Instructor:
Sorin Popescu (office: Math 3-109, tel. 632-8255, e-mail sorin at math.sunysb.edu)Office hours: Tu 2:00-3:00pm P-143, Thu 2:00-3:00pm Math 3-109
Numbers, Groups and Codes,
J. F. Humphreys, M. Y. Prest, (second edition), Cambridge University Press.
The class is an introduction to algebraic structures and applications. The above textbook moves from algebraic properties of integers, through other examples, to the beginnings of group theory. Applications to finite state machines, public key cryptography and to error correcting codes are also emphasised. Attention is also paid to the historical development of the mathematical ideas presented. The text should be easily accessible for both students of mathematics and computer science.
Here are a number of other good undergraduate books that you may perhaps find useful to consult during the semester (all of them available in our library):
| Topic | Sections in textbook | Week | Notes |
|---|---|---|---|
| Euclidean division algorithm, GCD/LCM, Induction | Sections 1.1 and 1.2 | 8/30-9/3 | [pretest sols] |
| Prime numbers, Unique factorization, Congruences | Sections 1.3 and 1.4 | 9/7-9/10 | |
| Solving linear congruences, Fermat's theorem | Sections 1.4 and 1.5 | 9/14 | |
| Euler's theorem | Section 1.6 | 9/20-9/24 | |
| Public key criptography | Section 1.6 | 9/27-10/1 | |
| Sets, functions | Sections 2.1 and 2.2 | 10/4-10/8 | Midterm 10/7 |
| Relations | Section 2.3 | 10/11-10/15 | |
| Permutations | Section 4.1 | 10/18-10/22 | |
| Order and signature of a permutation, transpositions and cycles | Section 4.2 | 10/25-10/29 | |
| Groups: defnition and examples | Section 4.3 | 11/1-11/5 | |
| Algebraic structures | Section 4.4 | 11/8-11/12 | Midterm 11/11 |
| Order of an element, generators, subgroups | Sections 5.1 and 5.2 | 11/15-11/19 | |
| Lagrange's theorem; finite groups of small order | Sections 5.2 and 5.3 | 11/23 | |
| Error detecting/correcting codes | Section 5.4 | 11/29-12/03 | |
| Error detecting/correcting codes (continued) | Section 5.4 | 12/6-12/10 | |
| Review | 12/9 | ||
| Final Exam | Tu 12/14 | 11:00am-1:30pm |
Note: Although students may take both MAT 312 and MAT 313, there is some nontrivial overlap in the material of these two courses.
Students are encouraged to do an individual special project or participate in a group special project. These could involve a historical report on material of the course, including perhaps a brief oral presentation or learning some topic in algebra not discussed in the course or writing a computer program for some algorithm. The choice of topic and the exact scope of the special project are to be determined after consultation with the instructor and the final form of a proposal must be presented in writing to the instructor.
Homework is an integral part of the course. Problems will be assigned periodically. You should try to solve them by yourself. You should also discuss them with your fellow students and you may work together on each problem set, but what you hand in must be your own writing and you should be able to answer questions about its content. The solutions of homework problems can be discussed (after the due date) in lectures and/or more appropriately in recitation sections. Some of the homework problems will be graded and solutions will be posted on the web. Problems marked with an asterisk (*) are for extra credit.
Late homeworks will not be accepted.
There will be two midterm examinations (on 10/7 and 11/11) and a final exam (on 12/14). All examinations are inclusive in the sense that they will cover all the material studied up to a specified date. The exact area of coverage of each examination will be posted on the web. No calculators, notes, or books, etc., will be allowed during the midterms or final exam. There will be NO makeup exams.
Your grade will be based on the weekly homeworks (20%), midterms (25% each), and the final exam (30%). The two lowest homework grades will be dropped before calculating the average. A special project and class participation may also contribute (up to 15%) toward the final grade (either as bonus, or as substitute for some of the homework).