Seminars

Fall Semester Postdoctoral
Seminar

Andres Contreras (The Fields Institute)
Local minimizers in GinzburgLandau theory
We present results concerning the existence of stable vortex configurations
in 2D and 3D GinzburgLandau. These results represent joint works
with R.L. Jerrard and S. Serfaty.

Tuesday. December 2, 2014
1:10 p.m.
Room 210

Fall Semester Members Seminar

YoungHeon Kim (University of British
Columbia)
Multimarginal optimal transport
We explain some recent progress on multimarginal optimal transport,
where a family of mass distributions are matched in an optimal way.
This is based on joint work with Brendan Pass.

Tuesday, December 2, 2014
2:10 p.m.
Room 230

Fall Semester Postdoctoral
Seminar

Fedor Soloviev (The Fields Institute)
Integrability of pentagram maps and Lax representations (Slides)
We discuss integrability of higher dimensional pentagram maps. These
maps provide an explicit example when a discrete map "jumps"
between different invariant tori leading to a generalized version
of ArnoldLiouville theorem. This is a joint work with Boris Khesin.

Wednesday.
December 3, 2014
2:10 p.m.
Room 210

Fall Semester Postdoctoral
Seminar

Jun Kitagawa (The Fields Institute /
University of Toronto)
The Aleksandrov estimate and its variants in MongeAmpère
equations
The Aleksandrov estimate plays a central role in the regularity theory
of weak solutions of the MongeAmp{`\e}re equation, which was pioneered
by Caffarelli in the early 90's. Rather than talk about the regularity
theory itself, I will focus on this one estimate and its variants,
for example which arise in regularity of the optimal transport problem.
I will give some elementary proofs and talk about the geometric intuition
in connection to convex geometry that lies behind this somewhat mysterious
looking estimate. Time permitting, I will also talk about an Aleksandrov
type estimate applicable to a new class of equations, which includes
problems in geometric optics that are not optimal transport problems
(joint work in progress with Nestor Guillen).

November 4, 2014
1:10 p.m.
Room 210

Analysis & Applied Math
Seminar

Slim Ibrahim (University of Victoria)
Asymptotic derivation of the classical MagnetoHydroDynamic
system from NavierStokesMaxwell
The incompressible MagnetoHydroDynamic (MHD) system is a classical
and fundamental model in plasma physics. Although well known, its
derivation from NavierStokes type equations has been so far formal.
In this talk and after reviewing the results about the wellposedness,
I show how an asymptotic analysis of such equations can rigorously
lead to a such a derivation. The key points is a precise study of
the weak stability in the Lorentz.
This is a joint work with D. Arsenio (Paris 7) & N. Masmoudi
(Courant)

Friday, October 31, 2014
1:10 p.m.
BA6183, Bahen Center, 40 St. George St.

Fall Semester Postdoctoral
Seminar

Cyril Joel Batkam (The Fields Institute)
A symmetric mountain pass theorem for strongly indefinite functionals
The symmetric mountain pass theorem of Ambrosetti and Rabinowitz (1973)
and its generalization by Bartsch (1993) are effective tools of finding
high energy solutions to many (partial) differential equations and
systems which are of variational nature and exhibit some symmetry
properties. A functional defined on a Hilbert space fits into the
framework of these critical point theorems only if its quadratic part
has a finite number of negative eigenvalues. In this talk, we present
a generalization of these results to the case where the quadratic
part has infinitely many negative eigenvalues (strongly indefinite
functional). As an application, we give a direct proof of the existence
of infinitely many solutions to the system
\begin{equation*}
\left\{
\begin{array}{ll}
\Delta u=g(x,v)\,\, \text{in }\Omega, & \hbox{} \\
\Delta v=f(x,u)\,\,\text{in }\Omega, & \hbox{} \\
u=v=0\text{ on }\partial\Omega, & \hbox{}
\end{array}
\right.
\end{equation*}
where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ ($N\geq3$),
$f(x,u)\backsim u^{p2}u$ and $g(x,v)\backsim v^{q2}v$, with
$2<p,q<2N/(N2)$. Part of the work is joint with Fabrice Colin.

