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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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].
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
--------------------------------------------------------------------------------
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.
Back to top
