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.)
Some resources for the first semester:
- Research papers, expository papers and lecture notes can
be downloaded from the web pages of
Tim Gowers,
Ben Green,
and Terry Tao.
- Number theory books:
- H. Davenport, Multiplicative Number Theory, 2nd ed. (revised
by H. Montgomery), Springer-Verlag 1980.
- G.A. Freiman: Foundation of a structural theory of set addition, AMS, 1973
- H. Montgomery, Ten Lectures on the Interface between Analytic
Number Theory and Harmonic Analysis, AMS, 1994.
- M. Nathanson, Additive Number Theory, I: The Classical Bases;
II: Inverse Problems and the Geometry of Sumsets,
Springer-Verlag, 1996.
- M. Nathanson, Elementary Methods in Number Theory,
Springer-Verlag, 2000.
- R.C. Vaughan, The Hardy-Littlewood Method, 2nd ed., Cambridge
Univ. Press, 1997 (note: the 2nd edition is substantially different
from the 1st).
- Papers and lecture notes on the Web:
- More papers, to be discussed in the second part of the course:
- On one problem
of Gowers, by I.D. Shkredov: analytic proof of Roth's theorem for
triangles in subsets of a lattice.
- Finite field models
in additive combinatorics, by B. Green: finite field versions of various
number-theoretic problems, e.g. Roth's theorem.
-
An argument of Shkredov in the finite field setting, by B. Green: an
addendum to the previous paper, concerning "Roth for triangles".
- Lectures in Harmonic Analysis,
by T. Wolff: download it here, or buy the book from AMS. Chapters 10 and 11
include an introduction to the Kakeya problem, in particular the Kakeya
finite fields problem is introduced in Chapter 11.
- A new
bound on partial sum-sets and difference sets, and applications to
the Kakeya conjecture, by N. Katz and T. Tao: a possible source
for the "arithmetic" approach to Kakeya, though the main ideas are
perhaps better explained in the next paper.
- New bounds
on Kakeya problems, by N. Katz and T. Tao: the "arithmetic approach"
developed further.
- Restriction and
Kakeya phenomena for finite fields, by G. Mockenhaupt and T. Tao: another
good starting point for the Kakeya finite fields problem.
- A new bound for
finite field Besicovitch sets in four dimensions, by T. Tao: more
finite-field Kakeya.
- A sum-product estimate
in finite fields and applications, by J. Bourgain, N. Katz, T. Tao.