Math 797AP

Spring 2014

** Course News:**

Monitor this space weekly for important course news, such as...

**Meeting times: ** MWF 1:25-2:15, in LGRT 1234.

**Instructor: **
Professor Farshid Hajir

**Office: **Lederle 1623C

**Phone: **545-6025

**Email: ** hajir atsymbol math.umass.edu

**Office Hours**: Office hours are MW 10-11. Students are also
welcome to set up an appointment outside of these hours by sending me
an e-mail or calling me on the phone.

**Prerequisites:** MATH 611 and 612 or equivalent. The basic
requirements are: Linear Algebra, group theory, knowledge of fields
and rings; some Galois theory is useful but not absolutely required,
so if MATH 612 is concurrently being taken, that could be enough
depending on how much extra work a particular student is willing to do
to pick up the necessary background material. Students should check
with the instructor if they have any doubt about whether they have
sufficient background knowledge to be successful in this course.

**Course Description:** MATH 797AP *Asymptotic Problems in Algebra,
Number Theory, and Discrete Mathematis* is a topics course. It is a
graduate-level introduction to certain types of extremal problems in
mathematics. Though the objects we study come from disparate parts of
mathematics (coding theory, graph theory, curves over finite fields,
algebraic number fields, lattices, ...), there is an underlying common
structure to all the problems, whereby upper bounds on the 'quality'
of the objects come from zeta functions, and the lower bound on
existence of optimal or near-optimal objects come from group theory
(or modular forms). The problems we study are often motivated by
applications to information-theory, engineering and other disciplines;
though the applications are not the focus of the course, we use them
as motivation for studying the intricate algebraic structures which
often achieve the optimal solutions for the problems studied,
especially in an asymptotic sense. We will use a number of survey
papers and textbooks available on codes (Van Lint, Algebraic Coding
Theory, GTM, Springer), Expander Graphs (AMS Bulletin survey by
Hoory-Linial-Wigderson and by Lubotzky), as well as notes provided by
the Instructor. There will be weekly problem sets and a Final
Project.

**Required Text: ** None, but a list of some recommended texts will
be provided below. Also, the instructor will
provide course notes as the semester
progresses.

**Optional Additional Resources:**

Some texts that might be useful for **Coding Theory** are as
follows, in rough order oof increasing hft. Note: This is just a
small selection of many good books on this topic. Feel free to seek
your own favorite book.

P. Garrett, The Mathematics of Coding Theory, Pearson, 2004.

R.J. McEliece, The Theory of Information and Coding, Cambridge, 2002.

W.W. Peterson, E.J. Weldon, Error-Correcting Codes, MIT, 1972.

R.M. Roth, Introduction to Coding Theory, Cambridge, 2006.

T. Richardson and R. Ubanke, Modern Coding Theory, Cambridge, 2007. ( pdf of preliminary draft available online)

F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1978.

W.C. Huffman, R.A. Brualdi, The Handbook of Coding Theory, 3
Volumes, Elsevier, 1998.

Some books/articles for our study of **Graph Theory** are as follows -- the first two being our main sources.

Some books/notes that are useful for the study of **Number Fields** are:

**Moodle Bulletin Board:** A virtual M797 discussion *may* be
set up via the Campus Moodle resource if there is student demand for
it. Stay tuned for details. Please use this service responsibly.
(Assuming I actually set it up, I will monitor it semi-regularly, but
if you want to direct a question specifically at me, the best way to
reach me is during office hours or by e-mail.

**Homework:** Homework will be posted on The
Homework Page and collected periodically. Collaboration on homework is
allowed and indeed encouraged; your collaborators should be cited.
**For more details, be sure to read and follow
the homework rules.**

**Attendance: ** Attendance is required during lectures.
I consider
attendance AND participation important ingredients for your success in
the course. Frequent absences will be reflected in your
grade.

**Exams:** Students will be asked to do a project in lieu of a final exam.

**Final Project:** A few weeks into the term, students will be
divided into small groups and will choose a project on which to
collaborate. Each student will be required to turn in an individual,
independently-written report, and the group will collectively produce
and deliver a presentation.
**Grading:**

homework - 50%

project and participationg - 50

A |
≥ 93% |

A- |
≥ 90% and < 93% |

B+ |
≥ 86% and < 90% |

B |
≥ 82% and < 86% |

B- |
≥ 78% and < 82% |

C+ |
≥74% and < 78% |

C |
≥ 70% and < 74% |

C- |
≥65% and < 70% |

D |
≥ 60% and < 65% |

F |
< 60% |

*Coding theory:* linear block codes and their parameters; rate vs. error correction, Manin's theorem on asymptotically good codes, Gilbert-Varshamov bound, algebro-geometric codes.

*Graph theory:* spectra of graphs and measures of connectivity, expanders and Ramnujan familes, Sipser-Spielman theorem.

*Number fields:* basics of algebraic number fields and class field towers, root discriminant bounds, Martinet function.

*Function fields:* Weil bound on number of points on a curve over a finite field, Drinfeld-Valdut theorem, Serre's lower bound on the Ihara function.

*Lattices:* Packing density, lattices coming from algebraic number fields.