Applied Topology

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.

Networks

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.

Computer Science

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.

Robotics

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.

 
The 2013-2014 thematic program at Institute for Mathematics and its Applications is Scientific and Engineering Applications of Algebraic Topology. I am honored to be able to participate in the entire program as a postdoctoral fellow.

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