CRM CAMP in Nonlinear Analysis

Preuves mathématiques assistées par ordinateur en analyse non linéaire

CRM CAMP in Nonlinear Analysis

Le principal objectif du projet CRM CAMP est de rassembler la communauté mondiale des chercheurs dans le domaine des méthodes de preuve assistées par ordinateur, en particulier ceux qui travaillent dans les domaines de la théorie des systèmes dynamiques et de l'analyse non linéaire. Cette communauté a connu une croissance spectaculaire au cours des trois dernières décennies, et a développé des méthodes pour résoudre un certain nombre de problèmes importants non résolus en mathématiques. Cependant, les chercheurs participants sont dispersés dans le monde entier et il existe un besoin croissant d'un forum régulier de discussion et de diffusion des résultats. Cela est particulièrement important en cette période d'interruption sans précédent des voyages.

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Activités scientifiques

6 octobre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

Séminaire présenté par Akitoshi Takayasu (University of Tsukuba, Japan)

In this talk, we introduce a numerical method for rigorously finding the monodromy matrix of hypergeometric differential equations. From a base point defined by fundamental solutions, we analytically continue the solution on a contour around a singular point of the differential equation using a rigorous integrator. Depending on the contour we obtain the monodromy representation of fundamental solutions, which represents the fundamental group of the equation. As an application of this method, we consider a Picard-Fuchs type hypergeometric differential equation arising from a polarized K3 surface. The monodromy matrix shows a deformation of homologically independent 2-cycles for the surface along the contour, which is regarded as a change of characterization for the K3 surface. This is joint work with Naoya Inoue (University of Tsukuba) and Toshimasa Ishige (Chiba University).

29 septembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Séminaire présenté par Piotr Nowak (Polish Academy of Sciences, Poland)

Property (T) was introduced in 1967 by Kazhdan and is an important rigidity property of groups. The most elementary way to define it is through a fixed point property: a group G has property (T) if every action of G by affine isometries on a Hilbert space has a fixed point. Property (T) has numerous applications in the form of rigidity of actions and operator algebras associated to the group, constructions of expander graphs or constructions of counterexamples to Baum-Connes-type conjectures. 

In this talk I will give a brief introduction to property (T) and explain the necessary group-theoretic background in order to present a computer-assisted approach to proving property (T) by showing that the Laplacian on the group has a spectral gap. This approach allowed us show that Aut(F_n), the group of automorphisms of the free group F_n on n generators, has property (T) when n is at least 5: the case n=5 is joint work with Marek Kaluba and Narutaka Ozawa, and the case of n at least 6 is joint work with Kaluba and Dawid Kielak. Important aspects of our methods include passing from a computational result to a rigorous proof and later obtaining the result for an infinite family of groups using a single computation. I will present an overview of these arguments.

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Série de séminaires hebdomadaires : tous les mardis de l'été à 10:00 (heure de Montréal/Miami).

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Séminaires à venir
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6 octobre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Computer-assisted proofs for finding the monodromy of hypergeometric differential equations

Séminaire présenté par Akitoshi Takayasu (University of Tsukuba, Japan)

In this talk, we introduce a numerical method for rigorously finding the monodromy matrix of hypergeometric differential equations. From a base point defined by fundamental solutions, we analytically continue the solution on a contour around a singular point of the differential equation using a rigorous integrator. Depending on the contour we obtain the monodromy representation of fundamental solutions, which represents the fundamental group of the equation. As an application of this method, we consider a Picard-Fuchs type hypergeometric differential equation arising from a polarized K3 surface. The monodromy matrix shows a deformation of homologically independent 2-cycles for the surface along the contour, which is regarded as a change of characterization for the K3 surface. This is joint work with Naoya Inoue (University of Tsukuba) and Toshimasa Ishige (Chiba University).

