2004 New Directions Short Course

It's very easy to just keep doing what I've been doing, but it can get stale. The New Directions program really has made me think in new ways and encouraged me to branch out.

— Kevin Knudson, 2004 New Directions short course participant

The New Directions program, introduced in 2003, is designed to help mid-career mathematicians move into new interdisciplinary research areas. The two components of the program are the New Directions Visiting Professorships, which involve participation in the full academic year program, and the annual New Directions Short Course, which is an intensive two-week summer short course in a developing area of application that shows particular promise for innovative mathematical investigation. Short course topics are carefully selected with the goal of maximizing benefit both for program participants and the greater research community. The area should have the potential for significant impact on the scientific community and be sufficiently well established that networking and funding opportunities exist, while offering ample unexplored or underdeveloped territory in which a newcomer can find important but accessible research projects.

In this article, we focus on the 2004 New Directions short course Computational Topology, taught by Herbert Edelsbrunner (Computer Science) and John L. Harer (Mathematics) of Duke University. Edelsbrunner and Harer characterize Computational Topology as "the development of algorithmic tools implementing topological concepts for use in the sciences and engineering." They presented "a broad picture in which algorithmic tools connect pure mathematics with scientific applications", with in-depth analyses of examples from structural molecular biology and geometric modeling. The course structure was a mixture of general lectures by the instructors, topical lectures by guest speakers, and daily multi-hour brain-storming sessions.

Even though the course took place only three months ago, it has already led several of the participants into new projects, reshaping their research, teaching, and overall scientific outlook. Three participants — Henry King (University of Maryland), Kevin Knudson (Mississippi State University), and Neza Mramor (University of Ljubljana) — have written a joint paper and implemented their algorithm for generating a discrete Morse function on a simplicial complex. Peter Saveliev (Marshall University) writes "My research interests have been significantly influenced by this course. Now they are firmly in the area of applications of topology. The two topics I am currently pursuing are 'Topology of networks' and 'Topology of proteins'." Robert Morse (University of Evansville) is writing a paper on computational homology and cohomology that will integrate ideas from the short course.

Many of the workshop participants are now introducing their students to ideas and applications from the short course. Some are designing new courses in computational topology, while others are incorporating key material and examples (e.g. molecular bioogy and 3D printing) into existing courses at both the graduate and undergraduate level. Mike Melko (Northern State University) is writing a Mathematica package to produce a solid model of Costa's minimal surface and, eventually, more complex minimal surfaces. He presented preliminary results at the 17th Annual International Conference on Technology in Collegiate Mathematics, and plans to write an NSF RUI grant proposal with a component on the use of discrete Morse theory in drug design.

Henry King expresses the sentiments of many of the short course participants: "I came in knowing essentially nothing of the subject. I enjoyed the whole experience immensely and I feel I can actually contribute something to the subject."