From June 15-26, 2009 the IMA will host an intensive short course designed to efficiently provide researchers in the mathematical sciences and related disciplines the basic knowledge prerequisite to undertake research in applied algebraic topology. The course will be taught by Gunnar Carlsson, Department of Mathematics, Stanford University and Robert Ghrist, Department of Electrical and Systems Engineering, Department of Mathematics, University of Pennsylvania. The primary audience for the course is mathematics faculty. No prior background in applied algebraic topology is expected. Participants will receive full travel and lodging support during the workshop.
Technological progress in data collection and storage brings with it the challenge of comprehension — to know the data is by no means the same as to understand the data. The 'big picture' requires an understanding that is global, as opposed to local. Such challenges are manifest across various disciplines: the difficulties of aggregating readings over a large sensor network, routing messages in a large communications network, finding periodic behavior in biological systems, or determining beliefs over a large social network, are all global in nature, requiring a degree of understanding that transcends local or combinatorial data.
Many of these same questions about the transition from local to global were asked in different guise a century ago at the dawn of algebraic topology. In this realm, one wishes to discern the global properties of a space given its local features (e.g., charts and overlaps, in the case of a manifold; cells and attaching maps in the case of a cell complex). The 20th century saw the creation of vast, elegant machinery for answering global questions. Recent developments in mathematics have made this machinery a useable computational tool, as well as a valuable tool for theoretical investigations. Spurred by advances in computation, classical and contemporary ideas in algebraic topology are emerging as tools for global problems in data analysis (in biology, image processing, biochemistry) as well as in engineering (robotics, communication systems, sensor networks).
This short course will balance theory and applications, with the dual goal of inspiring topologists to focus their skills on contemporary applications and informing practitioners of the panoply of available techniques for global challenges. An introduction to the theory will be followed by tutorials on diverse application domains, as well as specialized computational and theoretical techniques which make the methodology practical.