THEMATIC PROGRAMS

April 20, 2014

October 25- November 10, 2010
Workshop on Asymptotic Geometric Probability and Optimal Transportation

Talk Titles and Abstracts

Elements of Geometric Measure Theory in Wiener spaces
by
Luigi Ambrosio
Scuola Normale Superiore
Coauthors: Alessio Figalli

In the talk I will illustrate some recent developments of the theory of sets of finite perimeter in infinite-dimensional Gaussian spaces (Wiener spaces). Since in this context the usual differentiation theorems (based on Vitali or Besicovitch covering theorem) fail, the notion of measure-theoretic boundary and of codimension-one Hausdorff measure have to be properly understood. In particular I shall focus on recent papers by Hino and Ambrosio-Figalli on the extension of De Giorgi's representation theorem of the perimeter measure to infinite-dimensional spaces.


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Asymptotic behavior of smoothed-shock solutions in coating flows
by
Daniel Badali
University of Toronto
Coauthors: Marina Chugunova, Dmitry E. Pelinovski, Steven Pollack

The behavior of thin film of fluid on the inside of a rotating cylinder is examined. The physical parameters of the fluid (surface tension, viscosity, etc.) as well as the radius and angular frequency of the cylinder can be represented by the non-dimensional parameter e. For fluids of low surface tension, which are of interest practically, e is small, and so we consider the case when e << 1. Numerical results reveal the existence of an increasing number of steady states for e values down to 10-4, which is an order of magnitude smaller than previously published results. A surface locating the existence of steady states in a space of parameters (e and the mass and flux associated with each solution) was generated numerically. This surface was revealed to contain complicated loop formation, and a sophisticated turning-point algorithm was developed and implemented to push the limits of the parameter space. Spectral stability of shock solutions was analyzed numerically and unstable shock solutions were found to exist, contrary to what was previously thought.


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Wasserstein space over Hadamard space
by
Jerome Bertrand
University Paul Sabatier Toulouse
Coauthors: B. Kloeckner

In the talk, I will consider the quadratic Wasserstein space over a metric space of non-positive curvature (globally). Despite the fact that the Wasserstein space does not inherit the curvature property, I will show that some asymptotical properties extend to the Wasserstein space.


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Finite speed propagation of the interface and blow-up solutions for long-wave unstable thin-film equations
by
Marina Chugunova
University of Toronto
Coauthors: Mary Pugh and Roman Taranets

We study short-time existence, long-time existence, finite speed of propagation, and finite-time blow-up of nonnegative solutions for long wave unstable thin-film equations.


For 0 < n < 2 we prove the existence of a nonnegative, compactly supported, strong solution on the line that blows up in finite time. The construction requires that the initial data be nonnegative, compactly supported, and have negative energy. The blow-up is proven for a large range of (n,m) exponents and extends the results of [Indiana Univ Math J 49:1323-1366, 2000].


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On the size of the Navier - Stokes sungular set
by
Walter Craig
Department of Mathematics & Statistics, McMaster University

We consider the situation in which a weak solution of the Navier - Stokes equations fails to be continuous in the strong L^2 topology at some singular time t=T. We identify a closed set S_T in space on which the L^2 norm concentrates at this time T. The famous Caffarelli, Kohn Nirenberg theorem on partial regularity gives an upper bound on the Hausdorff dimension of this set. We study microlocal properties of the Fourier transform of the solution in the cotangent bundle T*(R^3) above this set. Our main result is a lower bound on the L^2 concentration set. Namely, that L^2 concentration can only occur on subsets of T*(R^3) which are sufficiently large. An element of the proof is a new global estimate on weak solutions of the Navier - Stokes equations which have sufficiently smooth initial data.


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Topological defects in the abelian Higgs model.
by
Magdalena Czubak
University of Toronto
Coauthors: Robert Jerrard

We provide a rigorous description of the dynamics of energy concentration sets in the abelian Higgs model.


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Remarks on the regularity of optimal transport
by
Philippe Delanoe
CNRS at University of Nice Sophia Antipolis
Coauthors: Yuxin GE (partly)

We will specify steps of the proof of the existence of a smooth optimal transportation map, via the continuity method, for the Brenier-McCann cost function on some closed Riemannian manifolds.


