Richard Palais
Adjunct Professor
Dept. of Mathematics
School of Physical Sciences
University of California at Irvine
Office: 288 MSTB
In 1997, after 37 years as a member of the Brandeis Department of Mathematics, I retired to leave myself more time to work in the area of Mathematical Visualization and more specifically to continue the development of my Macintosh program 3D-Filmstrip (now called 3D-XplorMath). In the Fall of 2004, my wife, Chuu-lian Terng, resigned from Northeastern Univ. to accept a position in the mathematics department at UCI (where she holds the Advance Chair) and we have now moved permanently to Irvine. I am continuing to work on mathematical visualization and in particular I am cooperating with David Eck of Hobart and William Smith College, helping with the design of his Java port of 3D-XplorMath, that will be called VMM---for The Virtual (or Visual) Mathematical Museum. However I have also partially "unretired" and I am now an Adjunct Professor of Mathematics at UCI, which means I will be teaching one or two courses per year.
Teaching
2004--2005
In the winter quarter of 2005 I helped Professor Chuu-lian Terng teach Math 162A (Introduction to Differential Geometry). In addition to covering the usual theoretical material, students were given basic training in the Matlab mathematical programming system, and they used this to carry out projects that created visually the objects (curves and surfaces) that theorems from the course proved to exist. We have found that students enjoy this dual approach, and that moreover it improves their comprehension of the theoretical material while at the same time teaching them valuable programming skills. I handled the Matlab teaching.
See the Math 162A Course Syllabus for details.
Professor Terng's Lecture Notes on Curves and Surfaces.
My Lecture Notes on Curves and Surfaces (with Matlab exercises).
2005--2006
In the Fall and Winter quarters, I will be teaching the Graduate Algebraic Topology course (250a and 250b). This has not been taught for several years, and I plan to teach a somewhat different course than that outlined in the current syllabus. I would like to explain the differences and explain some of the reasons for them.
First, it will be a two-quarter course, taught in the Fall and Winter, rather than a full year course as previously. This is in part because some of the material in the current syllabus is already covered in earlier courses---in particular, the Fundamental Group and Covering Spaces, and we shall review these very quickly. But, more importantly, the course will be aimed at providing students with a working knowledge of those parts of modern algebraic topology that have proved to be most useful in other fields. In particular we will de-emphasize some areas that would be important for students planning to do advanced work (and perhaps write a thesis) in algebraic topology and concentrate instead on topics such as vector bundles and Chern classes, that have played an increasingly important role in geometry and analysis. This seems to make sense, since given the current constitution of the math department, several members are working in areas that require a knowledge of these more "applicable" topics, while on the other hand it seems unlikely that a more advanced course in the subject will be given in the near future.
Here are the topics that I expect to cover:
1) Introduction to Categories and Functors.
2) The Fundamental Group and Covering Spaces.
3) Elements of Homotopy Theory.
4) Homology and Cohomology: Singular, Cellular, and Simplicial.
5) Morse Theory and Poincaré Duality.
6) Vector Bundles.
7) Classification of Vector Bundles.
8) Cohomology of Grassmanians and Characteristic Classes
9) De Rham Cohomology and Chern-Weil Theory.
The textbook for the course will be:
Algebraic Topology by Allen Hatcher, Cambridge University Press, Cambridge,
2002. ISBN 0-521-79540-0
This is an excellent and very complete (533 page!) introduction to algebraic
topology, and in fact it goes well beyond an introduction. Moreover the author
has made a pdf version available for free on his page:
<http://www.math.cornell.edu/~hatcher/AT/AT.pdf>
We will use Hatcher's book (and perhaps others) mainly as a reference though.
This will be primarily a lecture course, and while I may make printed versions
of (some of) my lecture notes available, you will be responsible for learning
the material either from my lectures or from elsewhere.
