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 fourthorder parabolic equations,
including the "thinfilm" equation with linear mobility
and the quantum driftdiffusion equation, can be seen as gradient
flows of firstorder integral functionals in the Wasserstein
space of probability measures. The aim of these lectures is
to present some general tools of the metricvariational 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 NavierStokes Equations
Over the next few weeks, the Fields Analysis Working Group (FAWG)
will survey some recent advances in the theory of the incompressible
NavierStokes 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 BennettCarberyTao multilinear Kakeya
estimate. 
Nov. 25
(Tuesday)
Fields
Room 210
12:10  1:00 pm 
Geordie Richards
The TomasStein 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. 383446] 
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. 383446] 
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), 330336.
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 unitlength 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 SzemerediTrotter 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 wellunderstood,
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 Linfinity
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 Linfinity 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 SemiGeostrophic Equations (Work in progress)
The semigeostrophic 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 semigeostrophic 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 CullenNorburyPurser 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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