Speaker Abstracts
Stefanella Boatto (Universidade Federal de Rio de
Janeiro)
N-vortex and N-body dynamics on closed surfaces : a common point of
view
Coauthors: Rodrigo Schaefer, David Dritschel
One of the today's challenges is the formulation of the N-body and N-vortex
dynamics on Riemann surfaces.In this article we address how the two problems
are strongly related one another when looking at them from the point of
view of the intrinsic geometry of the surface where the dynamics takes place.
This is rather different from what had been previously done by Florin Diacu
and collaborators where the dynamics over a sphere was deduced by viewing
the sphere as imbedded in R^3. In the vortex case there are already formulations
of point vortex dynamics over surfaces with constant curvature (see Y. Kimura
(1999), S. Boatto (2008)). In this article we address the following question
: given a surface M of metric g, the distribution of matter S -- viewed
as a set of point planets -- on M and its initial position and velocities,
is it possible to deduce the dynamics of the masses? We propose a formulation
of the point-bodies' dynamics directly in the intrinsic geometry of the
surface. Among other things, we find that in the plane the two masses problem
does not obeys to the known Kepler laws. Furthermore on a surface with not
constant Gaussian curvature, such as an ellipsoid of rotation, both a single
vortex would move and a single mass would accelerate, which is the classical
version of the relativistic Equivalence Principle.
Lia Bronsard (McMaster University)
Weak Anchoring for a Two-Dimensional Liquid Crystal
We study the weak anchoring condition for nematic liquid crystals in the
context of the Landau-De~Gennes model. We restrict our attention to two
dimensional samples and to nematic director fields lying in the plane, for
which the Landau-De~Gennes energy reduces to the Ginzburg--Landau functional,
and the weak anchoring condition is realized via a penalized boundary term
in the energy. We study the singular limit as the length scale parameter
tends to zero, assuming the weak anchoring parameter tends to infinity at
a prescribed rate. We also consider a specific example of a bulk nematic
liquid crystal with an included oil droplet and derive a precise description
of the defect locations for this situation.
Rustum Choksi (McGill University)
Self-Assembly: Variational models, Energy Landscapes, and Metastability
Self-assembly, a process in which a disordered system of preexisting components
forms an organized structure or pattern, is both ubiquitous in nature and
important for the synthesis of many designer materials. In this talk, we
will address three variational models for self-assembly from the point of
view of mathematical analysis and computation.
The first is a nonlocal perturbation of Coulombic-type to the well-known
Ginzburg-Landau/Cahn-Hilliard free energy. The functional has a rich and
complex energy landscape with many metastable states. I will present a simple
method for assessing whether or not a particular computed metastable state
is a global minimizer. The method is based upon finding a ``suitable"
global quadratic lower bound to the free energy.
The second model is a purely geometric and finite-dimensional paradigm
for self-assembly which generalizes the notion of centroidal Voronoi tessellations
from points to rigid bodies. Using a level set formulation, we a priori
fix the geometry for the structures and consider self-assembly entirely
dictated by distance functions. I will introduce a novel fast algorithm
for simulations in two and three space dimensions.
As time permits, I will present a third model based upon functionals with
competing attractive and repulsive algebraic potentials. I will address
questions about global existence of minimizers.
Michael Cullen (UK Met Office)
Free upper boundary value problems for the semi-geostrophic equations
Coauthors: B. Pelloni, T. Kuna, D. Gilbert
The semi-geostrophic system is widely used in the modelling of large-scale
atmo-spheric flows. In this paper, we prove existence of solutions of the
incompressible semi- geostrophic equations in a fully three-dimensional
domain with a free upper boundary condition. We show that, using methods
similar to those introduced in the pioneering work of Benamou and Brenier
[6], who analysed the same system but with a rigid boundary condition, we
can prove the existence of solutions for the incompressible free boundary
problem. The proof is based on optimal transport results as well as the
analysis of Hamiltonian ODEs in spaces of probability measures given by
Ambrosio and Gangbo [4]. We also show how these techniques can be modified
to yield the same result also for the compressible version of the system.
This presentation is an update on that given at MSRI in October 2013.