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The geometry of the Ma-Trudinger-Wang condition
by
Alessio Figalli
UT Austin
Coauthors: Ludovic Rifford, Cedric Villani

In this talk I'll first review some recent results concerning the question of finding necessary and sufficient conditions ensuring

continuity of optimal maps on Riemannian manifolds. Then we will see how the Ma-Trudinger-Wang condition, first introduced to prove regularity of

optimal transport maps, can be used as a tool to obtain geometric informations on the cut locus of the underlying manifold.


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Convolution inequalities for Boltzmann collision operators and applications
by
Irene M. Gamba
The University of Texas at Austin
Coauthors: Ricardo Alonso and Emanuel Carneiro

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in Lp we prove a Young's inequality for hard potentials, which is sharp for Maxwell molecules in the L2 case.

Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some Lsweak or Ls. The used method resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.

As an immediate application we obtain that distributional solution of the space inhomogeneous Boltzmann equation for singular (soft) potentials, for initial data near local Maxwellians states and integrable differential angular cross-section b ? La, are classical in the sense that propagate Lp-regularity in physical and velocity space and have Lp stability for a range of p depending on the space dimension dimension and the integrability exponent a.


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STICKY PARTICLE DYNAMICS WITH INTERACTIONS
by
Wilfrid Gangbo
Georgia Institute of Technology
Coauthors: Giuseppe Savare and Michael Westdickenberg

We consider compressible fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid. We explain how this flow can be described by a differential inclusion on the space of transport maps, when the sticky particle dynamics is assumed. We prove a stability result for solutions of this system. Global existence then follows from a discrete particle approximation.


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Homogenization, inverse problems and optimal control via selfdual variational calculus
by
Nassif Ghoussoub
The University of British Columbia

We use the theory of selfdual Lagrangians to provide a variational approach to certain inverse problems, to issues of optimal control, as well as to the homogenization of equations driven by a periodic family of monotone vector fields. The approach has the advantage of using weak and Gamma-convergence methods for corresponding functionals, as opposed to uniform and graph convergence methods which are normally used in the absence of standard potentials


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The Heat Flow as Gradient Flow
by
Nicola Gigli
University of Nice
Coauthors: Luigi Ambrosio (Scuola Normale Superiore - Pisa) Giuseppe Savaré (University of Pavia)

I will prove that in a generic metric measure space with Ricci curvature bounded below the Gradient Flow of the relative entropy w.r.t. W_2 coincides with the Gradient Flow of a naturally defined Dirichlet energy w.r.t. the L^2 structure.


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On Positivity of Steady States of the Thin Films Equation
by
Dan Ginsberg
University of Toronto

The thin film equation is a degenerate, nonlinear, fourth-order PDE which is interesting for both applied and theoretical reasons. I will provide a short introduction to the study of this equation, as well as present some results relating to the positivity of both classical and weak steady-state solutions. In addition, I will discuss some of the numerical techniques we have used (one-parameter continuation, Petviashvili's method) to study this equation.


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Adomian Decomposition Method, Cherruault Transformations, Homotopy Perturbation Method, and Nonlinear Dynamics: Theories and Comparative Applications to Frontier problems.
by
Tony Gomis
NBI
Coauthors: Yves Cherruault*, Université Paris 6 France * RIP in 2010

New global methods for solving complex, nonlinear, continuous and discrete,deterministic and stochastic,differential or integral, and combined functional equations, will be presented and compared. This talk will outline the Adomian Decomposition Method,and the Homotopy Perturbation Techniques, all offering solutions as convergent infinite functional series.In this talk , the Cherruault Alienor transformations based on a generalization of the space-filling curves theory(for quasi-lossless dimensionality compression, and for functional Global Optimization ) will be outlined and applied to real-world problems.

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Mass Transportation and Optimal Coupling of Brownian Motions
by
Elton P. Hsu
Department of Mathematics, Northwestern University
Coauthors: Theo Sturm (University of Bonn) Ionel Popescu (Georgia Tech)

It is well known that the well known mirror coupling is an optimal coupling of euclidean Brownian motions. In general optimal couplings are not unique. Using a simple uniqueness result from mass transportation theory with a concave cost function, we show that the mirror coupling is the unique optimal coupling among Markov couplings. Whether this result also holds for Riemannian Brownian motion on a compact Riemannian manifold is an unsolved and highly interesting problem. We show that the problem can be reduced to the uniqueness problem for a mass transportation problem for a cost function defined by the heat kernel. More generally, we attempt to develop a theory of coupling for manifold-valued semimartingales. It can be formulated as a theory of mass transportation theory in the (infinite dimensional) path space over the manifold. Various forms of cost function have been defined in the literature but so far none of them is completely satisfactory. We propose a new definition of cost function which we hope will lead to a more satisfactory theory of mass transportation for the path space. This talk is based in part on the joint work with

