# 2008-2009 FAWG Seminar

The Fields Analysis Working Group Seminar takes place (usually) at the Fields Institute Room 210 on Tuesdays.

The seminar also has a page at the Fields Institute.

## February 10, (2009) Tuesday 12:10--1:00. Fields Institute Room 210

Title: Recent Advances on the Navier-Stokes Equations
Abstract: The Fields Analysis Working Group will survey some recent advances in the theory of the incompressible Navier-Stokes equations. This talk will introduce the topics we plan to study. More information, including links to the relevant literature and some background sources, is posted here.
Lecture Notes: File:2009 02 10 Colliander FAWG Recent Advances Navier-Stokes.pdf

## December 2, (2008) Tuesday 12:10--1:00. Fields Institute Room 210

• Larry Guth (University of Toronto)
Title: A new proof of the Bennett-Carbery-Tao multilinear Kakeya estimates
Abstract: In a previous talk, Magda Czubak discussed a bilinear approach to the restriction problem in the plane. Similarly, there are bilinear and trilinear versions of the Kakeya problem. In 2006, Bennett, Carbery, and Tao proved a near-optimal n-linear Kakeya estimate in n-dimensional space. The n-linear Kakeya estimate is easier than the full conjecture because we only need to count n-tuples of tubes that meet each other transversely, but the result is still important because it's one of the few things we know sharply about Kakeya in higher dimensions. I recently gave a new proof of these estimates and sharpened them by proving an endpoint case. The new proof is based on Dvir's proof of the Kakeya conjecture over finite fields.
Reference: The article appears here on arXiv. See also Terry Tao's blog discussion of Larry's article. Colliand 20:58, 28 November 2008 (UTC)
Lecture Notes by J. Colliander:File:2008 12 02 Guth New Proof of BCT Multilinear Kakeya FAWG.pdf

## November 25, (2008) Tuesday 12:10--1:00. Fields Institute Room 210

• Geordie Richards (University of Toronto)
Title: The Tomas-Stein Restriction Theorem
Abstract: The restriction conjecture asserts that the estimate $\|\hat{f}\|_{L^{q}(S^{n-1})} \leq C\|f\|_{L^{p}(\mathbb{R}^{n})}$ holds if and only if $1\leq p<2n/(n-1)$ and $p' \geq (n+1)/(n-1)q$. We will discuss the restriction theorem of Tomas-Stein, which states that this conjecture is true for q = 2.
References:
1. E. Stein, Chapters 8 and 9 of "Harmonic Analysis", 1st Ed. Princeton University Press, NJ, 1993.
2. T. Tao, Lecture Notes for the restriction theorems and applications course.
Lecture Notes by J. Colliander: File:2008 11 25 Richards Stein-Tomas Restriction.pdf

## November 18, (2008) Tuesday 12:10--1:00. Fields Institute Room 210

Title: Stickiness, Graininess, Planiness and structure in the Kakeya problem
Reference:
KLT 2000 An Improved Bound on the Minkowski Dimension of Besicovitch Sets in ${\mathbb{R}}^3$, Nets Hawk Katz, Izabella Laba and Terence Tao, The Annals of Mathematics, Second Series, Vol. 152, No. 2 (Sep., 2000), pp. 383-446]
Lecture Notes by J. Colliander: File:2008 11 18 Katz Sticikness Plaininess Graininess.pdf

## November 11, (2008) Tuesday 12:10--1:00. Fields Institute Room 210

Title: Dvir's proof of the finite field Kakeya conjecture.
Lecture Notes written by J. Colliander: File:Guth Dvir'sProofKakeyaFiniteField Nov11.pdf

## November 4, (2008) Tuesday 12:10--1:00. Fields Institute Room 210

Title: Wolff's Hairbrush (continued)
Abstract: This talk describes some of the ideas in Tom Wolff's proof that any Besicovitch set in ${\mathbb{R}}^n$ contains a hairbrush. As a consequence, the dimension of any Besicovitch set is greater than or equal to (n + 2) / 2.

## October 28, (2008) Tuesday 12:10--1:00. Fields Institute Room 210

Title: Wolff's Hairbrush
Abstract: This talk describes some of the ideas in Tom Wolff's proof that any Besicovitch set in ${\mathbb{R}}^n$ contains a hairbrush. As a consequence, the dimension of any Besicovitch set is greater than or equal to (n + 2) / 2.
Lecture Notes written by David Reiss: File:Colliander Wolff Hairbrush Oct28.pdf

## October 21, (2008) Tuesday 12:10-1:00. Fields Institute Room 210

• Magdalena Czubak (University of Toronto)
Title: Restriction conjecture for the circle
Abstract: Restriction conjecture for the circle states the Fourier transform of an Lp function can be restricted to an Lq function on a circle with the following estimate

$\|\hat f\|_{L^q(S^1)}\lesssim \|f\|_{L^p(\mathbb R^2)},\quad p<\frac{4}{3}, q\leq\frac{p'}{3}$. The conjecture was first proven by C. Fefferman for $p<\frac{4}{3}, q<\frac{p'}{3}$ and by Zygmund for $p<\frac{4}{3}, q\leq \frac{p'}{3}$. We follow the proof as presented by Tao (see references below).

