Recent years have witnessed dramatic advances in the sophistication and efficiency of methods for algorithmically rendering extremely complex geometric objects. As the objects have become more complicated and the demands for accuracy have escalated, it has become increasingly important to guarantee that the computer generated object agrees with the actual object at ever more precise and subtle levels of comparison.

It is exactly these concerns that lie at the heart of the relatively new field of computational topology – a field offering new challenges and opportunities for meaningful collaboration between computer scientists and mathematicians. It is not enough to merely have an array of mathematical invariants that can be used to verify consistency of objects with regard to such notions as topological, embedding, and homotopy equivalence. To render objects algorithmically and simultaneously check consistency, it is necessary that the shape invariants be effectively computable, and better yet, that they can be computed efficiently.

Several fundamental concepts and questions in the field of computational topology will be discussed in this talk, primarily from the perspective of a special class of objects called swept volumes. These objects, representing the totality of points traversed by an initial region that is transported through space, have certain features that lend themselves to the efficient computation of shape invariants. The special features associated with the swept volume structure will be described, and recent progress in implementing a number of mathematical approaches will be summarized. Related open problems of current interest – ideally suited for joint mathematics and computer science research efforts – also will be discussed.

Denis Blackmore has been a Professor of Mathematical Sciences at the New Jersey Institute of Technology (NJIT) since 1982 and has been a visiting member of the Courant Institute of Mathematical Sciences on several occasions. Previously he taught at the Polytechnic University of New York. He is a co-founder and member of the Center for Applied Mathematics and Statistics at NJIT. While conducting his research in computational topology, dynamical systems, and differential topology, he has also devoted considerable time to engaging in collaborative research with scholars in various science and engineering disciplines. His research in fluid dynamics, mathematical physics, biomathematics, manufacturing science, granular flow dynamics, and metrology reflects his interests in applications of mathematics.


Biography:
Professor Blackmore received his Ph.D. in Mathematics in 1971 from the Polytechnic University of New York. He also earned an M.S. in Mathematics and a B.S. in Aerospace Engineering from the same institution. His research as a graduate student was in the areas of boundary layer theory and the qualitative theory of differential equations.

He has been invited to lecture on his research at universities and mathematical institutes in the U.S. and several other countries including China, Poland, Russia, Turkey and Ukraine. In addition, he has been an invited speaker at numerous national and international conferences, colloquia and symposia. Dr. Blackmore has organized or co-organized several national and international conferences including the 834th Meeting of the AMS in 1987, a minisymposium on Integrable Dynamical Systems and Their Applications at ICIAM 95, a minisymposium on Dynamical Systems in Manufacturing at ICIAM 99, the 1992 Japan-USA Symposium on Flexible Automation, the 1999 IUTAM Conference on Segregation in Granular Flows, and FACM'04 at NJIT.

Dr. Blackmore has co-written two books, co-edited one book, is co-writing a Springer monograph on integrable (infinite-dimensional) dynamical systems and has published more than 120 scientific papers in leading journals, books and conference proceedings. He has received grants from DARPA, NSF, ONR and the New Jersey Commission on Science and Technology. In addition to his research, for which he received the Harlan Perlis Research Award from NJIT in 1993, he is also devoted to instruction in mathematics and has won awards for his teaching. He has created more than a dozen graduate and undergraduate courses and was instrumental in developing the Ph.D. program in mathematical sciences at NJIT. He is a member of Sigma Gamma Tau, Pi Mu Epsilon, Tau Beta Pi, of SIAM, the AMS, GAMM, IAMP, the MAA, the AAAS, the NYAS, and the Society for Natural Philosophy.