3,627 Works

The mathematical meaning of the generalized Kahler potential

Marco Gualtieri
Recent advances have made it possible to finally understand the physicists' concept of generalized Kahler potential in precise mathematical terms. I will describe how this can be done once we understand the notion of symplectic Morita equivalence between Poisson manifolds. Another benefit of the study is a new formalism for thinking of generalized Kahler geometry as a whole.

Metrics on the collection of dynamic shapes

Facundo Memoli
When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of groups of animals. In a similar vein, when attempting to classify motion capture data according to action one is confronted with having to match/compare shapes that evolve with time. Motivated by these applications, we study the question of suitably metrizing the collection of all dynamic metric spaces (DMSs). We construct...

Quantization on a Riemannian manifold with application to air traffic control

Alice Le Brigant

Robust shape matching with optimal transport

Jean Feydy

Generalized H(div) geodesics and solutions of the Camassa-Holm equation

Andrea Natale

Gromov-Monge Quasimetrics and Distance Distributions

Tom Needham
In applications in computer graphics and computational anatomy, one seeks a measure-preserving map from one shape to another which preserves geometry as much as possible. Inspired by this, we consider a notion of distance between arbitrary compact metric measure spaces by blending the Monge formulation of optimal transport with the Gromov-Hausdorff construction. We show that the resulting distance is an extended quasi-metric on the space of compact mm-spaces. This distance has convenient lower bounds defined...

Solar models for Euler-Arnold equations

Stephen Preston
Many one-dimensional Euler-Arnold equations can be recast in the form of a central-force problem $\Gamma_{tt}(t,x) = -F(t,x) \Gamma(t,x)$, where $\Gamma$ is a vector in $\mathbb{R}^2$ and $F$ is a nonlocal function possibly depending on $\Gamma$ and $\Gamma_t$. Angular momentum of this system is precisely the conserved momentum for the Euler-Arnold equation. In particular this picture works for the Camassa-Holm equation, the Hunter-Saxton equation, and the Okamoto-Sakajo-Wunsch family of equations. In the solar model, breakdown comes...

Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics

Philipp Harms
We show that the functional calculus, which maps operators A to functionals f(A), is holomorphic for a certain class of operators A and holomorphic functions f. Using this result we are able to prove that fractional Laplacians depend real analytically on the underlying Riemannian metric in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics. (Joint work with Martins...

Semi-discrete unbalanced optimal transport and quantization

Bernhard Schmitzer
"Semi-discrete optimal transport between a discrete source and a continuous target has intriguing geometric properties and applications in modelling and numerical methods. Unbalanced transport, which allows the comparison of measures with unequal mass, has recently been studied in great detail by various authors. In this talk we consider the combination of both concepts. The tessellation structure of semi-discrete transport survives and there is an interplay between the length scales of the discrete source and unbalanced...

Normal coordinates and equivolumic layers estimation in the cortex (tentative)

Laurent Younes
TBA

Wasserstein for learning image regularisers

Carola-Bibiane Schönlieb
In this talk we will discuss the use of a Wasserstein loss function for learning regularisers in an adversarial manner. This talk is based on joint work with Sebastian Lunz and Ozan à ktem, see https://arxiv.org/abs/1805.11572

Analyze shape variability via deformations

Barbara Gris
I will present how shape registration via constrained deformations can help understanding the variability within a population of shapes.

Convnets, a different view of approximating diffeomorphisms in medical image registration

Dongyang Kuang
As with the heat of artificial intelligence, there are more and more researches starting to investigate the possible geometric transformations using data-driven methods such as convolutional neural networks. In this talk, I will start by introducing some existing work that learn 2D linear transformations in an unsupervised way. This then will be followed by an overview of some recent works focusing on nonlinear transformations in 3D volumetric data. Finally, I will present results from the...

Beyond Arnoldâ s geodesic framework of an ideal hydrodynamics

Boris Khesin
We discuss a ramification of Arnoldâ s group-theoretic approach to ideal hydrodynamics as the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms. We show such problems of mathematical physics as the motion of vortex sheets or fluids with moving boundary, have Lie groupoid, rather than Lie group, symmetries, and describe the corresponding geometry and equations. (This is a joint work with Anton Izosimov.)

Geometric modelling of uncertainties

Alexis Arnaudon
In mechanics, and in particular in shape analysis, taking into account the underlying geometric properties of a problem to model it is often crucial to understand and solve it. This approach has mostly been applied for isolated systems, or for systems interacting with a well-defined, deterministic environment. In this talk, I want to discuss how to go beyond this deterministic description of isolated systems to include random interactions with an environment, while retaining as much...

A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces

Marc Arnaudon
"Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. I will present a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula will allow to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. This result extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein in the context of discrete...

Semi-invariant metrics on diffeos

Klas Modin
We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy---a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity---under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method...

The $L^2$ exponential map in 2D and 3D hydrodynamics

Gerard Misiolek
In the 1960's V. Arnold showed how solutions of the incompressible Euler equations can be viewed as geodesics on the group of diffeomorphisms of the fluid domain equipped with a metric given by fluid's kinetic energy. The study of the exponential map of this metric is of particular interest and I will describe recent results concerning its properties as well as some necessary background.

Some ideas and results about gradient flows and large deviations

Christian Léonard
In several situations, the empirical measure of a large number of random particles evolving in a heat bath is an approximation of the solution of a dissipative PDE. The evaluation of the probabilities of large deviations of this empirical measure suggests a way of defining a natural ``large deviation cost'' for these fluctuations, very much in the spirit of optimal transport. Some standard Wasserstein gradient flow evolutions are revisited in this perspective, both in terms...

On some relations between Optimal Transport and Stochastic Geometric Mechanics

Ana Cruzeiro
We formulate the so-called Schrodinger problem in Optimal Transport on lie group and derive the corresponding Euler-Poincaré equations.

Parallel Simulation of Concentrated Vesicle Suspensions in 3D

Dhairya Malhotra
We will discuss a parallel boundary integral method for simulating highly concentrated vesicle suspensions in a Stokesian fluid. The simulation of high volume fraction vesicle suspensions which are representative of real biological systems (such as blood with 35% - 50% volume fraction for RBCs) presents several challenges. It requires computing accurate vesicle-vesicle interactions at length scales where standard quadratures are too expensive. The inter-vesicle separation can become arbitrarily small leading to vesicle collisions. Numerical errors...

A convergent evolving finite element algorithm for mean curvature flow of closed surfaces

Balázs Kovács
A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in Dziuk's method, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations for the normal vector and mean curvature and uses these evolving geometric...

On projection methods for large-scale Riccati equations

Valeria Simoncini
In the numerical solution of the algebraic Riccati equation $A^* X + XA â XBB^â X + C^â C = 0$, where $A$ is large, sparse and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton-Kleinman iteration. A robust implementation of these methods opens to new questions on the use of dissipativity properties of the given matrix $A$. In this talk we briefly...

Uniformly accurate methods for highly-oscillatory kinetic equations

Nicolas Crouseilles
In this talk, we consider the numerical solution of the highly-oscillatory Vlasov equations. Designed in the spirit of recent uniformly accurate methods, the scheme remains insensitive to the stiffness of the problem in terms of both accuracy and computational cost. The method is based on a careful ad-hoc reformulation of the equations. Some numerical results will be given to illustrate the behavior of the method.

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