
THE FIELDS
INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th
ANNIVERSARY
YEAR

Fields Analysis Working Group
201112



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.
Interested persons are welcome to attend either or both talks and
to propose talks to the organizers James Colliander (colliand<at>math.toronto.edu)
and Robert McCann (mccann<at>math.toronto.edu). Seminars
will be held on Thursdays at 12 noon in Room 210 unless stated otherwise.
Please check the website regularly for updates.
PAST SEMINARS

Friday
,
June 22, 2012 1:00 PM
Fields Institute,
library 
Oana
Silvia Serea, Univ. Perpignan Via Domitia
Differential games and Zubov's Method
(abstract) .....(slides) 
Thursday,
5 April, 2012 12:10 PM
Fields Institute,
Room 210 
Zhuomin Liu (University of Pittsburgh)
The Liouville Theorem on Conformal Mappings
The celebrated Liouville Theorem from 1850 states that in
dimension n?3, the only conformal maps are M\"obius transforms.
Liouville's proof, as well as many subsequent proofs, required
the mappings to be diffeomorphisms of class C3. However, since
C1 regularity is sufficient to define conformal maps, one
may inquire whether the Liouville theorem remains true under
that, or even weaker conditions, e.g. Sobolev functions. The
reduction from C3 regularity turned out to be very difficult.
It this talk we will discuss the development of the Liouville
Theorem under weaker and weaker regularity assumptions, including
results of Gerhing, Reshnetyak, Bojarski and Iwaniec, Iwaniec
and Martin on W1,nloc conformal mappings. Furthermore, Iwanice
and Martin proved that in even dimensions n?4, W1,n/2loc conformal
mappings are M\"{o}bious transforms and they conjectured
that it should also be true in odd dimensions. We also discuss
a proof of the Liouville Theorem f?W1,1loc in dimension n?3
under one additional assumption that the norm of the first
order derivative $

Thursday,
29 March 2012 12:10 PM
Fields Institute,
Room 210 
James Colliander, University of Toronto
http://www.math.toronto.edu/colliand
Big frequency cascades in the cubic nonlinear Schrödinger
flow on the 2torus
Smooth solutions of the cubic nonlinear Schrödinger
equation on the 2torus which display a big frequency cascade
were constructed in joint work with M. Keel, G. Staffilani,
H. Takaoka and T. Tao. These solutions start off oscillating
on long spatial scales. Over time, through nonlinear resonant
interactions, these solutions begins to oscillate on smaller
and smaller spatial scales exhibiting an arbitrary increment
in high regularity Sobolev spaces. A strategy has recently
been put forward by Z. Hani which aims to construct solutions
with an infinite cascade. Recent work by M. Guardia and V.
Kaloshin has quantified the speed of the transient cascade.
This talk will describe these developments.

Thursday,
9 March 2012 12:10 PM
Fields Institute,
Room 210 
Arick Shao, University of Torotno
A Generalized Representation Formula for Covariant Tensor
Wave Equations on Curved Spacetimes
In the Minkowksi space $\R^{1+3}$, the solution to an
inhomogeneous wave equation is given explicitly by Kirchhoff's
Formula. In this talk, we aim to extend Kirchhoff's Formula
into a local firstorder representation formula for covariant
tensorial wave equations on arbitrary curved $(1+3)$dimensional
Lorentzian spacetimes. This formula is a generalization of
an analogous KirchhoffSobolev parametrix derived by Klainerman
and Rodnianski, both in terms of the types of equations that
can be treated as well as the assumptions required. Furthermore,
the formula can be directly generalized to certain abstract
vector bundles on the spacetime.

