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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, currently
<colliand@math.toronto.edu> and <mccann@math.toronto.edu>).
2008
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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.
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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.
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August 13, 2008
Bahen Centre,
Room 6183
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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)
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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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