Gero Friesecke (Technische Universität München)
Alessandro Giuliani (Roma Tre University)
HEIGHT FLUCTUATIONS IN INTERACTING DIMERS
Perfect matchings of Z^2 (also known as non-interacting dimers on the square
lattice) are an exactly solvable 2D statistical mechanics model. It is known
that the associated height function behaves at large distances like a massless
gaussian field, with the variance of height gradients growing logarithmically
with the distance [Kenyon-Okounkov-Sheffield, 2006]. As soon as dimers mutually
interact, via e.g. a local energy function favoring the alignment among
neighboring dimers, the model is not solvable anymore and the dimer-dimer
correlation functions decay polynomially at infinity with a non-universal
(interaction-dependent) critical exponent. We prove that, nevertheless,
the height fluctuations remain gaussian even in the presence of interactions,
in the sense that all their moments converge to the gaussian ones at large
distances. The proof is based on a combination of multiscale methods with
the path-independence properties of the height function. Joint work with
V. Mastropietro and F. Toninelli.
Radu Ignat (University of Toulouse)
Pattern formation in micromagnetics
We will focus on the structure of topological singularities arising in
micromagnetics. Micromagnetics is the continuum theory of magnetic moments
based on a variational principle: an observed magnetization m is given as
a saturated (local) minimizer of a free energy functional, the micromagnetic
energy, which serves to describe magnetic patterns (e.g. domain patterns,
domain walls, vortices) on many different length scales. From the analytical
point of view, the main challenge in micromagnetics consists in the combination
of the nonconvex saturation constraint (i.e., |m| = 1) and the nonlocality
of the micromagnetic energy due to stray-field interaction, that becomes
singular in thin films and drives the formation of typical singular structures.
We will present several challenging problems generated by the competition
of these effects, an interplay of geometric analysis and harmonic maps,
elliptic regularity theory, variational methods and hyperbolic conservation
laws.
Ansgar Jüngel (Vienna University of Technology)
Discrete entropy methods for nonlinear diffusive evolution equations
Entropy methods are extremely useful tools for the analysis of nonlinear
diffusive evolution equations. The basic idea is to estimate the time derivative
of a so-called entropy functional, depending on the solution of the underlying
diffusion equation, in terms of (a function of) the entropy itself. This
yields not only stability estimates but also results on the large-time asymptotics
with sometimes sharp equilibration rates. For numerical purposes, it is
desirable to develop entropy-stable or etropy-dissipative approximations
for diffusive equations, in order to preserve numerically the continuous
structure. The analysis in discrete spaces started only very recently, see
e.g. the contributions of Caputo-DaiPra-Posta, Erbar-Maas, and Mielke.
In this talk, we will present some results on numerical discretizations
which preserve the entropy structure of some nonlinear diffusive equations.
Numerical methods include implicit Runge-Kutta and one-leg multi-step time
approximations and finite-volume space discretizations. The proofs are based
on novel discrete Beckner inequalities, systematic integration by parts,
and Dahlquist's G-stability theory. Porous-medium equations, cross-diffusion
systems, and fourth-order quantum diffusion equations will be considered.
Numerical simulations illustrate the theoretical results.
Malte Kampschulte (RWTH Aachen)
Gradient flows in the framework of Cartesian Currents
Cartesian Currents as introduced by Giaquinta, Modica and Soucek are well
suited to describe topological singularities, e.g. bubbling of spheres for
harmonic maps. While geared towards geometric variational problems, it would
be desirable to use this powerful tool also to describe the singularity
formation in dynamic problems, e.g. the bubbling in the harmonic map heat
flow or vortex annihilation in micromagnetics. One step in this direction
is looking at gradient flows of Cartesian Currents. For this, a surprisingly
general all-time existence result can be derived for minimizing movements.
Furthermore I will propose a generalization of the Wasserstein distance
as a candidate for a possible metric on (Cartesian) Currents, which is sensitive
for topological changes. This talk is part of my ongoing PhD work.