October 28, 2014
1:10 p.m.
Room 210

Fall Semester Members Seminar

Paul Woon Yin Lee (Chinese University
of Hong Kong)
Ricci curvature type lower bounds for subRiemannian structures
on Sasakian manifolds
In this talk, we introduce a type of Ricci curvature lower bound for
a natural subRiemannian structure on Sasakian manifolds and discussvarious
consequences under this condition.

Tuesday, October 28, 2014
2:10 p.m.
Stewart Library

Analysis & Applied Math
Seminar

Jochen
Denzler (University of Tennessee )
Existence and Regularity in the Oval Problem
The oval problem asks to determine, among all closed loops in ${\bf
R}^n$ of fixed length, carrying a Schrödinger operator ${\bf
H}= \frac{d^2}{ds^2}+\kappa^2$ (with curvature $\kappa$ and arclength
$s$), those loops for which the principal eigenvalue of ${\bf H}$
is smallest. A 1parameter family of ovals connecting the circle with
a doubly traversed segment (digon) is conjectured to be the minimizer.
Whereas this conjectured solution is an example that proves a lack
of compactness and coercivity in the problem, it is proved in this
talk (via a relaxed variation problem) that a minimizer exists; it
is either the digon, or a strictly convex planar analytic curve with
positive curvature. While the EulerLagrange equation of the problem
appears daunting, its asymptotic analysis near a presumptive singularity
gives useful information based on which a strong variation can exclude
singular solutions as minimizers.

Friday, October 24, 2014
1:10 p.m.
BA6183, Bahen Center, 40 St. George St.

Fall Semester Postdoctoral
Seminar

Arjun
Krishnan (The Fields Institute)
A stochastic homogenization approach to firstpassage percolation
(Slides)
Firstpassage percolation is a random growth model on the cubic lattice
Z^d. It models, for example, the spread of fluid in a random porous medium.
Quantitatively describing the ``average time'' required for the fluid
to percolate through the medium known as the timeconstant of firstpassage
percolation is a classical, but unsolved problem in probability. We
view firstpassage percolation as a problem of homogenization for a discrete
HamiltonJacobiBellman equation. By borrowing several tools from the
continuum theory of stochastic homogenization, we will derive an exact
variational formula (duality principle) for the timeconstant. Under a
symmetry assumption, we will give a constructive algorithm that produces
a minimizer of the formula. 
Tuesday, October 21, 2014
1:10 p.m.
Room 210

Fields Calculus of Variations
and Applications

Nader
Masmoudi (Courant Institute)
Gevrey spaces : Prandtl system and nonlinear inviscid damping
for 2D Euler.
We will discuss two recent applications of Gevrey spaces: The first
one is the local existence of the Prandtl system without analyticity
and without the Oleinik monotonicity assumptions. More precisely,
we assume Gevrey regularity in the horizontal variable (joint work
with David GerardVaret). The second one is the global asymptotic
stability of shear flows close to planar Couette flow in the 2D incompressible
Euler equations. Specifi cally, given an initial perturbation of the
Couette flow which is small in a suitable Gevrey space, we show that
the velocity converges strongly in L2 to another shear flow which
is not far from Couette. This strong convergence is usually referred
to as "inviscid damping" and is analogous to Landau damping
in theVlasovPoisson system (joint work with Jacob Bedrossian)

October 2123, 2014
10:10 a.m.
Room 210

Fall Semester Members Seminar

Nestor
Guillen (University of Massachusetts at Amherst )
On AleksandrovBakelmanPucci estimates for integrodifferential
equations
Convexity has played an important role in elliptic equations. One
such instance is its crucial appearance in the celebrated AleksandrovBakelmanPucci
estimate (ABP), which is the back bone of the regularity theory of
fully nonlinear elliptic equations. In this talk I will describe
the shortcomings of convexity when dealing with nonlocal operators
and will discuss alternatives as well as their applications. In particular,
I will describe a result obtained with Russell Schwab regarding pointwise
bounds (analogous to the ABP) for weak solutions of nonlocal elliptic
equations which is new even for nonlocal linear operators. Time permitting
I will also discuss an elementary new proof of the classical Aleksandrov
estimate obtained in joint work with Jun Kitagawa.

