## Mathematics 620F: Topics in Number Theory and Analysis

### Reading course, Fall 2004 - Spring 2005

Instructor: I. Laba. Office: Math Bldg 239. Phone: 822 2450. E-mail: ilaba(at)math.ubc.ca.

This semester, we meet on Wednesdays, 4-5 pm (note change of time), in MATX 1118. Click here for the schedule of meetings.

The plan is to discuss a certain group of problems at the interface of analytic and combinatorial number theory, harmonic analysis, combinatorics, ergodic theory, with ideas from other fields of mathematics also mixed in. These are all hot research topics - accordingly, we may often want to progress as soon as possible to reading current research papers, some not yet published in journals. Research problems, of varying levels of difficulty, will be suggested and discussed.

### Topics:

In the first semester, I would like to focus on Szemeredi's theorem on arithmetic progressions and other topics related to it. Szemeredi's theorem states that any subset of integers of positive density must contain arithmetic progressions of arbitrary length. It is considered to be one of the milestones of combinatorics, and an incredible array of methods and ideas, linking many different areas of mathematics, has been created in connection with it. Our topics will be selected from the following:
• Fourier analysis: "randomness" criteria for sets, finding structure in sets that are not random.
• combinatorial approach: Szemeredi's regularity lemma, its extensions and applications,
• ergodic-theoretic approach: background, multiple recurrence theorem, generalizations of Szemeredi's theorem,
• quantitative results for 3-term arithmetic progressions,
• the recent work of Green and Tao on arithmetic progressions in primes,
• other related results, e.g. finding patterns in subsets of the integer lattice.
For the second semester, I am thinking of a variety of problems, including:
• The Kakeya problem: the number-theoretic aspects. (The Fourier-analytic aspects require a fair bit of background in harmonic analysis, and hopefully will be included in a separate topics course some time in the future.)
• Sums vs. products: if A is a set of n numbers, must at least one of the sets {a+b: a,b in A} and {ab: a,b in A} have cardinality (almost) n2?
• Distance set problems (example: what is the minimum number of distinct distances between n points in the plane?).
• And many others of similar flavour: easy to state, difficult to solve, often require combining seemingly unrelated methods and ideas.
This is a tentative list only, expected to be modified in consultation with the participants, taking their background and interests into account. A background in number theory, harmonic analysis, and/or combinatorics will be helpful but not necessary.

### Schedule and format:

The meetings will be less structured than in most graduate courses, with emphasis on discussion, exchange of ideas, and problem-solving rather than formal lectures. Much of the learning is expected to take place during the meetings, in real time. I would like to have each meeting chaired by a designated person (not necessarily me...) who will be asked to read the current material in advance, present a brief introduction, answer questions, and moderate the discussion otherwise. Everyone is encouraged to participate.

The entire course (2 semesters) is worth 4 credits if you sign up. However, you do not have to sign up in order to attend. (Since this is a reading course, there is no minimum number of students required.)