|Office hours: MW 2:00-3:00pm, 4:30-5:30pm|
The goal of this course is to investigate the relationship between algebra and computation. The course is designed to expose students (sophomore/junior cs and math majors) to algorithms used for symbolic computation, as well as to the concepts from modern algebra which are applied to the development of these algorithms. This course provides a hands-on introduction to many of the most important ideas used in symbolic mathematical computation, which involves sproving combinatorial identities, solving system of polynomial equations, analytic integration, and solving linear difference equations.
The appropriate use of computer algebra systems (Mathematica) to support the teaching and learning of mathematics, and in related assessments, is incorporated throughout the course. This includes the use of Mathematica to assist in the development of mathematical ideas and concepts, as well as a tool for analysis, problem-solving and modelling work.
The course covers the following topics:
Proving Combinatorial Identities
What do you get when you add a couple of math professors, a Roman Catholic nun, and a computer? Wilf and Zeilberger have shown that computers can prove a certain class of combinatorial identities instantly and infallibly.
A Gröbner basis for a system of polynomials possesses a property that the set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials. Therefore, Gröbner bases are very useful for solving polynomial equations by elimination of variables.
We will describe algorithms to the problem of indefnite integration in finite terms. While the problem was formulated and studied by Abel and Liouville, only within the last 20 years the practical algorithms have been developed and implemented in the major computer algebra systems.
We will start with an introduction to experimental mathematics and what it is role in modern applied mathematics. We will cover a few modern tools, techniques and algorithms including inverse symbolic calculator and lattice basis reduction algorithm.
understanding major algorithms of symbolic computation
understanding some algorithms of experimental mathematics
understanding main ideas of a computer-asisted proof
understanding major algorithms for automated proving combinatorial identities
understanding Groebner bases algorithms in the context of automated theorem provers
understanding major tools for experimental mathematics and their use in modelling and discovery mathematical results
Assignments: There will be several written assignments during the semester. All students have to write these up individually. No late days are given except individual requests.
Exams: There will be three midterm exams. (no final exam-!)
|Please send corrections to Victor
Computer Science Department,
Carnegie Mellon University, Pittsburgh, PA.