MAT 332 — A journey into geometric complexity: from biological viruses to quantum computing
27 February 2006 — 3 March 2006
Old Main 403 Canisius College: 5:00 — 8:00 pm Daily
| Instructor: | Meera Sitharam Associate Professor CISE Department1 CSE Building University of Florida Gainesville, FL 32611-6120 sitharam@cise.ufl.edu http://www.cise.ufl.edu/˜sitharam |
Course Abstract: The primary goals of the course are to expose students to complexity theory through geometric problems; and to the process of extracting and formalizing geometric problems occurring in the real world. The secondary goal is to show how these geometric complexity problems both open up new areas of mathematics as well as provide fresh insights into classical areas.
The course will consist of the following topics, each requiring about 1 hour. Note that the topics could be interspersed. Course material and exercises will be provided on site and key unsolved research problems will be delineated. The primary emphasis will be on:
(a) motivational, geometric, usually computational problems occuring in real world scenarios,Geometric complexity of 2 and 3 dimensional structures
(b) on examples to build intuition in order to understand the elegance, depth and richness of these problems, and
(c) the important process of finding effective formalizations for them. We will only briefly touch upon
(d) their independent mathematical interest and relevant available classical and modern mathematical techniques for approaching them.
(1) Motivation 1: Geometric constraints in Virus and other Macromolecular Self-organizationHigher dimensional geometric complexity: embeddings and dimension reduction
(2) Motivation 2: Geometric constraints in Mechanical Computer Aided Design
(3) Rigidity characterizations and distance geometry
(4) Solution spaces and underlying algebraic geometry, tensegrity, unfolding linkages
(5) Polyhedral constructions, the role of symmetry
(6) The Game of geometric self-organization: robustness, complexity lower bounds and evolution
(1) Motivation 1: Mutually unbiased basis (MUB) problem in quantum cryptographyPrerequisites:
(2) Motivation 2: approximation of hard optimization problems, learning, codes, pseudorandom generation
(3) Dimension reduction: impossibility and complexity lower bounds
(4) The role of symmetry in dimension reduction
- Minimally; a good background in calculus together with some exposure to linear algebra and/or discrete mathematics.
Sitharam has had visiting positions at Purdue University (1997-98), and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers University (Spring 1998). Sitharam is currently a tenured associate professor in the CISE Department at the University of Florida, Gainesville.
Meera Sitharam’s research ranges over a variety of topics. She collaborates with mathematicians, theoretical physicists, structural biologists, mechanical engineers and economists. Philosophically, however, her sphere of interests is held together by a common, well-defined thread: accurately defining and capturing the elusive notion of complexity. Within this sphere, she enjoys finding interdisciplinary problems that give rise to rich new mathematical directions or to fresh insight into classical directions. Mathematically, she has noticed that she gravitates to an elegant doorway at the confluence of (asymptotic) geometry, algebra and combinatorics. Sitharam publishes research articles in the topics that currently intrigue her, her M.S. and Ph.D. students and collaborators. Sitharam has directed a number of Research Experiences for Undergraduates, and is currently sponsored by grants from the National Science Foundation.
1 Computer and Information Sciences and Engineering