T. Sturm and I. Popescu.


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Can you cut a convex body into five convex pieces with the same area and the same perimeter?
by
Alfredo Hubard
NYU
Coauthors: Boris Aronov

We use optimal transport and equivariant topology to show the next Borsuk Ulam/Ham sandwich type statement. Given a prime p smaller or equal than the dimension, an absolutely continuous probability measure m, a convex body K, and a continuous functional F from the space of convex bodies to the real numbers. It is always possible to partition the convex body K into p convex pieces K_1,K_2...K_p, such that m(K_1)=m(K_2)=...m(K_p)=1/p and F(K_1)=F(K_2)=...F(K_p). Simultaneously This is joint work with B.Aronov.


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Curvature of Random Metrics
by
Dmitry Jakobson
McGill university, Department of Mathematics and Statistics
Coauthors: Yaiza Canzani, Igor Wigman

We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension n>2, and for the Q-curvature of random Riemannian metrics.


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Optimal transport and geodesics for H1 metrics on diffeomorphism groups
by
Boris Khesin
University of Toronto

We describe the Wasserstein space for the homogeneous H1 metric which turns out to be isometric to (a piece of) an infinite-dimensional sphere. The corresponding geodesic flow turns out to be integrable, and it is a generalization of the Hunter-Saxton equation. The corresponding optimal transport can be used for the ßize-recognition", as opposed to the ßhape recognition". This is a joint work with J. Lenells, G. Misiolek, and S. Preston.


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Regularity of optimal transport maps on multiple products of spheres
by
Young-Heon Kim
University of British Columbia
Coauthors: Alessio Figalli and Robert McCann

Existence and uniqueness of optimal transportation maps is well known on Riemannian manifolds where the transportation cost of moving a unit mass is given by the distance squared function. However, regularity (such as continuity and smoothness) of such maps is much less known, especially beyond the case of the round sphere and its small perturbations. Moreover, if the manifold has a negative curvature somewhere, there are discontinuous optimal maps even between smooth mass distributions.

In this talk, we explain a regularity result of optimal maps on products of multiple round spheres of arbitrary dimension and size. This is a first such result given on non-flat Riemannian manifolds whose curvature is not strictly positive.

This is joint work with Alessio Figalli and Robert McCann.


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Regularity for the optimal transport problem with Euclidean distance squared cost on the embedded sphere
by
Jun Kitagawa
Princeton University
Coauthors: Micah Warren

We consider regularity for Monge solutions to the optimal transport problem when the initial and target measures are supported on the embedded sphere, and the cost function is the Euclidean distance squared. Gangbo and McCann have shown that when the initial and target measures are supported on boundaries of strictly convex domains in Rn, there is a unique Kantorovich solution, but it can fail to be a Monge solution. By using PDE methods, in the case when we are dealing with the sphere with measures absolutely continuous with respect to surface measure, we present a condition on the densities of the measures to ensure that the solution given by Gangbo and McCann is indeed a Monge solution, and obtain higher regularity as well.


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Positive definite functions and stable random vectors
by
Alexander Koldobsky
University of Missouri-Columbia

We say that a random vector X=(X1, ..., Xn) in Rn is an n-dimensional version of a random variable Y if for any a ? Rn the random variables ?aiXi and g(a) Y are identically distributed, where g:Rn? [0, 8) is called the standard of X. An old problem is to characterize those functions g that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L0. This result is almost optimal, as the norm of any finite dimensional subspace of Lp with p ? (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P.Lèvy. An equivalent formulation is that if a function of the form f(?·?K) is positive definite on Rn, where K is an origin symmetric star body in Rn and f:R? R is an even continuous function, then either the space (Rn, ?·?K) embeds in L0 or f is a constant function. Combined with known facts about embedding in L0, this result leads to several generalizations of the solution of Schoenberg's problem on positive definite functions.


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Sobolev regularity of optimal transportation
by
Alexander Kolesnikov
MSUPA and HSE (Moscow)

We present some global Sobolev a priori estimates for optimal transportation. Our approach is based on the above tangential formalism. In particular, we discuss several generalizations of Caffarelli's contraction theorem and discuss relations with the transportation inequalities.