Thursday, 1 March 2012 12:10
PM
Fields Institute,
Room 210 
Nobu Kishimoto, Kyoto University
Wellposedness and finitetime blowup for the Zakharov
system on twodimensional torus
We consider the Zakharov system on twodimensional torus.
First, we show the local wellposedness of the Cauchy problem
in the energy space by a standard iteration argument using
the $X^{s,b}$ norms. Our result does not depend on the period
of torus. Conservation laws and a sharp GagliardoNirenberg
inequality imply an a priori bound of solutions, which enables
us to extend the localintime solution to a global one if
its $L^2$ norm is less than that of the ground state solution
of the cubic NLS on $R^2$. We then show that the $L^2$ norm
of the ground state is actually the threshold for global solvability,
namely, that there exists a finitetime blowup solution to
the Zakharov system on 2d torus with the $L^2$ norm greater
than but arbitrarily close to that of the ground state. This
is joint work with Masaya Maeda (Tohoku University, Japan).

Thursday, 19 January 2012
12:10 PM
Fields Institute,
Room 210 
Jordan Bell, University of Toronto
The KAM theorem
The KolmogorovArnoldMoser (KAM) theorem tells us that if
the orbits of a Hamiltonian system are confined to low dimensional
tori, then many of the orbits will remain confined to low
dimensional tori if we perturb the Hamiltonian. Many presentations
of the KAM theorem do not give a precise formulation of the
theorem. I will give a precise statement of the KAM theorem
and then explain the method of it's proof.

Thursday, 05 January 2012
12:10 PM
Fields Institute,
Room 210 
Dorian Goldman, New York University
The Gammalimit of the OhtaKawasaki energy. Derivation
of the renormalized energy.
We study the asymptotic behavior of the screened sharp interface
OhtaKawasaki energy in dimension 2 using the framework of
convergence. In that model, two phases appear, and they interact
via a nonlocal Coulomb type energy. We focus on the regime
where one of the phases has very small volume fraction, thus
creating ``droplets" of that phase in a sea of the other
phase. We consider perturbations to the critical volume fraction
where droplets first appear, show the number of droplets increases
monotonically with respect to the perturbation factor, and
describe their arrangement in all regimes, whether their number
is bounded or unbounded. When their number is unbounded, the
most interesting case we compute the limit of the `zeroth'
order energy and yield averaged information for almost minimizers,
namely that the density of droplets should be uniform. We
then go to the next order, and derive a next order limit
energy, which is exactly the ``Coulombian renormalized energy
W" introduced in the work of Sandier/Serfaty, and obtained
there as a limiting interaction energy for vortices in GinzburgLandau.
The derivation is based on their abstract scheme, that serves
to obtain lower bounds for 2scale energies and express them
through some probabilities on patterns via the multiparameter
ergodic theorem. Without thus appealing at all to the EulerLagrange
equation, we establish here for all configurations which have
``almost minimal energy," the asymptotic roundness and
radius of the droplets as done by Muratov, and the fact that
they asymptotically shrink to points whose arrangement should
minimize the renormalized energy W, in some averaged sense.
This leads to expecting to see hexagonal lattices of droplets.

Thursday, 24 November 2011 at 12:10PM
Fields Institute,
Room 210

Robert McCann, University of Toronto
Higherorder time asymptotics of fast diffusion in Euclidean
space: a dynamical systems approach
With Jochen Denzler (UT Knoxville) and Herbert Koch (Bonn),
we quantify the speed of convergence and higher asymptotics
of fast diffusion dynamics on Euclidean space to the Barenblatt
(self similar) profile. The degeneracy in the parabolicity
of the equation is cured by reexpressing the dynamics on
a manifold with a cylindrical end, called the cigar. The nonlinear
evolution semigroup becomes differentiable with respect to
Hoelder initial data on the cigar. The linearization of the
dynamics is given by LaplaceBeltrami operator plus a drift
term (which can be suppressed by the introduction of appropriate
weights into the function space norm), plus a finitedepth
potential well with a universal profile. In the limiting case
of the (linear) heat equation, the depth diverges, the number
of eigenstates increases without bound, and the continuous
spectrum recedes to infinity. We provide a detailed study
of the linear and nonlinear problems in Hoelder spaces on
the cigar, including a sharp boundedness estimate for the
semigroup, and use this as a tool to obtain sharp convergence
results toward the Barenblatt solution. In finer convergence
results (after modding out symmetries of the problem), a subtle
interplay between convergence rates and tail behavior is revealed.
The difficulties involved in choosing the right functional
analytic spaces in which to carry out the analysis can be
interpreted as genuine features of the equation rather than
mere annoying technicalities.

