Colloquia, Seminars and Conference News
Title : Effective Computability of Equivalence of Stratified
Date : December 1, 2006. (11:00 am) Tea starts half an hour before each seminar
Location: ITEB 336
Speaker : Denis Blackmore
Abstract:
We first describe the category $CT^k$, $1 \le k \le \omega$, whose objects $V$ are locally compact subsets of a Euclidean space possessing a finite, $C^k$ Whitney regular stratification
$V = M_1 \cup M_2 \cup \cdots M_s$
with morphisms $\phi: V \rightarrow W$ that are $C^k$ maps respecting the stratifications of $V$ and $W$. It is shown that this category is especially well suited to the formulation and solution of a wide range of problems in computational topology involving geometric objects that are piecewise differentiable manifolds or non-manifolds (varieties).
For one thing, the category $CT^k$ includes just about all of the geometric objects that one is likely to encounter in computational topology applications. Moreover, this category enjoys the property of having analogs of many of the most important and useful structures and operations associated with differentiable manifolds. For example, it is demonstrated that the objects in $CT^k$ have tubular-like neighborhoods, which can be used to reduce various shape invariant analyses to local calculations (in these neighborhoods). More particularly, they can be employed - as will be explained in a brief description of current joint work with R. Kopperman and T. Peters - to simplify certain questions pertaining to embedding equivalence or ambient isotopy to questions of homeomorphism equivalence in tubular-like neighborhoods. We also shall show how recent results on effectively computable criteria for ambient isotopy of manifolds (with and without boundaries) by Peters et al. can be extended to objects in $CT^k$, and indicate how the tubular neighborhood structure can facilitate homology and cohomology calculations via the $T_0$ space approximation methods developed by Kopperman et al.
Bio: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. 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 computer-aided geometric design, fluid dynamics, granular flow dynamics, various engineering disciplines, mathematical physics, biomathematics, and metrology reflects his interests in applications of mathematics.
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. Dr. Blackmore has co-authored two books, co-edited two books, is co-writing a Springer monograph on integrable (infinite-dimensional) dynamical systems and has published scores of scientific papers in leading journals, books and conference proceedings. He has received research 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 hi
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