Instructor: Meera Sitharam

IMPORTANT: Lecture notes for Part I of course (offered in Spring 06)

IMPORTANT: Lecture notes for Part II of course (offered in Fall 06)

Goals:
-- The first goal of the course is to expose students to geometric complexity through the process of extracting and formalizing geometric problems occuring in various application domains.
-- The second goal is to show how these geometric complexity problems both open up new mathematical techniques as well as provide fresh insights into classical ones.
-- The final goal of the course is that by the end of the semester, each student would have involved themself deeply with at least one or two research problem that piqued their interest during the course. Collaborative efforts between students will be highly encouraged.

Emphases:

--motivational, geometric constraint and geometric complexity problems occuring in real world scenarios and the process of finding effective formalizations for them;
--examples and ``first-principle'' approaches to build intuition in order to understand the elegance, depth and richness of these problems;
-- their independent mathematical interest and relevant available classical and modern mathematical techniques for solving them.
-- delineating key unsolved research problems.

Topics: The course will consist of the following topics, each requiring about a total of 3-6 lecture hours. Note that the topics could be interspersed and not necessarily in the given order.

Geometric complexity of 2 and 3 dimensional structures (Mostly covered in Part I of course)

-- Motivation 1 : Geometric constraints in Virus and other Nanoscale, Macromolecular Self-organization
-- Motivation 2: Geometric constraints in Mechanical Computer Aided Design
-- Rigidity characterizations and distance geometry
-- Solution spaces and underlying algebraic geometry, tensegrity, unfolding linkages
-- Polyhedral constructions, the role of symmetry
-- The Game of geometric self-organization: robustness, complexity lower bounds and evolution

Higher dimensional geometric complexity: embeddings and dimension reduction (Mostly covered in Part II of course)

-- Motivation 1: Mutually unbiased basis (MUB) problem in quantum cryptography
-- Motivation 2: approximation of hard optimization problems, datamining/learning codes, pseudorandom generation
-- Dimension reduction: impossibility and complexity lower bounds
-- The role of symmetry in dimension reduction

Course Format and Material:
--We will spend the first 5 weeks or so giving a 1-2 hour introduction to each of the topics above, including problems to play with, using first principles. We will then proceed to revisit each topic in more detail (2-4 more hours each), during the remainder of the semester.
-- Generally, I will introduce each topic. In some cases, a student in the class who is familiar with the topic will do the introduction. (Typically, they would be doing their PhD research on the topic)
-- Urls (or references) to relevant papers or books will be posted on this webpage before the semester starts and will be updated throughout the course. Required reading will be indicated.
--Each student will be expected to give atleast one 2hr lecture presentation on student's choice of material selected from assigned list. See below on how this will contribute to your grade.
-- After each lecture, 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 twice 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 exercises.

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:
--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).
-- Discussing offline (study groups) with other students in the class, and communicating the results of your discussions to me and the remainder of the class. This is highly encouraged.
--Communicating your work to me on the assigned exercises and open problems. (see course format above)
--Timely preparation and posting of thorough, clear, complete lecture notes on the lecture days assigned to you. (see course format above)
--Atleast one 2hr lecture 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.
--Each month I will send an email to each student, giving feedback on how you are doing, what you need to work on, etc.