13 octobre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Rigorous computation of periodic solutions and Floquet multipliers in delay differential equations with time-forced discontinuities

Séminaire présenté par Kevin Church (McGill University, Canada)

I will present some recent work on rigorous computation of periodic solutions for delay differential equations with impulse effects. At fixed moments in time, the state of such a system is reset and solutions become discontinuous. Once a periodic solution of such a system has been computed, its Floquet spectrum can be rigorously computed by discretization of the monodromy operator (period map) and some technical error estimates. As an application, we compute a branch of periodic solutions in the pulse-harvested Hutchinson equation and examine its stability.

20 octobre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Details of the seminar of A. Mahboubi to come

Séminaire présenté par Assia Mahboubi (Inria, France & VU Amsterdam, Netherlands)

27 octobre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Details of the seminar of E. Sander to come

Séminaire présenté par Evelyn Sander (George Mason University, USA)

3 novembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Details of the seminar of G. Froyland to come

Séminaire présenté par Gary Froyland (UNSW Sydney, Australia)

10 novembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Details of the seminar of B. Barker to come

Séminaire présenté par Blake Barker (Brigham Young University, USA)

17 novembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Details of the seminar of F. Bartha to come

Séminaire présenté par Ferenc Bartha (University of Szeged, Hungary)

24 novembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation

Séminaire présenté par Gianni Arioli (Politecnico di Milano, Italy)

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. The uniqueness of stationary solutions is studied in dependence of the kinematic viscosity. For some particular forcing, it is shown that uniqueness persists on some continuous branch of stationary solutions, when the viscosity becomes arbitrarily small. On the other hand, for a different forcing, a branch of symmetric solutions is shown to bifurcate, giving rise to a secondary branch of nonsymmetric stationary solutions. Furthermore, as the kinematic viscosity is varied, the branch of symmetric stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

1 décembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Future Directions Series : Arnd Scheel

Séminaire présenté par Arnd Scheel (University of Minnesota, USA)

8 décembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

CRM-CAMP COLLOQUIUM: Jean-Pierre Eckmann

Séminaire présenté par Jean-Pierre Eckmann (University of Geneva, Switzerland)

Heure de début Titre Conférencier
2020-10-06 10:00 Computer-assisted proofs for finding the monodromy of hypergeometric differential equations Akitoshi Takayasu (University of Tsukuba, Japan)
2020-10-13 10:00 Rigorous computation of periodic solutions and Floquet multipliers in delay differential equations with time-forced discontinuities Kevin Church (McGill University, Canada)
2020-10-20 10:00 Details of the seminar of A. Mahboubi to come Assia Mahboubi (Inria, France & VU Amsterdam, Netherlands)
2020-10-27 10:00 Details of the seminar of E. Sander to come Evelyn Sander (George Mason University, USA)
2020-11-03 10:00 Details of the seminar of G. Froyland to come Gary Froyland (UNSW Sydney, Australia)
2020-11-10 10:00 Details of the seminar of B. Barker to come Blake Barker (Brigham Young University, USA)
2020-11-17 10:00 Details of the seminar of F. Bartha to come Ferenc Bartha (University of Szeged, Hungary)
2020-11-24 10:00 Symmetry breaking and Hopf bifurcations for the planar Navier-Stokes equation Gianni Arioli (Politecnico di Milano, Italy)
2020-12-01 10:00 Future Directions Series : Arnd Scheel Arnd Scheel (University of Minnesota, USA)
2020-12-08 10:00 CRM-CAMP COLLOQUIUM: Jean-Pierre Eckmann Jean-Pierre Eckmann (University of Geneva, Switzerland)

Séminaires passés
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29 septembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Séminaire présenté par Piotr Nowak (Polish Academy of Sciences, Poland)

Property (T) was introduced in 1967 by Kazhdan and is an important rigidity property of groups. The most elementary way to define it is through a fixed point property: a group G has property (T) if every action of G by affine isometries on a Hilbert space has a fixed point. Property (T) has numerous applications in the form of rigidity of actions and operator algebras associated to the group, constructions of expander graphs or constructions of counterexamples to Baum-Connes-type conjectures. 

