Mitch Richling: Mathematics Research Interests

I've had the opportunity to model some extremely complex systems using distributed Monte Carlo techniques in a large grid environment. Everything about this process is interesting:

• The final results of course!
• The statistical & machine learning techniques involved in analyzing the results.
• The mathematical problems associated with modeling itself.
• The related computer science problems – scalable network distributed PRNGs, sequence/PRNG coverage, etc…)

I have a long standing interest in a wide array of numerical linear algebra topics. Most recently, iterative and direct methods for sparse systems have captured my imagination. I've also been experimenting with various SMP parallelization approaches for things like GMRES, and with distributed parallelization for sparse elimination methods.

I am most interested in distributed, dependent systems of independent nodes as they are related to natural growth patterns. Coral is a good example – i.e. many tiny organisms working to build a large, organized structure. A less obvious example are things like the patterns on a sea shell.

My interest in nonlinear dynamical systems is largely practical in the sense that I enjoy building models of systems – both physical models and computer models. My interest in fractals is mostly in generating images, and I maintain a high performance raster graphics library designed for just this application.

The topic of my master's thesis was using ancient OOP techniques to develop very a generic implementation of B. Buchburger's algorithm in such a way that the same code was capable of working over various rings without modification. I have continued to be interested in this topic; however, my focus has become more theoretical over time.

I first became interested in the visualization of sections of 4-manifolds, 3-manifolds, and knot spaces in my undergraduate years. Back then the visualization of these objects was completely beyond the capabilities of existing OpenGL hardware. Thus my efforts were on writing very specialized software rasterisation tools. With the today's GPU technology, I'm more focused on the automated generation of representative simplicial complexes for various objects. These simplicial complexes may then be triangulated, and rendered on a modern graphics card.