Wednesday, 2008

16 April
11:10 AM

Almut Burchard, University of Toronto
On Caffarelli's C(1,alpha)regularity for MongeAmpere
equations
What is alpha?
I will summarize Caffarelli's geometric arguments (which Alessio
Figalli explained here Fall), and then estimate the Holder
exponent. The resulting lower bound on alpha shrinks dramatically
with the spatial dimension. Time permitting, I will discuss
a similar estimate of Forzani and Maldonado, and possible
improvements.
Relevant papers:
•L. Caffarelli, Some regularity properties of solutions
of MongeAmpère equation, Comm. Pure Appl. Math.
44 (1991) 965969.
• Liliana Forzani, Diego Maldonado, Properties of
the solutions to the MongeAmpère equation. Nonlinear
Analysis. 57 (2004), no. 56, 815829.

16 April
1:10 PM 
Paul Lee, University of Toronto
Optimal mass transportation with nonholonomic constraints
I will talk about my joint work with A. Agrachev on optimal
mass transportation with nonholonomic constraints. Nonholonomic
constraints are restrictions on velocity which do not arise
from the configuration. They can be imposed by fixing a plane
field or more generally a control system. The cost function
is given by an optimal control problem where we minimize certain
Lagrangian among curves which satisfies these constraints. We
prove, under certain assumption on the constraints, the existence
and uniqueness of solution to MongeKantorovich problem generalizing
previous work of Brenier, McCann and Bernard & Buffoni.
I will also talk about an example where the existence and uniqueness
theorem does not apply. 
2 April
11:10 AM 
Jim Colliander, University
of Toronto
Talk: TBA 
2 April
1:10 PM 
Walid Abou Salem, University
of Toronto
On the blind collision of fast solitons, II
Continuation of the previous talk on March 26. 
26 March
11:10 AM 
Robert Jerrard, University of Toronto
Dynamics of topological defects in semilinear hyperbolic
PDEs.
Over the past 30 years, numerous theorems have been provedestablishing
ways in which the behavior of some solutions of certain semilinear
elliptic partial differential equations arising in mathematical
physics are related to the minimal surface equation. Many
such results are also known for parabolic PDEs. I will discuss
a result that give the first rigorous proof (as far as I know)
of a result of this sort in the setting of nonlinear wave
equations. This is joint work with Alberto Montero.

26 March
1:10 AM 
Walid Abou Salem, University
of Toronto
On the blind collision of fast solitons
I discuss recent work on the collision of two fast solitons
for the nonlinear Schr\"odinger equation in the presence
of a spatially slowly varying external potential. For a high
initial relative speed $\v\$ of the solitons, one can show
that, up to times of order $\log\v\$ after the collision,
the solitons preserve their shape (in $L^2$norm), and the dynamics
of the centers of mass of the solitons is approximately determined
by the external potential, plus error terms due to radiation
damping and the extended nature of the solitons. I also remark
on how to obtain longer time scales under stronger assumptions
on the initial condition and the external potential. 
12 March
11:10 AM 
Hao Li (University of Toronto,
Economics) Copy of Slides
Equilibria in a sorting problem.
An interval of types is to be sorted into two equalsized
groups. Each type's payoff depends both on its percentile rank
in the group it joins, and on the average type of the group.
Equilibrium sorting pattern is considered in three settings.
In the first setting, each type chooses between the two groups,
with higher types having priority over lower types. In the second
setting, types bid competitively for ranks in the groups. In
the third setting, the two groups each choose how to allocate
a fixed amount of resources according to rank in order to maximize
the average type, and then types sort into the two groups as
in the first setting.
Note: The content of the talk will be drawn from two of my
recent papers,
"First in Village or Second in Rome?" and "Competing
for Talents." They can be downloaded from my website
http://www.chass.utoronto.ca/~haoli/research/index.html

12 March
1:10 AM 
Abdeslem Lyaghfouri (Fields
Institute)
Hoelder Continuity of Solutions to Quasilinear Elliptic Equations
Involving Measures.
I will present a paper by Tero Kilpela"inen on the
Ho"lder continuity of the solutions of the pLaplace equation
with righthand side measure. The relationship between the growth
of the measure and the Ho"lder continuity of the solutions
will be discussed.