Boris Khesin (University of Toronto)
KAM theory and the 3D Euler equation
We show that the dynamical system defined by the hydrodynamical Euler equation
on any closed Riemannian 3-manifold M is not mixing in the C^k-topology
(k>4 and non-integer) for any prescribed value of helicity and sufficiently
large values of energy. This can be regarded as a 3D version of Nadirashvili's
theorem showing the existence of wandering solutions for the 2D Euler equation.
On the way we construct a family of functionals on the space of divergence-free
C^1 vectorfields on the manifold, which are integrals of motion of the 3D
Euler equation. Given a vectorfield these functionals measure the part of
the manifold foliated by ergodic invariant tori of fixed isotopy types.
This allows one to get a lower bound for the C^k-distance between a divergence-free
vectorfield (in particular, a steady
solution) and a trajectory of the Euler flow.
This is a joint work with S.Kuksin and D.Peralta-Salas.
Elliott Lieb (Princeton University)
Indirect Coulomb Energy with Gradient Correction
Elliott Lieb, Princeton University (with Mathieu Lewin, C.N.R.S.)
We prove a Lieb-Oxford-type inequality on the indirect part of the Coulomb
energy of a general many-particle quantum state, with a lower constant than
the original statement but involving an additional gradient correction.
The result is similar to a recent inequality of Benguria, Bley and Loss,
except that the correction term is purely local, which is more usual in
density functional theory.
Mathieu Lewin (Cergy-Pontoise)
Derivation of nonlinear Gibbs measures from many-body quantum mechanics
Nonlinear Gibbs measures have recently become a useful tool to construct
solutions to time-dependent nonlinear Schrödinger equations with rough
initial data. In this talk I will explain how these measures can be obtained
from the corresponding many-particle quantum Gibbs states, in a mean-field
limit where the temperature $T$ diverges and the interaction behaves as
$1/T$. Our results cover the defocusing nonlinear Schrödinger case
on the circle, as well as smoother interactions in higher dimensions. Joint
work with P.T. Nam and N. Rougerie.
Michael Loss (Georgia Institute of Technology)
Kac particles interacting with a thermal reservoir
The Kac master equation coupled to a thermal reservoir offers a simple model
for studying equilibration. It describes a collection of interacting particles
that are allowed to collide with an infinite reservoir of thermal particles.
One can compute the first two gaps of the generator and also obtain reasonable
numbers for the decay rate of the entropy. This fact follows from a connection
of this model with the Ornstein Uhlenbeck process. If the reservoir is modeled
by a finite number of particles then the interaction will move this reservoir
out of equilibrium. Some results will be presented concerning the relation
of this system with the infinite reservoir. This is joint work with Federico
Bonetto, Ranjini Vaidyanathan and Hagop Tossounian.
Jianfeng Lu (Duke University)
Nonexistence of minimizers to some variational principles with nonlocal
repulsive interactions
Variational principles with competing attractive and repulsive interactions
often arise from physical applications. In this talk, we will consider variational
models with Coulomb repulsion coming from the theory of phase transition
and electronic structure theory. We will present some recent progress on
understanding the nonexistence of minimizers to these variational problems.
Christof Melcher (RWTH Aachen University)
Topological solitons in chiral magnetism
Magnets without inversion symmetry are a prime example of a solid state
system featuring topological solitons on the nanoscale, and a promising
candidate for novel spintronic applications. We prove existence of isolated
chiral skyrmions minimizing a ferromagnetic energy in a non-trivial homotopy
class. In contrast to the classical Skyrme mechanism from nuclear physics,
the stabilization is due to an antisymmetric exchange (Dzyaloshinskii-Moriya)
interaction term of linear gradient dependence, which breaks the chiral
symmetry.
Cy Maor
The emergence of torsion in the continuum limit of distributed dislocations
Coauthors: Raz Kupferman
In this talk I will present a rigorous homogenization theorem for distributed
dislocations, thus bridging between different approaches modeling them.
I will construct a sequence of locally-flat Riemannian manifolds with dislocation-type
singularities, and show that this sequence converges, as the dislocations
become denser, to a flat non-singular Weitzenböck manifold, i.e. a
manifold endowed with a metric connection with zero curvature and non-zero
torsion. In the process I will introduce a new notion of convergence of
Weitzenböck manifolds, which is applies to this class of homogenization
problems, and if time permits, discuss more general examples.