In this talk I will give a brief introduction to property (T) and explain the necessary group-theoretic background in order to present a computer-assisted approach to proving property (T) by showing that the Laplacian on the group has a spectral gap. This approach allowed us show that Aut(F_n), the group of automorphisms of the free group F_n on n generators, has property (T) when n is at least 5: the case n=5 is joint work with Marek Kaluba and Narutaka Ozawa, and the case of n at least 6 is joint work with Kaluba and Dawid Kielak. Important aspects of our methods include passing from a computational result to a rigorous proof and later obtaining the result for an infinite family of groups using a single computation. I will present an overview of these arguments.

22 septembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Computing and validating collisions, ejections, and homoclinics for the three body problem

Séminaire présenté par Shane Kepley (Rutgers University, USA)

Understanding connecting and collision/ejection orbits is central to the study of transport in Celestial Mechanics. The atlas algorithm combines the parameterization method with rigorous numerical techniques for solving initial value problems in order to find and validate connecting orbits. However, difficulties arise when parameterizing orbits passing near a singularity such as “near miss” homoclinics or ejection/collision orbits. In this talk we present a method of overcoming this obstacle based on rigorous Levi-Civita regularization which desingularizes the vector field near the primaries. This regularization is performed dynamically allowing invariant manifolds to be parameterized globally, even near singularities.

15 septembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

Séminaire présenté par Elena Queirolo (Rutgers University, USA)

We prove the existence of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE. For this, we rewrite the Kuramoto–Sivashinsky equation into a desingularized formulation near the Hopf point via a blow-up approach and we apply the radii polynomial approach to validate a solution branch of periodic solutions. Then this solution branch includes the Hopf bifurcation point.

8 septembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon

Séminaire présenté par Isaia Nisoli (Universidade Federal do Rio de Janeiro, Brazil)

In this talk I will present a Computer Aided Proof of Noise Induced Order (NIO) in a model associated with the Belousov-Zhabotinsky reaction: when studying the random dynamical system with additive noise associated to the BZ map, as the noise amplitude increases the Lyapunov exponent of the model transitions from positive to negative. The proof is obtained through rigorous approximation of the stationary measure using Ulam method.

I will also show a sufficient condition for the existence of NIO in a wide family of one-dimensional examples.

[1] S. Galatolo, M. Monge, I. Nisoli "Existence of Noise Induced Order: a computer aided proof", Nonlinearity 33(9)
[2] I. Nisoli "Sufficient Conditions for Noise Induced Order in 1-dimensional systems", arXiv:2003.08422

1 septembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Computer assisted proofs of Arnold Diffusion

Séminaire présenté par Maciej Capiński (AGH University of Science and Technology, Poland)

We will present three methods that can be used for computer assisted proofs of Arnold diffusion in Hamiltonian systems. The first is the classical Melnikov method; the second is based a shadowing lemma in the setting of the scattering map theory; the last is based on topological shadowing using correctly aligned windows and cones. We will also discuss an application in the setting of the Planar Elliptic Restricted Three Body Problem.

25 août 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

Séminaire présenté par Maxime Breden (École Polytechnique, France), Maximilian Engel (Freie Universität Berlin, Germany)

In this talk, we discuss a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains, we reformulate the problem into the rigorous computation of eigenvectors of some elliptic PDEs, namely the Kolmogorov/Fokker-Planck equations describing distributions of the underlying stochastic process, and are thus able to prove that  the first Lyapunov exponent is positive for certain parameter regimes.

18 août 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Rigorously validated estimation of statistical properties of expanding maps

Séminaire présenté par Caroline Wormell (University of Sydney, Australia)

Full-branch uniformly expanding maps and their long-time statistical quantities serve as common models for chaotic dynamics, as well as having applications to number theory. I will present an efficient method to compute important statistical quantities such as physical invariant measures, which can obtain rigorously validated bounds. To accomplish this, a Chebyshev Galerkin discretisation of transfer operators of these maps is constructed; the spectral data at the eigenvalue 1 is then approximated from this discretisation. Using this method we obtain validated estimates of Lyapunov exponents and diffusion coefficients that are accurate to over 100 decimal places. These methods may also fruitfully be extended to non-uniformly expanding maps of Pomeau-Manneville type, which have largely been altogether resistant to numerical study.