13 February
11:10 AM 
Tristan Roy (U.C.L.A.)
Global wellposedness for the defocusing cubic wave equation

6 February
11:10 AM 
Shuanglin Shao (U.C.L.A.)
Restriction estimates for Paraboloids in the Cylindrically
Symmetric Case
In this talk, we will discuss the linear and bilinear restriction
estimates for paraboloids when functions are assumed to be cylindrically
symmetric. For dyadically supported functions, further estimates
are available and sharp up to endpoints, which prove to be very
useful in establishing the global wellposedness result of certain
critical nonlinear Schrödinger equations in the radial
case. We also derive that the Restriction Conjecture for paraboloids
is true in the cylindrically symmetric case. 
6 February
13:10 PM 
Gideon Simpson (Columbia University)
The Mathematics of Magma Migration: Nonlinearity, Degeneracy,
and Dispersion
Geologic processes occur on time scales that introduce nonstandard
rheologies. In particular, magma migration is modeled as a poro
viscous flow. We will see that such models lead to nonlinear,
nonlocal, dispersive wave equations with the potential for degeneracy.
This talk will discuss the wellposedness of these equations,
the stability of their solitary waves, and associated open problems.

30 January
11:10 AM 
Marina Chugunova (University
of Toronto)
Spectral Properties of the NonSelfAdjoint Operator Associated
with the Periodic Heat Equation
The periodic heat equation has been derived as a model of the
dynamics of a thin viscous fluid on the inside surface of a
cylinder rotating around its axis. It is well known that the
related Cauchy problem is generally
illposed. We study the spectral properties of the nonselfadjoint
operator associated with this equation. Some open questions
will be stated.
(joint work with D. Pelinovsky) 
23 January
13:10 PM 
Abdeslem Lyaghfouri (Fields
Institute)
Hoelder Continuity of Solutions to Quasilinear Elliptic Equations
Involving Measures.
I will present a paper by Tero Kilpeläinen on the Hölder
continuity of the solutions of the pLaplace equation with righthand
side measure. The relationship between the growth of the measure
and the Hölder continuity of the solutions will be discussed. 
16 January
11:10 AM 
Kiumars Kaveh (University
of Toronto)
Isoperimetric inequality, its generalizations and applications
The classical isoperimetric inequality states that if P
is the perimeter of a closed simple curve in the plane and A
is the area of the region enclosed by the curve, then 4\pi A
is less than or equal to P^2 and the equality holds if and only
if the curve is a circle. It follows that among all the curves
with given perimeter, circle encloses the biggest area. The
origin of the inequality goes back to the antiquity. We will
discuss this inequality and its generalizations to arbitrary
dimensions namely BrunnMinkowski and AlexanderovFechel inequlaities.
They invlove mixed volumes of convex bodies in R^n. If there
is
time, we mention connecion with algebaric geometry. 
9 January
11:10 AM 
Larry Guth (Stanford University)
Packing widths and isoperimetric inequalities 
2007

5 December
11:10 AM 
Yuxin Ge (Université
Paris XII and University of Washington)
On the $\sigma_2$scalar curvature and its application
In this talk, we establish an analytic foundation for a fully
nonlinear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds
with positive scalar curvature. This equation arises from conformal
geometry. As application, we prove that, if a compact 3dimensional
manifold $M$ admits a riemannian metric with positive scalar
curvature and $\int \sigma_2\ge 0$, then topologically $M$ is
a quotient of sphere. 
5 December
1:10 AM 
Alessio Figalli (CNRS Nice)
Caffarelli's Holder regularity theory of MongeAmpere equations.
Part II.
This is a continuation of the talk on Nov. 28. 
28 November
11:10AM 
Benjamin Stephens (University
of Toronto)
Parallel transport in Wasserstein Space
We'll look at John Lott's derivations of the basic formulas
of curvature and parallel transport in Wasserstein Space for
a smooth compact Riemannian manifold.
We'll also describe what parallel transport looks like on the
real line and relate it to a nonlinear diffusion problem. 
28 November
1:10PM 
Alessio Figalli (CNRS Nice)
Caffarelli's Holder regularity theory of MongeAmpere equations
We will explain Caffarelli's Holder regularity theory of classical
MongeAmpere equations. 
21 November
11:10AM 
Alessio Figalli, CNRS Nice
and SNS Pisa
A mass transportation approach to quantitative isoperimetric
inequalities.
In this talk I will show how one can prove a sharp quantitative
version of the anisotropic isoperimetric inequality by exploiting
mass transportation theory, especially Gromov's proof of the
isoperimetric inequality and the BrenierMcCann Theorem. This
is a joint work with F. Maggi and A. Pratelli. 
21 November
1:10PM 
YoungHeon Kim, University of Toronto
An a priori second order derivative estimate for a MongeAmp\'ere
type
equation 2.
This is a continuation of the talk (with the same title) last
week.