References:
1. T. Tao Lecture #5 for the Restriction theorems and applications course.
2. C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36.
3. A. Zygmund, On Fourier coefficients and transforms of functions of two variables., Studia Math. 50 (1974), 189–201.
Lecture Notes written by David Reiss: File:Czubak KakeyaRestrictionConjecture Oct21.pdf

## October 7, (2008) Tuesday 12:10-1:00. Fields Institute Room 210

• Hiro Oh (University of Toronto)
Title: Multiplier problem for the ball
Abstract: In this talk, we will discuss C. Fefferman's disproof of the Disc Conjecture "the characteristic function for the unit ball is an Lp multiplier in $\mathbb{R}^n$ for 2n / (n − 1) < p < 2n / (n + 1)." First, we will show that the Fourier multiplier operator T corresponding to the characteristic function for the unit ball is unbounded in Lp for the values of p outside the range described in the Disc Conjecture, using the asymptotic behavior of the Bessel functions. Then, using the construction of Besicovitch sets in $\mathbb{R}^2$, we will show that T is bounded only in $L^2 ({\mathbb{R}}^2)$ (which immediately implies that T is bounded only in $L^2 ({\mathbb{R}}^n)$.) The details can be found in my notes.
References:
1. C. Fefferman, The Multiplier Problem for the Ball, Ann. of Math. 94 (1971), 330-336.
2. L. Grafakos, Section 10.1 (also see Sec 4.5) in Classical and Modern Fourier Analysis, 1st ed. Prentice Hall, NJ, 2004.
Note that the 2nd ed. is coming out in 2008 in two volumes Classical Fourier Analysis and Modern Fourier Analysis.
3. Note that Hiro prepared on this subject. File:Ballmultiplier.pdf

## September 30, (2008) Tuesday 12:10-1:00. Fields Institute Room 210

• Ben Stephens (University of Toronto) 12:10-100
Title: Besicovitch Sets
Abstract: A Besicovitch set (also called a Kakeya set), is a compact set in Rn that contains a unit-length line segment pointing in every direction and has Lebesgue measure 0. In this talk we construct such sets for $n \geq2$. When n = 2 we show that any Besicovitch set has Hausdorff dimension 2.
Lecture Notes written by David Reiss: File:StephensFAWGLectureKakeya-Besicovitch Sets29Sept.pdf
Lecture Notes written by Jim Colliander: File:StephensFAWGLectureKakeya-Besicovitch Sets29Sept jim.pdf

## September 23, (2008) Tuesday 12:10-1:00. Fields Institute Room 210

Title: Combinatorial problems related to the Kakeya conjecture
Abstract: Last week, we introduced the Kakeya conjecture and discussed its relationship to Fourier analysis. This week, we give an overview of some combinatorial problems related to Kakeya. The highlights are the Kakeya problem over finite fields, the Szemeredi-Trotter theorem, and estimates for sum sets and difference sets.
The Kakeya problem over finite fields was proposed by Wolff in the late 90's, and it was solved last spring by Dvir. Dvir's proof is strikingly short and elegant. It was one of the things that made me excited to study the Kakeya problem at this time.
The Szemeredi-Trotter theorem deals with the intersection patterns of lines in the plane: given L distinct lines in the plane, how many points can there be at which D lines intersect?
If A and B are sets of real numbers, the sum set A+B is defined to be the set of all sums of an element from A with an element from B. If A is a generic set with cardinality N, then the sum set A+A has cardinality around N2. For many sets A, the sum set A+A has cardinality much less than N2 - an important example is an arithmetic progression. In combinatorial number theory, there are a number of results following the philosophy that if the sum set A+A is small, then the set A should have some structural properties. For example, one can prove that if A+A is small enough, then the difference set A-A must also be small. In the late 90's, Bourgain realized that combinatorial estimates of this kind can be used to get estimates for the Kakeya problem.
Lecture Notes written by David Reiss: File:GuthFAWGLecture KakeyaCombinatorics 23Sept08.pdf

## September 16, (2008) Tuesday 12:10-1:00. Fields Institute Room 210

Title: Introduction to the Kakeya conjecture and related topics
Abstract: The Kakeya conjecture is a geometric problem about overlapping rectangles in the plane - or about overlapping cylinders in higher dimensions. The planar version is well-understood, and the higher dimensional version is a major open problem in mathematics. Over time, mathematicians have found that this problem is connected to a wide variety of other problems, including problems in Fourier analysis, PDE, and number theory.
In this talk, I will introduce the conjecture and some things connected to it. I will discuss the ball multiplier and the restriction problem from Fourier analysis. I will discuss an analogue of the Kakeya problem using finite fields instead of real numbers - this analogue was recently solved. I will discuss a combinatorial problem about points and lines in the plane solved by Szemeredi and Trotter. I will discuss some combinatorial number theory involving sum sets, difference sets, and product sets.
The main goal is to lay out a sequence of cool results, each of which can be proven in a later talk.
Lecture Notes written by David Reiss: File:GuthFAWGLecture KakeyaConjecture 16Sept08.pdf

## August 13, (2008) Wednesday 11:10-12:00. BA 6183

Title: Spectral theory and convergence rates for the fast diffusion equation in weighted Hölder spaces

(PLEASE NOTE UNUSUAL LOCATION this week --- BA 6183)

Abstract: For the fast diffusion equation in the mass preserving parameter range, we
obtain sharp asymptotic convergence rates to the Barenblatt solution with respect to
the relative $L^\infty$ norm from spectral gaps by
establishing a nonlinear differentiable semiflow in Hölder spaces on a Riemannian
manifold called the cigar manifold. On this manifold, the equation becomes uniformly
parabolic. It is possible to obtain faster rates than O(1 / t) when the
reference Barenblatt solution is appropriately scaled. To this end, the
interplay between weights in the function space, the spectrum of the
linearized operator and growth of its (formal) eigenfunctions needs to be
investigated carefully, leading to estimates in appropriately weighted
relative $L^\infty$ norms.
(joint work with Herbert Koch and Robert McCann)