Tuesday, October 21, 2014
2:10 p.m.
Room 210

Fall Semester Postdoctoral
Seminar

Mircea
Petrache (The Fields Institute)
The quest for global duality for geometric massminimization
problems
The classical Plateau problem consists in minimizing the area of a
2D surface in R^3 under the constraint of fixed boundary. This question
can be interpreted in several ways. The corresponding rigorous formulations
led naturally to the introduction of the notion of integral currents
and the more general on of flat chains with coefficients in a normed
group G, starting from the 60's. The general tools for the global
control of minimizers are mostly limited to the case of currents (i.e.
G = Z), where a duality structure is present and we have the notion
of a calibration, which in 1D is related to Kantorovich duality. I
will describe some cornerstone results and several partial generalizations
obtained since the 80's. For the 1D case I will describe a recent
result obtained in collaboration with Roger Zuest where a natural
nonlinear duality structure can be recovered for chains with coefficients
in G=Z/2Z. It can be interpreted as an unoriented analogue of Kantorovich
duality or as a maxflowmincut duality with coefficients taken modulo
2. As a motivation for future work I will point out 1Dproblems with
three other choices of G which give direct applications respectively
to the theory of branched transport, to the study of dislocations
in crystals and to a new kind of "optimal information transport"
problem.

Tuesday, October 14, 2014
1:10 p.m.
Stewart Library

Fields Calculus of Variations
and Applications

Martial
Agueh (University of Victoria)
Uniqueness of the compactly supported weak solution to the
relativistic VlasovDarwin system (Slides)
The relativistic VlasovDarwin (RVD) system is a kinetic model that
describes the evolution of a collisionless plasma whose particles
interact through their selfinduced electromagnetic field and move
at a speed ``not too fast'' compared with the speed of light. In contrast
with the VlasovPoisson system, it is an approximation of the VlasovMaxwell
system which also takes into account the magnetic effect of the particles.
In this work, we prove uniqueness of weak solutions to the RVD system
under the assumption that the solutions remain compactly supported
at all times. Our proof exploits the formulation of the RVD system
in terms of the "generalized" space and momentum variables.
This formulation permits to rewrite the system in terms of a scalar
and vector potentials, which allows to view it as a generalization
of the VlasovPoisson system. We then use optimal transport techniques
to study uniqueness of weak solutions for this system. This is a joint
work with R. SospedraAlfonso.

Tuesday, October 14, 2014
2:10 p.m.
Stewart Library

Fall Semester Postdoctoral
Seminar

Ihsan
Topaloglu (The Fields Institute)
Existence of minimizers of nonlocal interaction energies
In this talk I will consider the minimization of nonlocal
interaction energies of the form $$E[\mu]=\int_{\mathbb{R}^n}\!\int_{\mathbb{R^n}}
W(xy)\,d\mu(x)d\mu(y)$$ over the space of probability measures.
This type of energies arise naturally in descriptions of systems of
interacting particles, as well as continuum descriptions of systems
with longrange interactions. By taking a direct variational approach
I will present sharp conditions for the existence of minimizers for
a broad class of nonlocal interaction energies. This broad class includes,
but is not limited to, energies defined via attractiverepulsive potentials
used in modelling collective behavior of manyagent systems, granular
media and selfassembly of nanoparticles. Finally, I will discuss
the close relation between this sharp condition and the notion of
$H$stability of pairwise interaction potentials in statistical mechanics.
This is a joint work with R. Simione and D. Slepcev.