11 août 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Torus knot choreographies in the N-body problem

Séminaire présenté par Renato Calleja (Universidad Nacional Autonoma de Mexico, Mexico)

N-body choreographies are periodic solutions to the N-body equations in which equal masses chase each other around a fixed closed curve. In this talk I will present a systematic approach for proving the existence of spatial choreographies in the gravitational body problem with the help of the digital computer. These arise from the polygonal system of bodies in a rotating frame of reference. In rotating coordinates, after exploiting the symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies.

4 août 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE

Séminaire présenté par Daniel Wilczak (Jagiellonian University, Poland)

We give a computer-assisted proof of the existence of symbolic dynamics for a certain Poincaré map in the one-dimensional Kuramoto-Sivashinsky PDE. In particular, we show the existence of infinitely many (countably) periodic orbits (POs) of arbitrary large principal periods. We provide also a study of the stability type of some POs and show the existence of a countable infinity of geometrically different homoclinic orbits to a periodic solution. The proof utilizes pure topological results (variant of the method of covering relations on compact absolute neighbourhood retracts) with rigorous integration of the PDE and the associated variational equation. This talk is based on the recent results [1,2].

[1] D. Wilczak and P. Zgliczyński. A geometric method for infinite-dimensional chaos: symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line, Journal of Differential Equations, Vol. 269 No. 10 (2020), 8509-8548.
[2] D. Wilczak and P. Zgliczyński. A rigorous C1-algorithm for integration of dissipative PDEs based on automatic differentiation and the Taylor method, in preparation.

28 juillet 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

A modification of Schiffer's conjecture, and a proof via finite elements

Séminaire présenté par Nilima Nigam (Simon Fraser University, Canada)

Approximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues for the Dirichlet Laplacian in certain domains. If these are to be used as part of a proof, care must be taken to ensure each step of the computation is validated and verifiable. In this talk we present a long-standing conjecture in spectral geometry, and its resolution using validated finite element computations.  Schiffer’s conjecture states that if a bounded domain Ω in R^n has any nontrivial Neumann eigenfunction which is a constant on the boundary, then Ω must be a ball. This conjecture is open. A modification of Schiffer’s conjecture is: for regular polygons of at least 5 sides, we can demonstrate the existence of a Neumann eigenfunction which does not change sign on the boundary. In this talk, we provide a recent proof using finite element calculations for the regular pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains of the triangle with mixed Dirichlet and Neumann boundary conditions. We use a learning algorithm to find and optimize this sequence of subdomains, and use non-conforming linear FEM to compute validated lower bounds for the lowest eigenvalue in each of these domains. The linear algebra is performed within interval arithmetic. This talk is based on the following paper, which is a joint work with Bartlomiej Siudeja and Ben Green at University of Oregon.

21 juillet 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains

Séminaire présenté par Xuefeng Liu (Niigata University, Japan)

We consider the solution verification for the stationary Navier-Stokes equation over a bounded non-convex 3D domain Ω. In 1999, M.T. Nakao, et al., reported a solution existence verification example for the 2D square domain.  However, it has been a difficult problem to deal with general 2D domains and 3D domains, due to the bottleneck problem in the  a priori error estimation for the linearized NS equation. Recently, by extending the hypercircle method (Prage-Synge's theorem) to deal with the divergence-free condition in the Stokes equation, the explicit error estimation is constructed successfully based on a conforming finite element approach [arXiv:2006.02952]. Further,  we succeeded in the solution existence verification for the stationary NS equation in several nonconvex 3D domains.  In this talk, I will show the latest progress on this topic, including the rigorous estimation of the eigenvalue of Stokes operator in 3D domains.

