My research falls in a branch of mathematics known as applied topology. The field of applied topology is a young and exciting area of mathematics research. In recent years, researchers have made much progress in applying topological methods to solve problems in engineering and computer science. Much of the success in these applications results from the use of topological methods to organize and analyze data.
My own work involves classification theorems for certain topological and geometric integrals called Hadwiger Integrals. I am working to prove theorems and develop computational techniques that will be useful in fields such as sensor networks, signal processing, cell dynamics, and astronomy. The following paragraphs describe some particular applications of topology.
Topology can be used to aggregate data from a network of sensors. For example, the Euler integral can produce a global count of objects of interest given local counts at various sensors along with connectivity information. Topology is relevant for other problems such as network coverage, surveillance, pursuit evasion, and network coding.
When multiple processors share common resources, topology can help describe the space of deadlock-free execution paths. Topology is also useful in image processing, extracting useful information from noisy images. In particular, algorithms involving Euler characteristic are of great utility for recognizing objects that appear in images.
Engineers building a robot that moves in some way can use topology to describe the robot's configuration space—that is, the space of possible configurations of the robot's joints. This helps solve motion planning and robot stability problems.
Want to read more?
- Read about topology on Wikipedia
- Robert Ghrist (my thesis advisor) has written notes on applied topology.
- Stanford University's Applied and Computational Algebraic Topology research group has links to many resources on their website.
- See my list of papers and presentations