Math 47a - Introduction to Research

Class meets at: Goldsmith 116, MW 2:10 - 3:30 PM
Lecturer: Dmitry Kleinbock
Office: Goldsmith 207, phone 6-3059
Office Hours: Th 1-2 PM or by appt.
E-mail address: kleinboc@brandeis.edu
Teaching assistant: Anish Ghosh, ghosh@brandeis.edu,
office hours Tue 1-2 PM in Goldsmith 306 or by appt.

Final projects:

• On quandles and knot invariants, by Ilya Bronshtein


• Mathematics in Sudoku, Fall 2005, by Richard Frank


• On splitting up piles of stones, by Gregory Igusa

Course materials:

• a PDF file with the first four homework assignments and their deadlines


• the research-paper version of the same material, in TeX and PDF, with another two homework assignments.

In the first half of the course basic mathematical problem-solving skills and strategies will be surveyed and then applied to a wide variety of situations. Many of the problems will be essentially background-free, that is, the key to a solution will lie in a creative idea rather than in use of "heavy machinery" of higher mathematics. There exist numerous examples of fun (Olympiad-style) problems which require only elementary school background but lead to some serious mathematics. There will be weekly problem assignments preceded by instructor's explanations, in some cases reviewing the background needed for problems, and discussion sessions where students will present and discuss their solutions (after one or two weeks of work). Possible topics include: games and algorithms, combinatorics and counting principles, induction, invariants, divisibility, elementary algebra, geometry and analysis.

In the second half of the course, selected problems mentioned in the first part will be gradually transformed into research projects, which will involve generating data, making conjectures and eventually writing a mathematical paper in LaTeX (we plan to spend part of the class time in a computer lab learning basic typesetting skills). As the course satisfies the writing-intensive requirement, there will be several writing assignments, including a mid-term project and the final paper. Each will need to be revised. Students will also give oral presentations on their projects.

Reading materials will be distributed in class whenever relevant; in the beginning of the course we will closely follow the book Problem-Solving Strategies by A. Engel, which is placed on reserve at the library. Many useful links related to LaTeX can be found at the web site of this course as taught in Spring 2003 by Ira Gessel.