Thursday
November 17, 2011
12:10 PM
Fields Institute,
Room 210 
Marina Chugunova (University of Toronto)
On uniqueness of the waitingtime type solution of the
thin film equation
By formal asymptotic methods we analyze persistence and loss
of zeros in generalized weak solutions of the thin film equation.
We use a new dissipated functional recently constructed by
Laugesen to prove an auxiliary energy equality that leads
to the proof of the uniqueness of the waitingtype thin film
solution under some additional assumptions about the initial
data and regularity. Joint work with John King and Roman Taranets.

Thursday
November 10, 2011
12:10 PM
Fields Institute,
Room 210 
Brendan Pass (University of Alberta)
Optimal transportation with infinitely many marginals
I will discuss work in progress on an optimal transportation
problem with infinitely many prescribed marginals. After formulating
the problem and stating the main result, I will show that
this result can be interpreted as a type of rearrangement
inequality for stochastic processes. I will then discuss some
connections with parabolic PDE, derivative pricing in mathematical
finance and phase space bounds in quantum physics.

Thursday
November 3, 2011
at 12:10 PM
Fields Institute,
Room 210 
Christian Seis (University of Toronto)
On the coarsening rates in demixing binary viscous liquids
We consider the demixing process of a binary mixture of
two liquids after a temperature quench. In viscous liquids,
demixing is mediated by diffusion and convection. The typical
particle size grows as a function of time t, a phenomenon
called coarsening. Simple scaling arguments based on the assumption
of statistical selfsimilarity of the domain morphology suggest
the coarsening rates: from ? t^{1/3} for diffusionmediated
to ? t for flowmediated coarsening. In joint works with Y.
Brenier, F. Otto, and D. Slepcev, we derive the crossover
of both scaling regimes in the sense of lower bounds on the
energy, which scales, heuristically at least, as an inverse
length. The mathematical model is a CahnHilliard equation
with an additional convection term. The velocity of the convecting
fluid is determined by a Stokes equation. Our analysis follows
closely a method proposed by R. V. Kohn and F. Otto, which
is based on the gradient flow structure of the evolution.

Thursday
October 27, 2011 at 12:10 PM
Fields Institute,
Room 210

Spyros Alexakis (University of Toronto)
Loss of compactness and bubbling for complete minimal surfaces
in hyperbolic space
We consider the Willmore energy on the space of complete
minimal surfaces in H3 and study the possible loss of compactness
in the space of such surfaces with energy bounded above. This
question has been extensively studied for various energy functionnals
for closed manifolds. The first such study was that of Sacks
and Uhlenbeck for harmonic maps. The key tools to study the
loss of compactness in that case are epsilonregularity and
removability of singularities; the loss of compactness can
then occur due to bubbling at a finite number of points where
energy concentrates. We find the analogous results in our
setting of complete surfaces. These are the first results
in this direction for surfaces with a free boundary. joint
with R. Mazzeo.
More information on the speaker can be found at: http://www.claymath.org/fas/research_fellows/Alexakis/

Thursday
October 20, 2011
12:10 p.m.
Fields Insitute,
Room 210 
Amir Moradifam (University of Toronto) (slides)
Conductivity imaging from one interior measurement in the
presence of perfectly conducting and insulating inclusions
We consider the problem of recovering an isotropic conductivity
outside some perfectly conducting or insulating inclusions
from the interior measurement of the magnitude of one current
density field $J$. We prove that the conductivity outside
the inclusions, and the shape and position of the perfectly
conducting and insulating inclusions are uniquely determined
(except in an exceptional case) by the magnitude of the current
generated by imposing a given boundary voltage. We have found
an extension of the notion of admissibility to the case of
possible presence of perfectly conducting and insulating inclusions.
This makes it possible to extend the results on uniqueness
of the minimizers of the least gradient problem $F(u)=\int_{\Omega}
a\nabla u$ with $u_{\partial \Omega}=f$ to cases where
u has flatregions (is constant on open sets). This is a joint
work with Adrian Nachman and Alexandru Tamasam.