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Central Limit Theorem and geometric probability
by
Shaghayegh Kordnoori
Msc of statistics, Islamic Azad University North Tehran Branch

In this paper convergence of geometric functional analysis and classical convexity which create asymptotic geometric analysis was investigated;Moreover,a Central Limit Theorem for convex sets and Long-concave measures was studied.

It was shown that expressing the geometric probability as sums of stabilizing functionals can make the rates of convergence of CLT optimal For graphs in computational geometry.


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Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian Manifolds
by
Lee, Paul Woon Yin
UC Berkeley
Coauthors: Andrei Agrachev

Measure contraction property (MCP) is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. However, the definition of MCP is not computable in general. In this talk, I'll discuss computable sufficient conditions for a three dimensional contact subriemannian manifold to satisfy such property.


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Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds
by
Lee, Woon Yin (Paul)
University of California, Berkeley
Coauthors: A.Agrachev

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.


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A stochastic formula for the entropy and applications
by
Joseph Lehec
Université Paris-Dauphine

We prove a variational stochastic formula for the Gaussian relative entropy of a measure. As an application we give unified and short proofs of a number of well-known inequalities.


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Global regularity of the reflector problem
by
Jiakun Liu
Princeton University
Coauthors: Neil S. Trudinger

In this talk we study a reflector system which consists of a point light source, a reflecting surface and an object to be illuminated. Due to its practical applications in optics, electro-magnetics, and acoustic, it has been extensively studied during the last half century. This problem involves a fully nonlinear partial differential equation of Monge-Ampere type, subject to a nonlinear second boundary condition. In the far field case, it is related to the reflector antenna design problem. By a duality, namely a Legendre type transform, Xu-Jia Wang has proved that it is indeed an optimal transportation problem. Therefore, the regularity results of optimal transportation can be applied. However, in the general case, the reflector problem is not an optimal transportation problem and the regularity is an extremely complicated issue. In this talk, we give necessary and sufficient conditions for the global regularity and briefly discuss their connection with the Ma-Trudinger-Wang condition in optimal transportation.


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NUMERICAL SIMULATION OF RESONANT TUNNELING OF FAST SOLITONS FOR THE NONLINEAR SCHRODINGER EQUATION
by
Xiao Liu
Coauthors: WALID K. ABOU SALEM and CATHERINE SULEM

In this work, we show numerically the phenomenon of resonant tunneling for fast solitons through large potential barriers for the cubic Nonlinear Schrödinger equation in one dimension with external potential. We consider two classes of potentials, namely the `box' potential and a repulsive 2-delta potential under certain conditions. We show that the transmitted wave is close to a soliton, calculate the transmitted mass of the solution and show that it converges to the total mass of the solution as the velocity is increase.


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Hardy-Littlewood-Sobolev Inequalities via Fast Diffusion Flows
by
Michael Loss
School of Mathematics, Georgia Tech
Coauthors: Eric A. Carlen and Jose A. Carrillo

We give a simple proof of the l = d-2 cases of the sharp Hardy-Littlewood-Sobolev inequality for d = 3, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for d=2 via a monotone flow governed by the fast diffusion equation.


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Optimal and Better Transport Plans
by
Gabriel Maresch
TU Vienna
Coauthors: Mathias Beiglböck, Martin Goldstern and Walter Schachermayer

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value infinity. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=8} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a "better" notion of optimality called robust optimality.


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The magnitude of a metric space
by
Mark Meckes
Case Western Reserve University

Magnitude is a partially defined numerical invariant of metric spaces introduced recently by Tom Leinster, motivated by considerations from category theory, which generalizes the cardinality of a finite set. I will discuss some of what is known and not known about magnitude, highlighting connections with harmonic analysis, intrinsic volumes (in both convex and Riemannian geometry), and biodiversity. This is work of Tom Leinster, Simon Willerton, and myself.