Tuesday Sept 30, 2014
1:10 p.m.
Room 230

Fall Semester Postdoctoral
Seminar

Ihsan
Topaloglu (The Fields Institute)
Existence of minimizers of nonlocal interaction energies
In this talk I will consider the minimization of nonlocal
interaction energies of the form $$E[\mu]=\int_{\mathbb{R}^n}\!\int_{\mathbb{R^n}}
W(xy)\,d\mu(x)d\mu(y)$$ over the space of probability measures.
This type of energies arise naturally in descriptions of systems of
interacting particles, as well as continuum descriptions of systems
with longrange interactions. By taking a direct variational approach
I will present sharp conditions for the existence of minimizers for
a broad class of nonlocal interaction energies. This broad class includes,
but is not limited to, energies defined via attractiverepulsive potentials
used in modelling collective behavior of manyagent systems, granular
media and selfassembly of nanoparticles. Finally, I will discuss
the close relation between this sharp condition and the notion of
$H$stability of pairwise interaction potentials in statistical mechanics.
This is a joint work with R. Simione and D. Slepcev.

Tuesday Sept 30, 2014
1:10 p.m.
Room 230

Fields Calculus of Variations
and Applications

Olivier
Kneuss (University of Zurich )
BiLipschitz Solutions to the Prescribed Jacobian Inequality
in the Plane and Applications to Nonlinear Elasticity
(Slides)
We show that the prescribed Jacobian inequality in the plane admits
 unlike the prescribed Jacobian equation  a biLipschitz solution
in case of righthand sides of class L8 (with identity boundary conditions).
We then apply our result to a model functional in nonlinear elasticity,
the integrand of which blows up as the Jacobian determinant of the
map in consideration drops below a certain positive threshold. For
such functionals, the derivation of the equilibrium equations for
minimizers requires an additional regularization of test functions,
which is provided by our newly constructed maps. This is a joint work
with J. Fischer.

Tuesday, September 30 2014
2:10 p.m.
Room 230

Fields Calculus of Variations
and Applications

Monica
Musso (Pontificia Universidad Católica de Chile)
Nondegeneracy of entire nonradial nodal solutions to Yamabe
problem in $\mathbb{R}^n$
We provide the first example of a sequence of {\em nondegenerate},
in the sense of DuyckaertsKenigMerle, nodal nonradial solutions
to the critical Yamabe problem $ \Delta Q= Q^{\frac{2}{n2}} Q,
\ \ Q \in {\mathcal D}^{1,2} (\mathbb{R}^n). $
This is a joint result with J. Wei.

Tuesday Sept 23, 2014
2:10 p.m.
Stewart Library

Fall Semester Postdoctoral
Seminar

Manuel
Gnann (The Fields Institute)
The moving contact line in viscous thin films: a singular
free boundary problem
We are interested in the thinfilm equation with quadratic mobility
and zero contact angle, modeling the height of a viscous thinfilm
with a linear Navierslip condition at the liquidsolid interface.
This degenerate parabolic fourthorder problem has the contact line
(the triple junction between the three phases liquid, gas, and solid)
as a free boundary. Starting with the analysis of sourcetype selfsimilar
solutions, we conclude that solutions cannot expected to be smooth
and explicitly characterize the singular expansion of such solutions
at the free boundary. With this understanding, we are able to prove
a wellposedness result of the corresponding full parabolic problem.
We conclude the talk with an overview of other questions and results,
such as the generalization to thinfilm equations with general mobility,
higher regularity, and convergence to the sourcetype selfsimilar
solution. Many of the presented results are joint with Lorenzo Giacomelli,
Hans Knüpfer, and Felix Otto.

Tuesday Sept 23, 2014
1:10 p.m.
Stewart Library

Fall Semester Postdoctoral
Seminar

Jean
Louet (The Fields Institute)
Sobolev spaces with respect to a measure and applications to optimal
transport problems with Sobolev penalization
In this talk, I will recall some known and give several new results about
the Sobolev spaces with respect to a measure in an Euclidean and variational
framework. We give a complete description of the tangent space to a generic
measure in 1D, and an original compactness result which stays open in
higher dimension. This allows to show the existence of solutions to an
optimal transport problem with Sobolevlike penalization. Some counterexamples
and pathological cases in dimension 2 will also be discussed. 
Tuesday Sept 9, 2014
1:10 p.m.
Room 230