14 juillet 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Uniqueness of Whitham's highest cusped wave

Séminaire présenté par Javier Gómez-Serrano (Brown University, USA & University of Barcelona, Spain)

Whitham’s equation of shallow water waves is a non-homogeneous non-local dispersive equation. As in the case of the Stokes wave for the Euler equation, non-smooth traveling waves with greatest height between crest and trough have been shown to exist. In this talk I will discuss uniqueness of solutions to the Whitham equation and show that there exists a unique, even and periodic traveling wave of greatest height, that moreover has a convex profile between consecutive stagnation points, at which there is a cusp. Joint work with Alberto Enciso and Bruno Vergara.

7 juillet 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains

Séminaire présenté par Michael Plum (Karlsruhe Institute of Technology, Germany)

Many boundary value problems for semilinear elliptic partial differential equations allow very stable numerical computations of approximate solutions, but are still lacking analytical existence proofs. In this lecture, we propose a method which exploits the knowledge of a "good" numerical approximate solution, in order to provide a rigorous proof of existence of an exact solution close to the approximate one. This goal is achieved by a fixed-point argument which takes all numerical errors into account, and thus gives a mathematical proof which is not "worse" than any purely analytical one. A crucial part of the proof consists of the computation of eigenvalue bounds for the linearization of the given problem at the approximate solution. The method is used to prove existence and multiplicity statements for some specific examples, including cases where purely analytical methods had not been successful.

30 juin 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance

Séminaire présenté par Jonathan Jaquette (Boston University, USA)

In this talk we study the nonlinear Schrödinger equation   posed on the 1-torus. Based on their numerics, Cho, Okamoto, & Shōji conjectured in their 2016 paper that: (C1) any singularity in the complex plane of time arising from inhomogeneous initial data is a branch singularity, and (C2) real initial data will exist globally in real time. If true, Conjecture 1 would suggest a strong incompatibility with the Painlevé property, a property closely associated with integrable systems. While Masuda proved (C1) in 1983 for close-to-constant initial data, a generalization to other initial data is not known. Using computer assisted proofs we establish a branch singularity in the complex plane of time for specific, large initial data which is not close-to-constant.

Concerning (C2), we demonstrate an open set of initial data which is homoclinic to the 0-homogeneous-equilibrium, proving (C2) for close-to-constant initial data. This proof is then extended to a broader class of nonlinear Schrödinger equation without gauge invariance, and then used to prove the non-existence of any real-analytic conserved quantities. Indeed, while numerical evidence suggests that most initial data is homoclinic to the 0-equilibrium, there is more than meets the eye. Using computer assisted proofs, we establish an infinite family of unstable nonhomogeneous equilibria, as well as heteroclinic orbits traveling between these nonhomogeneous equilibria and the 0-equilibrium.

23 juin 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Séminaire présenté par Jan Bouwe van den Berg (VU Amsterdam, Netherlands)

In this talk we discuss a mathematically rigorous computational method to compare local minimizers of the Ohta-Kawasaki free energy, describing diblock copolymer melts. This energy incorporates a nonlocal term to take into account the bond between the monomers.

Working within an arbitrary space group symmetry, we explore the phase space, computing candidates both with and without experimentally observed symmetries. We validate the phase diagram, identifying regions of parameter space where different spatially periodic structures have the lowest energy. These patterns may be lamellar layers, hexagonally packed cylinders, body-centered or close-packed spheres, as well as double gyroids and 'O70' arrangements. Each computation is validated by a mathematical theorem, where we bound the truncation errors and apply a fixed point argument to establish a computer-assisted proof. The method can be applied more generally to symmetric space-time periodic solution of many partial differential equations.