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A generalization of Caffarelli's Contraction Theorem via heat-flow
by
Emanuel Milman
University of Toronto
Coauthors: Young-Heon Kim (UBC)

A theorem of L. Caffarelli implies the existence of a map T, pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map is a contraction in this case). This theorem has found numerous applications pertaining to correlation inequalities, isoperimetry, spectral-gap estimation, properties of the Gaussian measure and more. We generalize this result to more general source and target measures, using a condition on the third derivative of the potential. Contrary to the non-constructive optimal-transport map, our map T is constructed as a flow along an advection field associated to an appropriate heat-diffusion process. The contraction property is then reduced to showing that log-concavity is preserved along the corresponding diffusion semi-group, by using a maximum principle for parabolic PDE. In particular, Caffarelli's original result immediately follows by using the Ornstein-Uhlenbeck process and the Prékopa-Leindler Theorem. We thus avoid using Caffarelli's regularity theory for the Monge-Ampère equation, lending our approach to further generalizations. As applications, we obtain new correlation and isoperimetric inequalities.


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Isolated Singularities of Polyharmonic Inequalities
by
Amir Moradifam
University of Toronto
Coauthors: Marius Ghergu and Steven Taliaferro

We study nonnegative classical solutions u of the polyharmonic inequality
-Dmu = 0 in B1(0) - {0} ? Rn,

where D is the Laplacian operator. We give necessary and sufficient conditions on integers n = 2 and m = 1 such that these solutions u satisfy a pointwise a priori bound as x? 0. In this case we show that the optimal bound for u is
u(x) = O(G(x)) as x? 0

where G is the fundamental solution of Laplacian in Rn.


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Densities from Geometry to Poincaré
by
Frank Morgan
Williams College

The concept of density plays important roles in probability, in geometry and isoperimetric problems, and in Perelman's 2003 proof of the Poincaré Conjecture. The talk will include open questions and progress by undergraduates.


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Icicles, washboard road and meandering syrup
by
Stephen W. Morris
Dept. of Physics, University of Toronto
Coauthors: Antony Szu-Han Chen, Anne-Florence Bitbol, Nicolas Taberlet, Jim N. McElwaine, Jonathan H. P. Dawes, Neil M. Ribe, and John R. Lister.

This talk will describe three recent experiments on emergent patterns in three diverse physical systems. The overall shape and subsequent rippling instability of icicles is an interesting free-boundary growth problem. It has been linked theoretically to similar phenomena in stalactites. We grew laboratory icicles and determined the motion of their ripples. Washboard road is the result of the instability of a flat granular surface under the action of rolling wheels. The rippling of the road, which is a major annoyance to drivers, sets in above a threshold speed and leads to waves which travel down the road. We studied these waves, which have their own interesting dynamics, both in the laboratory and using 2D molecular dynamics simulation. A viscous fluid, like syrup, falling onto a moving belt creates a novel device called a “fluid mechanical sewing machine.” The belt breaks the rotational symmetry of the rope-coiling instability, leading to a rich zoo of states as a function of the belt speed and nozzle height.


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The multi-marginal optimal transportation problem.
by
Brendan Pass
University of Toronto

We consider an optimal transportation problem with more than two marginals. We use a family of semi-Riemannian metrics derived from the mixed, second order partials derivatives of the cost function to provide upper bounds on the dimension of the support of the solution.


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Optimal mass transportation and billiard scattering by rough bodies
by
Alexander Plakhov
University of Aveiro, Department of Mathematics, Aveiro 3810-193, Portugal

The law of elastic reflection by a smooth surface is well known: the angle of incidence is equal to the angle of reflection. In contrast, the law of elastic scattering by a rough surface is not unique, but rather depends on the shape of microscopic pits and groves forming the roughness. In the talk a characterization for laws of scattering by rough surfaces will be given. We will also consider some problems of optimal resistance for rough bodies which can be naturally interpreted in terms of optimal roughening of artificial satellites’ surface on low Earth orbits. We will show that these problems can be reduced to optimal mass transportation (OMT) on the sphere with quadratic cost, and then solve a special OMT problem of this kind.


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About the approximation of orientation-preserving homeomorphisms via piecewise affine or smooth ones
by
Aldo Pratelli
Pavia (Italy)
Coauthors: Sara Daneri (Sissa, Italy) Carlos Mora-Corral (BCAM, Spain)

An old important question is to approximate W1, 2 orientation-preserving homeomorphisms by means of smooth ones, or at least of piecewise affine ones. This is particularly important in the context of non-linear elasticity.

However, this is not easy to do, since classical results only allow to approximate with piecewise affine functions, but in the L8 sense instead of the W1, 2 sense that one would need.

In this talk, we will describe some history of the problem and the state-of-the-art, then we will present some very recent contributions.


