SCIENTIFIC PROGRAMS AND ACTIVITIES

Andres Contreras (The Fields Institute)
Local minimizers in Ginzburg-Landau theory

We present results concerning the existence of stable vortex configurations in 2D and 3D Ginzburg-Landau. These results represent joint works with R.L. Jerrard and S. Serfaty.

Tuesday. December 2, 2014

1:10 p.m.
Room 210

Young-Heon Kim (University of British Columbia)
Multimarginal optimal transport


We explain some recent progress on multi-marginal 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

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 Arnold-Liouville theorem. This is a joint work with Boris Khesin.

Wednesday. December 3, 2014

2:10 p.m.
Room 210

Jun Kitagawa (The Fields Institute / University of Toronto)
The Aleksandrov estimate and its variants in Monge-Ampère equations

The Aleksandrov estimate plays a central role in the regularity theory of weak solutions of the Monge-Amp{`\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

Slim Ibrahim (University of Victoria)
Asymptotic derivation of the classical Magneto-Hydro-Dynamic system from Navier-Stokes-Maxwell


The incompressible Magneto-Hydro-Dynamic (MHD) system is a classical and fundamental model in plasma physics. Although well known, its derivation from Navier-Stokes type equations has been so far formal. In this talk and after reviewing the results about the well-posedness, 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.

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 $\mathbb{R}^N$ ($N\geq3$),
$f(x,u)\backsim |u|^{p-2}u$ and $g(x,v)\backsim |v|^{q-2}v$, with
$2<p,q<2N/(N-2)$. Part of the work is joint with Fabrice Colin.

October 28, 2014

1:10 p.m.
Room 210

Paul Woon Yin Lee (Chinese University of Hong Kong)
Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds


In this talk, we introduce a type of Ricci curvature lower bound for a natural sub-Riemannian structure on Sasakian manifolds and discussvarious consequences under this condition.

 

Tuesday, October 28, 2014

2:10 p.m.
Stewart Library

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 1-parameter 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 Euler-Lagrange 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.

Tuesday, October 21, 2014

1:10 p.m.
Room 210

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 Gerard-Varet). 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 theVlasov-Poisson system (joint work with Jacob Bedrossian)

October 21-23, 2014

10:10 a.m.
Room 210

Nestor Guillen (University of Massachusetts at Amherst )
On Aleksandrov-Bakelman-Pucci estimates for integro-differential equations

Convexity has played an important role in elliptic equations. One such instance is its crucial appearance in the celebrated Aleksandrov-Bakelman-Pucci estimate (ABP), which is the back bone of the regularity theory of fully non-linear elliptic equations. In this talk I will describe the shortcomings of convexity when dealing with non-local 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 non-local elliptic equations which is new even for non-local 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

Mircea Petrache (The Fields Institute)
The quest for global duality for geometric mass-minimization 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 maxflow-mincut duality with coefficients taken modulo 2. As a motivation for future work I will point out 1D-problems 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

Martial Agueh (University of Victoria)
Uniqueness of the compactly supported weak solution to the relativistic Vlasov-Darwin system (Slides)

The relativistic Vlasov-Darwin (RVD) system is a kinetic model that describes the evolution of a collisionless plasma whose particles interact through their self-induced electromagnetic field and move at a speed ``not too fast'' compared with the speed of light. In contrast with the Vlasov-Poisson system, it is an approximation of the Vlasov-Maxwell 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 Vlasov-Poisson system. We then use optimal transport techniques to study uniqueness of weak solutions for this system. This is a joint work with R. Sospedra-Alfonso.

Tuesday, October 14, 2014

2:10 p.m.
Stewart Library

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 $$E[\mu]=\int_{\mathbb{R}^n}\!\int_{\mathbb{R^n}} W(|x-y|)\,d\mu(x)d\mu(y)$$ 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 long-range 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 attractive-repulsive potentials used in modelling collective behavior of many-agent systems, granular media and self-assembly 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

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 $$E[\mu]=\int_{\mathbb{R}^n}\!\int_{\mathbb{R^n}} W(|x-y|)\,d\mu(x)d\mu(y)$$ 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 long-range 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 attractive-repulsive potentials used in modelling collective behavior of many-agent systems, granular media and self-assembly 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

Olivier Kneuss (University of Zurich )
Bi-Lipschitz 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 bi-Lipschitz solution in case of right-hand 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

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 Duyckaerts-Kenig-Merle, nodal nonradial solutions to the critical Yamabe problem $ -\Delta Q= |Q|^{\frac{2}{n-2}} 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

Manuel Gnann (The Fields Institute)
The moving contact line in viscous thin films: a singular free boundary problem


We are interested in the thin-film equation with quadratic mobility and zero contact angle, modeling the height of a viscous thin-film with a linear Navier-slip condition at the liquid-solid interface. This degenerate parabolic fourth-order 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 source-type self-similar 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 well-posedness result of the corresponding full parabolic problem. We conclude the talk with an overview of other questions and results, such as the generalization to thin-film equations with general mobility, higher regularity, and convergence to the source-type self-similar 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

Tuesday Sept 9, 2014

1:10 p.m.
Room 230

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

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 8-10 Distinguished Lecture Series at Fields.

Wednesday Sept. 3, 2014

2:10 p.m.
Room 230

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

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 steady-steady competitive equilibria turn out to be characterized as the solutions to an infinite-dimensional 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 pyramid-like structure with the potential to produce a phase transition separating singular from non-singular wage gradients.

Wages are determined by supply and demand. In a steady-state 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