Variational Problems Seminar

Codina
Cotar (University College London)
Gradient interfaces with and without disorder (Slides)
Gradient interface models are an important class of models in statistical
mechanics arising in the study of random interfaces and of the Gaussian
Free Field (harmonic crystal). Recently their study has attracted
a lot of attention as they are approximations of critical physical
systems and natural models for a macroscopic description of elastic
systems in material sciences, surface charges in dipole gases as well
as for fluctuating phase interfaces. In addition, the contour lines
of the Gaussian Free Field converge to forms of Schramm Loewner Evolution
(an active field of modern mathematics for understanding critical
phenomena  Fields Medal in 2006). Of further interest is the fact
that gradient models have long range correlations, which are a mathematical
frontier. This makes gradient models exciting new ground for mathematics,
attracting people with very different backgrounds, such as analysis,
probability, applied mathematics, material sciences and mathematical
physics.
In this talk I will give an overview of known results and open problems
for gradient interface models with and without disorder.

Tuesday Sept 9, 2014
2:10 p.m.
Room 230

Fall Semester Members Seminar

Simon
Brendle (Stanford University)
An introduction to Geometric Flows
Parabolic flows have become a fundamental tool in the study of many
geometric problems. In this talk, I will give a brief introduction
to the classical works of Hamilton and Huisken on Ricci flow and mean
curvature flow which started the subject.
Background material:
This talk is intended to be an informal introduction to the subject
for young researchers, as a prelude to the speaker's upcoming Sept
810 Distinguished Lecture Series at Fields.

Wednesday Sept. 3, 2014
2:10 p.m.
Room 230

Fall Semester Members Seminar

Katy
Craig (University of California
at Los Angeles)
A blob method for the aggregation equation
The aggregation equation describes the motion of particles according
to the minimization of a nonlinear interaction energy. Often, the
interaction between particles is chosen to scale according to a power
law potential, leading to aggregation or repulsion, depending on the
sign of the potential. In the case of the Newtonian potential, the
aggregation equation shares many important features with the vorticity
formulation of the Euler equations. In this talk, I will present joint
work with Andrea Bertozzi on a new numerical method for the aggregation
equation, inspired by vortex blob methods for the Euler equations.
I will present quantitative results on the convergence of the method
along with many numerical examples exploring its qualitative behavior.

Wednesday Sept. 3, 2014
3:30 p.m.
Room 230

Fall Semester Members Seminar

Robert
McCann (University of Toronto)
Academic wages, singularities, phase transitions and pyramid schemes
In this lecture we introduce a mathematical
model which couples the education and labor markets, in which steadysteady
competitive equilibria turn out to be characterized as the solutions
to an infinitedimensional linear program and its dual. In joint work
with Erlinger, Shi, Siow and Wolthoff, we use ideas from optimal transport
to analyze this program, and discover the formation of a pyramidlike
structure with the potential to produce a phase transition separating
singular from nonsingular wage gradients.
Wages are determined by supply and demand.
In a steadystate economy, individuals will choose a profession, such
as worker, manager, or teacher, depending on their skills and market
conditions. But these skills are determined in part by the education
market. Some individuals participate in the education market twice,
eventually marketing as teachers the skills they acquired as students.
When the heterogeneity amongst student skills is large, so that it
can be modeled as a continuum, this feedback mechanism has the potential
to produce larger and larger wages for the few most highly skilled
individuals at the top of the market. We analyze this phenomena using
the aforementioned model. We show that a competitive equilibrium exists,
and it displays a phase transition from bounded to unbounded wage
gradients, depending on whether or not the impact of each teacher
increases or decreases as we pass through successive generations of
their students.
We specify criteria under which this equilibrium
will be unique, and under which the educational matching will be positive
assortative. The latter turns out to depend on convexity of the equilibrium
wages as a function of ability, suitably parameterized.

July 31, 2014
12:00 p.m.
Room 230
