Seminar in Mathematics

I. Kra

On different ways of counting:

combinatorics and number theory

Course meets Tu, Th 2:20-3:40 in Physics P124.

I. Kra office hours (Tu, Sept. 5- F, Dec. 15) in Math 4-111: Tu 11-12, W
3-4, Th 9-10 (in Undergraduate Mathematics office), and by
appointment.

The seminar will proceed along the following lines:

1. We will identify good combinatorial and number theoretic problems; for
example, in how many
ways can a positive integer be written as a sum of four
squares? This will mostly involve talks by the lecturer and discussions with
students.

2. If a readily accessible proof, usually combinatorial or number theoretic,
is available that uses no more than the
course prerequisites (some knowledge of algebra (for example, MAT 310) and
analysis (MAT 320)), students will be
assigned the topic for presentation. Proofs requiring extensive preparation
will be presented by the lecturer. Students familiar with complex analysis
will be requested to talk on topics requiring that background.

3. Alternate analytic proofs of important combinatorial and number theory
results will be explored, mostly by the lecturer.

4. An introduction of a new topic will be followed by a student search of
the literature and an opportunity for student presentations of what they
learned and ideas that occur to them for tackling the problem. At the
beginning of the semester, student talks will be optional. As the semester
progresses more will be required of the students.

Topics that will be covered:

1. Counting the rationals.

2. The prime number theorem.

3. The twin prime conjecture.

4. Partitions.

5. The divisor function.

6. Function theoretic identities that have number theoretic content.

I am interested in the interaction between combinatorics, number theory and
analysis and its applications to computing and physics. The above list of
topics can be modified on the basis of student interest.

The seminar will require active student participation and will
encourage student discoveries of known (and perhaps unknown) mathematics.

A problem list for the course will be maintained on the web. Students should attempt both the routine and the challenging problems. Each student is required to hand in three ``scholarship/research papers'' and deliver one or two (depending on class size) 20-30 minute presentations. The papers can be solutions to selected problems on the list or results of investigations on topics discussed in lecture. At least one paper must be on such a topic (not a solution to a problem on the list). Each paper should consist of 2 to 3 typed well written pages. The first paper is due by November 1; the second by November 15; the last by December 1. The topics for the oral presentations will be assigned by the lecturer and posted on the web. The first lecture will be on a topic related to the lectures or the problem list; the second will be a short report on a paper from accessible literature.

The grade on the course will be based on the work by the students described above and class participation.

I have placed the following seven books on reserve:

1. J.A. Anderson, *Discrete Mathematics with Combinatorics*,
Prentice-Hall, 2001.

2. G.E. Andrews, *The Theory of Partitions*, Addison-Wesley, 1976.

3. G.E. Andrews, *q-series: Their Development and Application in
Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra*,
Amer. Math. Soc., 1985.

4. J. Bak and D.J. Newman, *Complex Analysis*, Springer-Verlag, 1982.

5. R.A. Brualdi, *Introductory Combinatorics*, Prentice-Hall, 1999.

6. G.H. Hardy and E.M. Wright, *An Introduction to the Theory of
Numbers* (fifth edition), Clarendon Press, 1995.

7. F.S. Roberts, *Applied Combinatorics*, Prentice-Hall, 1984.

References 1, 5 and 7 are junior level texts in Combinatorics; they have many
problems; 2 and 3 are modern solid treatments of Partitions and related
topics; 6 is a classic. I have included 4, since I am particularly
interested in the interaction of Combinatorics and Complex Analysis. It has
good background material for many of the topics to be discussed. Students
should also read regularly __The American Mathematical
Monthly__.

We will have occasional guest lectures in the course.

GUEST LECTURES

Date | Topic | Speaker |

9/28/00 | Computers in research | S. Sutherland |

**Special needs** If you have a physical, psychiatric, medical or
learning disability that may impact on your ability to carry out assigned
course work, you may contact the Disabled Student Services (DSS) office
(Humanities 133, 632-6748/TDD). DSS will review your concerns and determine,
with you, what accommodations may be necessary and appropriate. I will take
their findings into account in deciding what alterations in course work you
require. All information on
and documentation of a disability condition should be supplied to me in
writing at the earliest possible time AND is strictly confidential. Please
act early, since I will not be able to make any retroactive course changes.

STUDENT TALK ASSIGNMENTS

Date | Topic | Speaker |

9/21 | The Goldbach conjecture | Lynch |

10/17 | The pigeon hole principle | Simon |

10/12 | e is irrational | Prince |

10/24 | Generating functions | Nesterenko |

10/26 | Quadratic reciprocity | Lloyd |

11/7 | Partition congruences | Nikes |

11/16 and 30 | Zagier; Amer. Math. Monthly, 97, 1990 |
Simon |

12/7 | Hirschorn; Discrete Math, 131, 1994 |
Prince |

11/21 | Winquist; J. Comb. Th. 6, 1969 |
Lynch |

11/28 | Drost; Amer. Math. Monthly, 104, 1997 |
Nikes |

12/5 | Carlitz and Subbarao; Proc. Amer. Math. Soc. 32, 1972 |
Nesterenko |

12/12 | Dyson; in Ramanujan Revisited |
Lloyd |