University of Massachusetts, Amherst
Math 797AP

Asymptotic Problems in Algebra, Number Theory, and Discrete Mathematics
Spring 2014

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

Instructor: Professor Farshid Hajir
Office: Lederle 1623C
Phone: 545-6025
Email: hajir atsymbol

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.

  • A long survey article on Expander Graphs by Hoory, Linial and Wigderson.
  • Elementary Number Theory, Group Theory, and Ramanujan Graphs by G. Davidoff, P. Sarnak, A. Valette - London Mathematical Society, Student Texts 55, Cambridge University Press, 2003. If you google this book, you should be able to find long excerpts from the book online. Try searching "Davidoff, Sarnak, Valette" for example.
  • Spectral Graph Theory by Fan R. K. Chung (CBMS Regional Conference Series in Mathematics, number 92 - AMS, 1997).
  • Spectres de Graphes by Yves Colin de Verdiere. Cours Specialises, collection SMF, numero 4, 1998 (distributed by the AMS).
  • Random Walks on Infinite Graphs and Groups by Wolfgang Woess. Cambridge Tracts in Mathematics, number 138, Cambridge University Press, 2000.
  • Graph Theory by Reinhard Diestel. Springer-Verlag, electronic edition, 1997, 2000. ( Free download)

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

  • A few pages of notes introducing algebraic number fields and rings.
  • Introduction to Algebraic Number Theory by Pierre Samuel.
  • Algebraic Number Fields by Juergen Neukirch, Springer.
  • Algebraic Number Theory by Serge Lang, Springer.
  • Algebraic Number Theory Free online book by J.S. Milne.

  • 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

    Grading Scale


    ≥ 93%


    ≥ 90% and < 93%


    ≥ 86% and < 90%


    ≥ 82% and < 86%


    ≥ 78% and < 82%


    ≥74% and < 78%


    ≥ 70% and < 74%


    ≥65% and < 70%


    ≥ 60% and < 65%


    < 60%

    Course topics and learning goals: Students will become familiar with the basic ideas of several important subfields of mathematics and understand how they are linked together via the concept of asymptotically good families.

    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.