SCIENTIFIC PROGRAMS AND ACTIVITIES

October 20, 2014

Fields Analysis Working Group 2008-09

A working group seminar and brown bag lunch devoted to nonlinear dynamics and the calculus of variations meeting once a week for three hours at the Fields Institute. The focus will be on working through some key papers from the current literature with graduate students and postdocs, particularly related to optimal transportation and nonlinear waves, and to provide a forum for presenting research in progress.
The format will consist of two presentations by different speakers, separated by a brown bag lunch.
More information will be linked to http://tosio.math.toronto.edu/pdewiki/index.php/Main_Page as it becomes available.

Interested persons are welcome to attend either or both talks and to propose talks to the organizers Robert McCann and James Colliander <colliand@math.toronto.edu> and <mccann@math.toronto.edu>.

2009

May 12
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
Giuseppe Savare (Universita di Pavia)
Nonnegative solutions to 4th order evolution equations by optimal transport
Some interesting nonlinear fourth-order parabolic equations, including the "thin-film" equation with linear mobility and the quantum drift-diffusion equation, can be seen as gradient flows of first-order integral functionals in the Wasserstein space of probability measures. The aim of these lectures is to present some general tools of the metric-variational approach to gradient flows which are useful to study this kind of equations and their asymptotic behavior. (Joint works in collaboration with U.Gianazza, R.J. McCann, D. Matthes, G. Toscani)
Feb. 10
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
J. Colliander, University of Toronto
Recent Advances on the Navier-Stokes Equations
Over the next few weeks, the Fields Analysis Working Group (FAWG) 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 at: http://tosio.math.toronto.edu/pdewiki/index.php/Fluids_References

2008

Dec. 2
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
Larry Guth, University of Toronto
A new proof of the Bennett-Carbery-Tao multilinear Kakeya estimate.
Nov. 25
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
Geordie Richards
The Tomas-Stein theorem


Nov. 19
(Wednesday)

Bahen Centre
BA6183
3:10 - 4:00 pm
Nets Katz (Indiana University)
Advanced additive combinatorics and structure in the Kakeya problem

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]
Nov. 18
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
Nets Katz (Indiana University)
Stickiness, Graininess, Planiness and structure in the Kakeya problem
Reference:
KLT 2000 An Improved Bound on the Minkowski Dimension of Besicovitch Sets in , Nets Hawk Katz, Izabella Laba and Terence Tao, The Annals of Mathematics, Second Series, Vol. 152, No. 2 (Sep., 2000), pp. 383-446]
Nov. 11
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
Larry Guth
Dvir's proof of the finite field Kakeya conjecture
Nov. 4
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
J. Colliander (University of Toronto)
Wolff's Hairbrush II
This talk describes some of the ideas in Tom Wolff's proof that any Besicovitch set in contains a hairbrush. As a consequence, the dimension of any Besicovitch set is greater than or equal to (n + 2) / 2.
Oct. 28
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
J. Colliander (University of Toronto)
Wolff's Hairbrush
This talk describes some of the ideas in Tom Wolff's proof that any Besicovitch set in contains a hairbrush. As a consequence, the dimension of any Besicovitch set is greater than or equal to (n + 2) / 2.
Oct. 21
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
Magdalena Czubak (University of Toronto)
Restriction conjecture for the circle
Restriction conjecture for the circle states the Fourier transform of an L^p function can be restricted to an L^q 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.
Oct. 7
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
Hiro Oh (University of Toronto)
Multiplier problem for the ball
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 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 , we will show that T is bounded only in (which immediately implies that T is bounded only in . 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 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.
Sept. 30
(Tuesday)

Fields
Room 210
12:10 - 1:00 pm
Ben Stephens (University of Toronto)
Besicovitch Sets
A Besicovitch set (also called a Kakeya set), is a compact
set in R^n 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>=2. When n=2 we show that any Besicovitch set has
Hausdorff dimension 2.
Sept. 23 (Tuesday)
Fields
Room 210
12:10 - 1:00 pm
Larry Guth (University of Toronto)
Combinatorial problems related to the Kakeya conjecture.
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.
Sept. 16 (Tuesday)
Fields
Room 210
12:10 - 1:00 pm
Larry Guth (University of Toronto)
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.

August 13, 2008
Bahen Centre,
Room 6183

Jochen Denzler (University of Tennessee)
Spectral theory and convergence rates for the fast diffusion equation in weighted Hoelder spaces
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-infinity norm from spectral gaps by establishing a nonlinear differentiable semiflow in Hoelder 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-infinity norms.
(joint work with Herbert Koch and Robert McCann)

July 2, 2008
11:10 a.m.

Dorian Goldman (University of Toronto)
Existence of Weak Lagrangian Solutions to a One Dimensional Model of the Moist Semi-Geostrophic Equations (Work in progress)

The semi-geostrophic equations are an approximation of the Navier Stokes equations that filter out noise and are better for the purposes of modelling large scale atmospheric dynamics. Currently only existence in Lagrangian varaibles is known for these equations (due to Cullen/Feldman) and no rigorous mathematics at all has been done when the effects of MOISTURE convection in the atmosphere are included in the model.

In this talk, I study the semi-geostrophic equations with additional terms incorporating the effects of moisture. A one dimensional model of these equations which encompasses the effects of moisture is studied and a weak formulation of this model is then defined. This model is used as a numerical scheme by forecasters for the purpose of predicting the formation of rain storms, and so it is desirable to know that this scheme is in some sense well posed. A new stability condition that STRENGTHENS the Cullen-Norbury-Purser stability condition is introduced which encompasses the effects of moisture on the dynamics. A time stepping procedure is used to construct difference equations in discrete space and time which are shown to converge to weak solutions as the size of the mesh tends to zero.

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