**
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.