Instructor: Prof. William Stein

This is a course in the practical use of computation as an aid to mathematical research. Topics will include programing in Python with Sage (http://sagemath.org), creating and querying object-oriented and relational databases, setting up and running distributed computations, and writing optimized compiled code. We will also discuss the meaning of proof in computational mathematics and standards of ethics and verifiability in the context of computer-assisted mathematical research. Students will gain a general understanding of some of the capabilities of Maple, Matlab, Mathematica, and Magma, and have basic introduction to at least the following open source software and libraries: Sage, Singular, Macaulay2, GAP, PARI, GMP, NTL, and Maxima. Students must have taken another course that involved rigorous mathematical proofs, and should have prior exposure to a computer programming language.

Prerequisites: Students must have taken another course that involved rig\ orous mathematical proofs, such as Math 402 or 424, as well as Math 310, and should have prior exp\ osure to a computer programming language.

For more information, please visit the course website: http://wiki.wstein.org/2008/480a

Instructor: Prof. Rekha Thomas

This will be a research oriented class that will survey the modern tools used to solve systems of polynomial equations and inequalities both over the complex numbers and real numbers. The subject area is almost as old as mathematics itself with myriad applications in science and engineering. The tools needed come from many different parts of mathematics such as algebra, geometry, combinatorics, topology, optimization and analysis. Along with learning the theory, students will be expected to work on a research question that will require experimenting with computational packages and possibly writing some code.

Prerequisite: Math 308

Instructor: Prof. Gerald Folland

The object of this course is to develop the mathematical model for the hydrogen atom (and, in a more approximate sense, other atoms) that explains the structure of the electron shells and leads to a theoretical basis for the periodic table of elements. The emphasis is on the exploitation of the rotational symmetry of the problem to guide the way to the solution. This goal provides an opportunity to see how algebra (mostly linear algebra) and analysis can be used together to analyze a situation, and to introduce some more advanced mathematical subjects such as Hilbert spaces and Lie groups.

We will work through as much as possible of the book Linearity, Symmetry, and Prediction in the Hydrogen Atom, by Stephanie Frank Singer.

Prerequisites: The required prerequisites are (i) Math 308 or Math 136; (ii) Math 324 and 327 or Math 334 and 335; (iii) Physics 121-2-3. Also recommended are at least one 400-level math course (402 is particularly relevant) and/or at least one 200 or 300 level physics course.