Thursday
October 13, 2011
12:10 p.m.
Fields Insitute, Room 210 
Jonathan Korman (University of
Toronto)
Optimal transportation with capacity constraints
The classical problem of optimal transportation can be formulated
as a linear optimization problem on a convex domain: among all
joint measures with fixed marginals find the optimal one, where
optimality is measured against a cost function. Here we consider
a variant of this problem by imposing a constraint on the joint
measures: among all joint measures withfixed marginals and which
are dominated by a given measure find the optimal one. We show
uniqueness of the solution, and compute a surprising example.

Thursday
October 6, 2011
12:10 p.m.
Fields Insitute,
Room 210 
Ehsan Kamalinejad (University of
Toronto)
Gradient flow methods for thinfilm and related higher
order equations
We will discuss recent results on a class of higherorder
evolution equations that can be viewed as gradient flows on
the space of probability measures with respect to the Wasserstein
metric. The simplest of these equations is the thinfilm equation
$\partial_tu=\partial_x(u \partial_x^3u)$, which corresponds
to the Dirichlet energy. We will consider questions of existence
and uniqueness of these gradient flows. A key probem in the
analysis is the lack of convexity of the relevant energy functionals.
This is a joint work with Almut Burchard.

Thursday
September 22, 2011
12:10 p.m.
Fields Insitute,
Room 210

Christian Seis (University of Toronto)
RayleighB\'enard convection. A new bound on the Nusselt
number
This talk may be viewed as a followup to the UofT analysis
seminar from Sept. 16, but also aspires to be selfcontained!
We consider RayleighB\'enard convection as modeled by the
Boussinesq equations in the infinitePrandtlnumber limit.
We are interested in the scaling of the average upward heat
transport, the Nusselt number $Nu$, in terms of the nondimensionalized
temperature forcing, the Rayleigh number $Ra$. Experiments,
asymptotics and heuristics suggest that $Nu \sim Ra^{1/3}$.
This work is mostly inspired by two earlier rigorous work
on upper bounds of $Nu$ in terms of $Ra$: 1.) The work of
Constantin and Doering establishing $Nu \sim Ra^{1/3}\ln^{2/3}Ra$
with help of a (logarithmically failing) maximal regularity
estimate in $L^{\infty}$ on the level of the Stokes equation.
2.) The work of Doering, Reznikoff and Otto establishing $Nu
\sim Ra^{1/3}\ln^{1/3}Ra$ with help of the background field
method. We present a new bound on the Nusselt number: Etimates
behind the background field method can be combined with the
maximal regularity in $L^{\infty}$ to yield $Nu \sim Ra^{1/3}\ln^{1/3}\ln
Ra$  an estimate that is only a double logarithm away from
the supposed optimal scaling.
This is joint work with Felix Otto.

Thursday September 29, 2011
12:10 p.m.
Fields Insitute, Room 210 
Codina Cotar (University of Toronto)
Density functional theory and optimal transportation
with Coulomb cost
We present here novel insight into exchangecorrelation
functionals in density functional theory, based on the viewpoint
of optimal transport. We show that in the case of two electrons
and in the semiclassical limit, the exact exchangecorrelation
functional reduces to a very interesting functional of novel
form, which depends on an optimal transport map T associated
with a given density . Since the above limit is strongly correlated,
the limit functional yields insightinto electron correlations.
We prove the existence and uniqueness of such an optimal map
for any number of electrons and each , and determine the map
explicitly in the case when is radially symmetric. This is joint
work with Gero Friesecke and Claudia Klueppelberg. 
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