Heure de début Titre Conférencier
2020-09-29 10:00 A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups Piotr Nowak (Polish Academy of Sciences, Poland)
2020-09-22 10:00 Computing and validating collisions, ejections, and homoclinics for the three body problem Shane Kepley (Rutgers University, USA)
2020-09-15 10:00 Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE Elena Queirolo (Rutgers University, USA)
2020-09-08 10:00 A proof of Noise Induced Order in the BZ map, and some remarks on the phenomenon Isaia Nisoli (Universidade Federal do Rio de Janeiro, Brazil)
2020-09-01 10:00 Computer assisted proofs of Arnold Diffusion Maciej Capiński (AGH University of Science and Technology, Poland)
2020-08-25 10:00 Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems Maxime Breden (École Polytechnique, France), Maximilian Engel (Freie Universität Berlin, Germany)
2020-08-18 10:00 Rigorously validated estimation of statistical properties of expanding maps Caroline Wormell (University of Sydney, Australia)
2020-08-11 10:00 Torus knot choreographies in the N-body problem Renato Calleja (Universidad Nacional Autonoma de Mexico, Mexico)
2020-08-04 10:00 Rigorous numerical investigation of chaos and stability of periodic orbits in the Kuramoto-Sivashinsky PDE Daniel Wilczak (Jagiellonian University, Poland)
2020-07-28 10:00 A modification of Schiffer's conjecture, and a proof via finite elements Nilima Nigam (Simon Fraser University, Canada)
2020-07-21 10:00 Solution verification for the stationary Navier-Stokes equation over bounded non-convex 3D domains Xuefeng Liu (Niigata University, Japan)
2020-07-14 10:00 Uniqueness of Whitham's highest cusped wave Javier Gómez-Serrano (Brown University, USA & University of Barcelona, Spain)
2020-07-07 10:00 Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains Michael Plum (Karlsruhe Institute of Technology, Germany)
2020-06-30 10:00 An overabundance of breathers in a nonlinear Schrödinger equation without gauge invariance Jonathan Jaquette (Boston University, USA)
2020-06-23 10:00 Stable periodic patterns in 3D for the Ohta-Kawasaki problem Jan Bouwe van den Berg (VU Amsterdam, Netherlands)
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Vidéos

15 septembre 2020

Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

Elena Queirolo

1 septembre 2020

Computer assisted proofs of Arnold Diffusion

Maciej Capiński

11 août 2020

Torus knot choreographies in the N-body problem

Renato Calleja

14 juillet 2020

Uniqueness of Whitham's highest cusped wave

Javier Gómez-Serrano

23 juin 2020

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg

Série Open Problems

This is a series of talks focusing on either open problems concerning techniques of computer-assisted proof, or more broadly open problem in mathematics where the speaker believes there could be a computer-assisted solution. Talks range from 5 minutes to an hour, and can be proposed at any level. When an open problem is solved, or when substantial progress is made, we provide citation and links to the relevant work.

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Autres activités scientifiques

Ici, seront regroupées les différentes activités scientifiques reliées au groupe de recherche. D'abord, les activités à venir, puis les activités passées.

Activitiés scientifiques à venir

1 décembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

Future Directions Series : Arnd Scheel

Séminaire présenté par Arnd Scheel (University of Minnesota, USA)

8 décembre 2020 de 10 h 00 à 11 h 00 (heure de Montréal/Miami) Réunion Zoom

CRM-CAMP COLLOQUIUM: Jean-Pierre Eckmann

Séminaire présenté par Jean-Pierre Eckmann (University of Geneva, Switzerland)

À propos

La série de séminaires du CAMP CRM explore l'interaction entre le calcul scientifique et l'analyse mathématique rigoureuse, en mettant l'accent sur la recherche dans des domaines tels que la théorie des systèmes dynamiques et l'analyse non linéaire. Ce domaine s'est rapidement développé au cours des dernières décennies et, les chercheurs étant répartis dans le monde entier, il est de plus en plus nécessaire de mettre en place un forum régulier pour partager les résultats, poser des questions intéressantes et discuter de nouvelles orientations. Ce projet est envisagé comme une sorte de centre communautaire en ligne pour les rassemblements hebdomadaires, ainsi que comme un dépôt de matériel pédagogique. En plus de la série de conférences hebdomadaires, le programme sert également de mécanisme pour organiser des ateliers, des tutorats et d'autres activités scientifiques. Nous espérons qu'en augmentant la visibilité de cette recherche, le projet stimulera les collaborations entre les groupes existants et entre notre communauté et les mathématiciens travaillant dans d'autres domaines.

