Topics on: Geometric, PDE, and Variational Methods

Math 8600 (graduate course), Spring, 2002
MWF: 1:25-2:15pm, at Vincent Hall 2
Proposed and taught by: Prof. Jackie Shen

Homepage for Miva-I (fall, 2001): syllabus, projects, and photos…
Syllabus for Miva-II, as of January 22, 2001.
Mission Statement (The wildest and widest version. Also see Miva-I):

For millions of years, life has distinguished itself from non-life by the most fundamental activity:
Sense and Sensibility. Dolphins send sound signals and process their echoes and reflections to
navigate the 3-dimensional aquatic world, while humans mostly rely on the light (or photon) signals
and visual perception to understand the context of the surrounding 3-dimensional world.
Only with sense and perception, intelligence emerges, civilization grows, and society evolves.

The beginning of this new millennium, driven by the digital revolution, has opened up
a completely new dimension to sense and intelligence. Life science and digital technology are
gradually heading to the same destination. The needs are growing stronger day by day to
have a precise understanding on how DNA signals pass on life information, how cells and
organs communicate and coordinate, and even on a large scale, how thousands of doctors
on this planet make their diagnostics based on the signals and information they gather from
patients.

With the telecommunication and remote sensoring technology, on the other hand, both the
dimensions and the degrees of the dimensions, of our real and virtual world, are growing
in an unprecedented manner. The information ``highways'' carry much more involved
structure and interaction than the most developed USA transportation system. The dimension
of the Internet virtual data world (or The Matrix) is far more than that of our real world,
which is believed to be 3, or at most 4 if time is to be included. With all digital ``eyes", our
naked vision is no longer limited to the distance from the Empire State to the Hudson River.
We are now often pushed to generate a perception and make a decision, based on what we
``see" on the remote northern pole of the Mars or inside the organs of a patient's abdomen.

Efficiency, accuracy, intelligence (learning, self-correcting, and adapting), and generality,
are thus becoming demanding qualities on (1-D sequential) signals, (2-D) images or visual data,
and (multi-dimensional) general data processing and analysis, for the civilized unfolding of this new millennium.

Reviewing the history of natural science and technology (total differentials in thermal dynamics,
Navier-Stokes in fluid dynamics, Schroedinger equation in quantum mechanics, just to name a few),
we (applied mathematicians) are clearly presented with the God-given golden opportunity of
making a wave, in this conventionally more computer science, cognitive, and engineering field.

And this course (or its future siblings) has been created and will be developed in the
light of this vision, though perhaps very naive as it sounds now…

Created on Jan 22, 2002, by Microsoft Word. Last updated: Jan 22, 2002.