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Jordan-Kinderlehrer-Otto scheme for a relativistic cost.
by
Marjolaine Puel
IMT Toulouse, France
Coauthors: R. McCann.

In a paper written in collaboration with R. McCann, we prove existence of solution of a relativistic heat equation via the Jordan-Kinderlehere-Otto scheme.

This method is strongly based on the existence of an optimal map for the Monge-Kantorovich problem with a relativistic cost.

In the study of the Jordan-Kinderlehrer-Otto scheme, we obtain this optimal map using a double minimization process but the general problem of the existence of an optimal map for such a cost is still open.

I will also present this problem which is the heart of a new collaboration with Jerome Bertrand still in progress.


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Triangulation and discretizations of metric measure spaces
by
Emil Saucan
Department of Mathematics, Technion, Haifa, Israel

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition CD(K, N). Moreover, we show that the triangulation can be modified to become a thick one and that, in consequence, such manifolds admit weight-sensitive quasiregular mappings on Sn, with applications to information manifolds. The application of the existence of thick triangulation to estimating length and index of closed geodesics on the considered spaces is also explored.

Furthermore. we extend to weak CD(K, N) spaces the results of Kanai regarding the discretization of manifolds, and show that the volume growth of such a space is the same as that of any of its discretizations.


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Evolution variational inequalities and heat flows in metric measure spaces
by
Giuseppe Savaré
Department of Mathematics, University of Pavia
Coauthors: Luigi Ambrosio, Nicola Gigli

We introduce a metric definition of the heat flow on general metric measure spaces in terms of an evolution variational inequality satisfied by the entropy functional in the Wasserstein space of probability measures and we study the properties of its solutions, in particular concerning linearity, stability, tensorization, and contraction.

The links with the theory of metric-measure spaces with lower Ricci curvature bound by Sturm and Lott-Villani and with the theory of Dirichlet forms will also be discussed.


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Orthogonal and symplectic matrix models: universality and other properties
by
Mariya Shcherbina
Institute for Low Temperature Physics Ukr. Ac. Sci, Kharkov, Ukraine

Orthogonal and symplectic matrix models with real analytic potentials and multi interval supports of the equilibrium measures will be discussed. For these models universality of local eigenvalue statistics and bounds for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics are obtained


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Singularity formation under the mean-curvature flow
by
Israel Michael Sigal
University of Toronto

I will review recent results on singularity formation under the mean-curvature flow. In particular, l will discuss the phenomena of neck-pinching and collapse of closed surfaces.


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Coherent Structures in the Nonlinear Maxwell Equations
by
Gideon Simpson
University of Toronto
Coauthors: Michael I. Weinstein Dmitry Pelinovsky

The primitive equations governing wave propagation in spatially varying optical fibers are the nonlinear Maxwell equations, though this process is often modeled using the nonlinear coupled mode equations (NLCME). NLCME describe the evolution of the slowly varying envelope of an appropriate carrier wave. They are known to possess solitons, which may be of use in optical transmission. In this talk, we numerically study the evolution the NLCME soliton in the primitive equations, and find them to be robust. This is highly non-trivial, as the nonlinear Maxwell equations are a non-convex hyperbolic system, requiring careful treatment of the Riemann problem. Furthermore, we consider extensions of NLCME to a system of infinitely many nonlinear coupled mode equations and present some results suggesting this new system also possesses localized solutions.


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The Aleksandrov-Fenchel inequalities of k+1-convex domains
by
Yi Wang
Princeton University
Coauthors: Sun-Yung Alice Chang

In this paper, we obtain the Aleksandrov and Fenchel inequalities for quermassintegrals of k+1-convex domains. In our proof, the optimal transport map is an important tool to build up connections between different geometric quantities.


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Parabolic optimal transport equations on compact manifolds.
by
Micah Warren
Princeton University
Coauthors: Young-Heon Kim, Jeffrey Streets

We write down a parabolic optimal transport equation and prove that, in almost all of the cases where regularity is known in the elliptic case, the solutions exists for all time and converge to a solution of the elliptic optimal transport equation. Using a metric motivated by special Lagrangian geometry, exponential convergence follows quite easily from an argument of Li-Yau. We will discuss this result, as well as some motivations and analogies to special Lagrangian geometry. We will focus on joint work with Young-Heon Kim and Jeffrey Streets, and may also mention work with Kim and Robert McCann.


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