Biographie

Jean-Philippe Lessard is an associate professor at McGill University since 2017. He obtained his Ph.D. from Georgia Tech in 2007 under the supervision of Konstantin Mischaikow. He spent some time as a postdoctoral researcher at Rutgers University, at VU University Amsterdam, was awarded a fellowship from the IAS in Princeton and was a group leader at the Basque Center for Applied Mathematics. He then became a professor at Laval University, where he stayed for six years. In 2016, he was awarded the CAIMS/PIMS Early Career Award in Applied Mathematics and is currently CRM’s deputy director of scientific programs. In his research, he combines numerical analysis, topology and functional analysis to study finite and infinite dimensional dynamical systems.

Biographie

Jason D. Mireles James received his Ph.D. from the University of Texas at Austin in 2009, where he worked with Rafael de la Llave. He moved to Rutgers University where he was first a postdoc from 2010 to 2011, and then a Hill Assistant Professor in the Mathematics Department from 2011-2014. During this time, he worked closely with the group of Konstantin Mischaikow. In 2014 he joined the Department of Mathematics at Florida Atlantic University, where he currently holds the rank of associate professor. His research focuses on problems in nonlinear analysis, drawing on tools from computational mathematics, approximation theory, and functional analysis.

 

Biographie

Jan Bouwe van den Berg is a full professor at Vrije Universiteit Amsterdam since 2007. He obtained his Ph.D. from Leiden University in 2000 under the supervision of Bert Peletier. He spent two years as a postdoc in Nottingham and has held visiting positions at Simons Fraser University and at McGill. He was awarded an NWO-Vici grant in 2012 and he was a CRM-Simons visiting professor in 2019. Jan Bouwe’s research revolves around dynamical systems and nonlinear partial differential equations, where he use techniques ranging from topological and variational analysis to (rigorous) computational methods to study the dynamics of patterns.

Vidéos

23 juin 2020

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg

Vidéos Youtube

15 septembre 2020

Validating Hopf bifurcations in the Kuramoto-Sivashinsky PDE

Elena Queirolo

1 septembre 2020

Computer assisted proofs of Arnold Diffusion

Maciej Capiński

11 août 2020

Torus knot choreographies in the N-body problem

Renato Calleja

14 juillet 2020

Uniqueness of Whitham's highest cusped wave

Javier Gómez-Serrano

23 juin 2020

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Jan Bouwe van den Berg

Livres

Articles d'enquête

SeMA, 76, pages 459–484, 2019

Computer-assisted proofs in PDE: a survey

Javier Gómez-Serrano

Notices of the American Mathematical Society, Volume 62 (9), pages 1057-1061, 2015

Rigorous Numerics in Dynamics

Jean-Philippe Lessard, Jan Bouwe van den Berg

Acta Numerica, Volume 19, pages 287-449, 2010

Verification methods: rigorous results using floating-point arithmetic

Siegfried M. Rump

SIAM Review, Volume 38 (4), pages 565-604, 1996

Computer-assisted proofs in analysis and programming in logic: a case study

Hans Koch, Alain Schenkel, Peter Wittwer

Écoles

1 août 2018 https://mym.iimas.unam.mx/renato/curso.html

Computer-Assisted Proofs in Nonlinear Dynamics

Jason D. Mireles James, Jean-Philippe Lessard

The main question addressed in this course is: suppose we have computed a good numerical approximation of a solution of nonlinear equation -- can we establish the existence of a true solution nearby? Combining tools from functional analysis, complex analysis, numerical analysis, and interval computing, we see that for many of the problems mentioned above the answer is yes. A broad and example driven overview of the field of validated numerics is given.