorem for semiprojective C*-algebras (the commutative proof does not adapt to the noncommutative case).

In the course of the work, I also identified and began to study a property of both spaces and C*-algebras somewhat similar to the ANR property (or semiprojectivity), called e-openness, which appears to be new and interesting even in the commutative case.

Outline of the talk:

1. What are ANR's and why are they interesting?

2. What does the Borsuk Homotopy Extension Theorem say, and how is it proved?

3. What do C*-algebras have to do with topology?

4. What are semiprojective C*-algebras and why are they interesting?

5. What does the version of Borsuk's Theorem for semiprojective C*-algebras say and how is it proved?

6. What are e-open and e-closed spaces and why are they interesting?

A number of examples will be discussed.