801 - Topics in Applied Mathematics - Section 001

This is a second offering of the new course that I introduced in 2008, with sets of lecture
notes and improved presentations. The principal aims are: (1) Systematic treatment of
topics in the emerging interface of applied/computational mathematics and
topology/geometry, and (2) Introducing new problems and an exposition of the exciting
and rapid research progress in computational mathematics of modeling complex systems
based on very large data sets. Some representative topics are:
(a) To introduce the students to selected geometric methods in current research in pattern
recognition;
(b) To provide a practical guide to mathematical modeling of large dynamical systems.
The course consists of lectures and hands-on computation exercises using MATLAB. The
students are encouraged to learn the materials in the course of exploring one of the
suggested research topics in modeling complex dynamical systems (such as in biological
systems, learning theory, social and other networks.)
The lectures will discuss the background from topology, differential geometry and PDE
(such as the Laplace operator on data point clouds, graphs, …), and how they are applied
in pattern recognition, vision, etc. In the computational part, I will also review the
relevant background from probability theory, statistics and analysis of large data sets that
are cornerstones of current mathematical modeling research. There will be tutorials for
students who need to get started with hands-on computation with MATLAB as needed
for term projects. The course grade is based on a term project. Sample projects could be
obtained from the instructor.