NSF/CBMS Workshop

Algebraic Topology in Applied Mathematics

The conference was held at Cleveland State University from August 3, 2009 to August 7, 2009. Photos from the conference may be found below.

Topology is an ideal mathematical discipline for contemporary problems in science and engineering. Indeed, Topology was conceived by Poincare as a necessary ingredient for understanding when differential equations can be solved and this has led, via de Rham and Morse theories to profound connections between Topology and Analysis. Topological results, being based not on precise distances or displacements but rather upon global features, tend to be very robust and insensitive to noise and various errors, a useful feature for real-world problems. Now, through applications to areas such as sensor networks, statistics and robot motion-planning, it is possible to \emph{see} Topology in the down-to-Earth (and practical) problems of modern technology. This CBMS Conference presented 10 lectures by Professor Robert Ghrist focusing on very recent applications of algebraic topology and its methods to problems in modern technological areas. In particular, lectures described how topological results may be used to show whether or not a sensor network has full coverage over a designated area, an issue important to national (or industrial) security. A different application is to the statistics of huge data sets (such as genetics data). New topological methods strive to identify global topological characteristics of data sets whose natural homes are in 30, 40 or 1000 dimensions. Traditional statistical methods are quite limited when dealing with such monstrous data sets and these new topological methods hold out hope for a completely new type of statistical analysis. Finally, in recent years, the subject of configuration spaces has received much attention in Topology because there are deep interactions between geometry and homotopy theory. But configuration spaces are also important in practical areas involving, for instance, robot motion-planning; areas (from space exploration to industrial work) that are essential for future research and development. This type of symbiosis between topological methods and practical application is emblematic of the lecture series. Most topological concepts and tools have remained ensconced within Mathematics departments, walled off from scientists and engineers by a formidable history of specialized terminology and subtle but non-intuitive algebraic constructs. The broad goal of these lectures was to interest younger mathematicians and engineers in pursuing the connections between Topology and applications in science and engineering, and to demonstrate forcefully the particular blend of efficacy and beauty that characterizes this budding relationship. Such interdisciplinary interactions provide a perfect pathway for the achievements of pure mathematics to make their way from the world of abstraction to the world of people!

Organizers Peter Bubenik and John Oprea:

--- Oprea Lecture on LS Category and its Applications at CBMS Workshop

Group Photo

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