Nov. 14, 2007
11:10 a.m. 
Maria Sosio, University of
Toronto
An application of the continuity method.
I'll present an application of the continuity method that
allows to reduce the solution of an elliptic differential equation
to that one of the Poisson equation. In this proof the Schauder
estimates, which I'll take for granted, play an important role
for switching from the Laplacian to the elliptic differential
operator. The main reference for this talk will be the book:
Jurgen Jost, Partial Differential Equations, Springer (2007). 
Nov. 14, 2007
1:10 p.m. 
YoungHeon Kim, University of Toronto
An a priori second order derivative estimate for a MongeAmpere
type equation.
This is a learning seminar talk, and can be regarded as a
continuation of the talk by Maria Sosio at 11h10. We will
discuss the a priori second order derivative estimate given
in the paper by Ma, Trudinger, and Wang `Regularity of Potential
Functions of the Optimal Transportation Problem'(Arch. Rational
Mech. Anal. 177(2005) (51183); see Section 4.

Nov. 7
13:00 
Chad Groft, University of Toronto
Isoperimetric inequalities and universal covers
The Dehn function of a finitely generated group G relates the
algebraic structure of G to the geometry of the universal cover
of a finite 2complex X where \pi_1(X) = G. In 1999 Alonso,
Wang, and Pride introduced qdimensional analogues of the Dehn
function for groups G of type F_q and proved a similar theorem
relating these functions to the universal covers of finite,
highly acyclic q complexes, and in 1992 Epstein introduced
similar functions with chains in place of sphere and disk maps.
It is natural to ask whether these latter functions are the
same, and what role the "highly acyclic" condition
plays if any. I will give brief definitions of Alonso's and
Epstein's functions, as well as generalizations to maps M \to
X where M is a compact oriented manifold with boundary, and
present partial results relating the functions and further generalizing
AWP's results. 
October 31
11:10  12:00 
Almut Burchard, University
of Toronto
Eternal solutions to the Ricci flow on R^2.
I will discuss recent work of Daskalopoulos and Sesum on eternal
solutions to the Ricci flow on R^2. My goal is to provide some
context for recent work of Robert McCann, Aaron Smith and myself
on the evolution of a particular family of asymmetric cigarshaped
surfaces under Ricci flow.

October 10
11:30  2:00* 
Fields Analysis With Geometry: Working Group
Organizational Meeting
The Fields Analysis Working Group has agreed to join forces
with last year's Geometry Seminar for the coming year. A brief
organizational meeting to discuss plans for the semester, to
be followed by a brown bag lunch.
*Note exceptional time!

August 16
13:00  14:00 
Brendan Pass (University of Toronto)
Optimal Transportation on Alexandrov Spaces
I will present a paper by J. Bertrand on the existence and uniqueness
of optimal maps on Alexandrov spaces with curvature bounded
below. 
11:10  12:10 
Nathan Killoran (University of Toronto)
Supports of Extremal Doubly and Triply Stochastic Measures
Doubly stochastic measures are Borel probability measures
on the unit square which push forward via the projection maps
to the Lebesgue measure on each axis. The set of doubly stochastic
measures is convex, so its extreme points are of particular
interest. I will examine the necessary and sufficient conditions
for a set to support an extremal doubly stochastic measure,
and demonstrate that such a set can be decomposed into a countable
collection of Graphs of functions, called a `limbnumbering
system.' I will also show how this structure generalizes to
triply stochastic measures on the unit cube.

July 5 
Abdeslem Lyaghfouri (King Fahd University of Petroleum
and Minerals)
On the Dam Problem with Two Fluids Governed by a Nonlinear
Darcy's Law
In joint work with S. Challal,we consider the problem
of two fluids flow through a porous medium governed by a nonlinear
law. We establish the existence of a weak solution and prove
that it is locally Lipschitz continuous in the zone above
the lower fluid. Then we prove the continuity of the upper
free boundary. In the rectangular case, we prove the existence
of a monotone solution with respect to the vertical variable,
and the continuity of the lower free boundary. Finally, we
prove that there is a unique monotone solution with respect
to x and y, when the dam is rectangular and the flow is obeying
to the linear Darcy law.

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