Cyrill Muratov (New Jersey Institute of Tehnology)
Low density phases in a uniformly charged liquid with homogeneous neutralizing
background
This talk is concerned with the macroscopic behavior of global energy minimizers
in the three-dimensional sharp interface unscreened Ohta-Kawasaki model
of diblock copolymer melts. This model is also referred to as the nuclear
liquid drop model in the studies of the structure of highly compressed nuclear
matter found in the crust of neutron stars, and, more broadly, is a paradigm
for energy-driven pattern forming systems in which spatial order arises
as a result of the competition of short-range attractive and long-range
repulsive forces. We are interested in the large volume behavior of minimizers
in the low volume fraction regime, in which one expects the formation of
a periodic lattice of small droplets of the minority phase in a sea of the
majority phase. Under periodic boundary conditions, we prove that the considered
energy ?-converges to an energy functional of the limit “homogenized”
measure associated with the minority phase consisting of a local linear
term and a non-local quadratic term mediated by the Coulomb kernel. As a
consequence, asymptotically the mass of the minority phase in a minimizer
spreads evenly across the domain. We also prove that the energy density
distributes uniformly across the domain as well, with the energy density
approaching that of the minimizers of the volume constrained problem in
the whole space. This suggest that in the microscopic limit the minimizers
should appear as a uniformly distributed array of droplets which minimize
the energy density for the volume constrained whole space problem. This
is joint work with H. Knuepfer and M. Novaga.
Phan Thanh Nam (Cergy-Pontoise)
Interpolation inequalities with non-local terms and Lieb-Thirring inequalities
for interacting bosons
I will discuss the relation between a class of interpolation inequalities
with non-local terms first proved by Frank, Bellazzini and Visciglia and
the Lieb-Thirring inequalities for interacting bosons developed recently
by Solovej, Lundholm and Portmann. The problem on the existence of optimizers
is open.
Michiel Renger (WIAS Berlin)
Connecting Particle Systems to entropy-driven gradient flows
Coauthors: Hong Duong, Vaios Laschos, Alexander Mielke, Mark Peletier, Marco
Veneroni, Johannes Zimmer
The statistical mechanics programme has provided us a deep understanding
of the connection between stochastic particle systems at the microscopic
level and thermodynamics on the macro level. In particular, the entropy
functional can be connected to particle systems by its large deviations.
I try to extend this principle to the non-equilibrium case, and connect
entropic gradient flows to particle systems in a similar fashion. Such connection
can reveal a large class of gradient flow structures, among which are the
Wasserstein-Entropy structure, as well as a previously unknown structure
for discrete Markov chains. I also briefly consider the inverse problem:
which microscopic fluctuations are needed to retrieve a given gradient flow
structure from the large deviations?
Etienne Sandier (Université Paris 12 )
Critical point of the Ginzburg-Landau functional in N dimensions
(Joint work with Yuxin Ge and Peng Zhang, UPEC)
"In this talk I will describe recent results on the limits of minimizers
and non minimizing critical points of the Ginzburg-Landau functional in
N dimensions, and a construction of such non-minimizing sequence."
Marc Sedjro (RWTH Aachen)
On the Almost Axisymmetric Flows
Coauthors: Michael Cullen
Almost axisymmetric flows are obtained as approximations to the inviscid
Boussinesq equations. They bring about an unusual Monge Ampere equations
for which we prove the existence and the uniqueness of a variational solution.
I will discuss how overcoming a regularity problem could help construct
a solution for these flows.
I.M. Sigal (Univeristy of Toronto)
Singularity formation in the mean curvature flow
The mean curvature flow appears naturally in the motion of interfaces in
material science, physics and biology. It also arises in geometry and has
found its applications in topological classification of surfaces. In this
talk I will discuss recent results on formation of singularities under this
flow. In particular, I will describe the 'spectral' picture of singularity
formation and sketch the proof the neck pinching results obtained jointly
with Zhou Gang and Dan Knopf.
Back to top
