Instructor: Meera Sitharam
Combinatorial rigidity is a topic has been investigated for nearly a century in a wide variety of classical mathematical areas including: Asymptotic geometric analysis; Computational algebraic geometry, geometric algebra and invariant theory; and Asymptotic, analytic, extremal and probabilistic combinatorics. In addition, it has spurred exciting new theorems and techniques requiring a combination of these areas, in some cases leading to solutions of long open problems. Approximation algorithms for combinatorial optimization, Machine Learning, Compression, Dimension Reduction etc. all involve the topic of asymptotically efficient geometric embedding of structures. Furthermore, (non)embeddability results have an immediate bearing on Complexity theory, Coding and Cryptography.
The course will a emphasize
(A) A foundational treatment of the topic of asymptotically efficient geometric embedding and nonembeddability , starting with key classical theorems from over a century ago, leading up to recently proved theorems -- in the mathematical areas mentioned in Item (II) above. These include: Schoenberg's theorem on embedding into Hilbert space; Cayley-Menger's theorems; and their connection to rigidity and configuration space theorems
(B) A demonstration of how these theorems are relevant to key problems in the CS topics mentioned in Item (I) above.
Course Format and Evaluation:
-- Urls (or references) to relevant papers or books will be posted on this webpage and will be updated throughout the course. Required reading will be indicated.
--Each student will be expected to give atleast one week's worth of lecture presentation on student's choice of material selected from an assigned list. See below on how this will contribute to your grade.
-- For each week, one assigned student will be responsible for preparing lecture notes, typing them up, discussing them with me, and posting them by the end of the week, so that students will be able to go over it before the next week's classes. (Each student should expect to come up more than once in this rotation; see below on how this will contribute to your grade).
-- Some of the interesting questions that arise during the lectures will be assigned as discussion exercises (see below).
What are the students expected to do: The assumption is that you are taking this class because you are really interested in it (not for any other reason such as getting enough credits, seeking an easy grade, etc.). Some of you may be doing research related to the area, others could have a strong curiosity about the material. If either of these is true, I would expect you to automatically have the sustained motivation necessary to put in adequate effort on a large chunk of the topics, and you would consequently have no trouble getting a good grade in the course.
Your grade will be based on the following:
--Timely preparation and posting of thorough, clear, complete lecture notes on the lecture days assigned to you. (see course format above)
--One week's worth of presentation on student's choice of material selected from assigned list (see course format above). Students will be expected to put in significant effort on reading, organiziton and delivery of this presentation. They will be asked questions during the presentation which they are expected to answer.
--Active class participation (which will show me to what extent you are reading the relevant material, reading to patch up holes in your background, actively going over and keeping up your understanding of the lecture material currently being presented). YOu can do this by working on problems assigned in class and communicating the results to me. YOu can work in groups, in which case, communicate the results of your discussions.
Lecture notes for Part I of a somewhat related course (offered in the past)
Lecture notes for Part II of a somewhat related course (offered in the past)
Lecture notes for this course