UvA-VU Dynamic Analysis Seminar talks from previous years

2017

 

22 November: Jeremie Joudioux (Nijmegen), 16:00-17:00, WN-F612

Title:The vector-field method for geometric transport equations with applications to stability problems in General Relativity 

Abstract: The vector-field method, developed by Klainerman in 80's, was an important tool to understand the global existence of solutions to the Cauchy problems for nonlinear wave equations. This tool was a key step in the development of PDE tools to handle the stability problem of Minkowski space-time, solution to the Einstein equations in vacuum. In a work in collaboration with D. Fajman (Vienna) and J. Smulevici (Orsay), this method was extended to the relativistic collisionless Boltzmann equation (Vlasov equation). In this talk, I review the vector-field method for the wave equation, explain its extension to geometric transport equations, and, finally after introducing the Minkowski stability problem, argue how this vector-field method is used to prove the stability of Minkowski space-time as a solution to Einstein-Vlasov system.


25 October @UvA: Daan de Groot (VU), 16:00-17:00, SP-F1.15

Title: Searching for the fundamental constraints in microbial growth: minimally complex solutions and open questions

Abstract: Unicellular organisms could in principle be seen as simple bags of enzymes making more enzymes. Convincing experimental evidence however shows that these ‘simple bags’ are capable of working in the fastest way possible in an impressive range of environments, thereby apparently solving an optimisation problem of great complexity. During my PhD I investigate what simple (mathematical) rules could govern microbes such that this optimisation problem is solved.
We started this investigation by trying to identify the optimal steady state for a cell’s metabolism in a static environment, i.e. how should substrates be used to create energy and cellular building blocks. The solution space of this problem is high-dimensional (there exist billions of metabolic pathways from substrates to cell synthesis), but we mathematically show that the actual solution will be composed of only a few pathways. This number is bounded by the number of fundamental constraints that a microbe runs into (e.g. limits on solvent capacity of the cytosol, membrane area, or the synthesising capacity of ribosomes). The nature of these constraints might be different for each organism and even for each environment, which brings us to the question: “Can we propose a generally applicable experimental procedure that identifies these constraints?” In this talk I will show you part of the answer and state some open questions.


11 October @VU: Andres Pedroza (Colima), 16:00-17:00, WN-P631

Title: Lagrangian submanifolds in the one-point blow-up of CP^2

Abstract:We will show how a Lagrangian submanifold in CP^2 that is Hamiltonian isotopic to RP^2, lifts to a Lagrangian submanifold in the symplectic one-point blow up of CP^2 such that is no longer Hamiltonian isotopic to the lift of RP^2. We show this by computing the Lagrangian Floer homology of the pair of Lagrangian submanifolds in the symplectic blow up in terms of the Lagrangian Floer homology of the pair in CP^2.

 

27 September @VU: Emmanuel Opshtein (Strasbourg), 16:00-17:00, WN-P631

Title: C^0 rigidity of Lagrangian submanifolds

Abstract:Classical symplectic geometry deals with those applications that preserve a differential structures given by a certain 2-form. Eliashberg-Gromov discovered however that a diffeomorphism that is a C^0-limit of symplectic diffeomorphisms is itself symplectic. As a result, the C^0-closure of the symplectic group in the Homeomorphism group of a manifold gives an interesting object. We are led to a natural question: which objects are of interest in this "C^0-symplectic geometry" ? In this talk, I will discuss the relevance of Lagrangian submanifolds (which are the most famous objects in classical symplectic geometry) in this C^0-symplectic geometry. We will see that these submanifolds enjoy very strong robustness with respect to these C^0-limits.


13 September @VU: Thomas Rot (Köln), 16:00-17:00, WN-P631

Title: The classification of proper Fredholm maps up to proper homotopy

Abstract:In the fifties Pontryagin showed that homotopy classes of maps into spheres are in one to one correspondence with framed cobordism classes of the domain. This correspondence enabled him to compute the homotopy groups $\pi_{n+k}(S^n)​$ for small values of k. In this talk I will discuss extensions of these ideas to infinite dimensions. This is joint work with Alberto Abbondandolo.


16 August @VU: Tom van den Bosch (VU), 16:00-17:00, WN-M639

Title: Stochastics of growing and persisting cell populations

Abstract: Some cellular populations contain subpopulations which exhibit a phenotypic switch to become dormant. These persister cells do not grow, but are able to survive outside stress to reduce the risk of total populations extinction. However, the rates at which cells switch their phenotype is a random variable, and as such the number of persisters shows stochasticity. Because of this, there is a nonzero probability that a population contains no persisters, thus being at risk. In this paper we aim to study this risk. We first derive a master equation for the persister cell model. Since fractions of persister cells are usually low, the persister model is very similar to the exponential growth process, of which we derive an exact probability distribution. Using this, we derive the probability generating function of a simplified model for persisters, which we show to be accurate for low fractions. We then fit distributions to data generated by the Gillespie algorithm to show the distribution of persister cells. Lastly, we solve the master equation for the first and second moments to derive expressions for the fraction of persister cells and the noise in the number of cells. With this fraction we derive an exact probability distribution for a model in which the fraction is constant.


21 June @VU: Tomas Dohnal (Dortmund), 16:00-17:00, Room S-623

Title: Rigorous Asymptotics of Moving Pulses for Nonlinear Wave Problems in Periodic Structures

Abstract: The possibility of moving, spatially localized pulses of constant or time periodic form in periodic media, e.g. in photonic crystals, is interesting from the mathematical as well as the applied point of view. An example is optical computing where such pulses could function as bit carriers. Pulses in the form of asymptotically small and wide wavepackets can be studied with the help of envelope approximations. Hereby the envelope satisfies an effective equation with constant coefficients. Rigorous results of such approximations in one spatial dimension on long time intervals for the periodic nonlinear Schrödinger equation will be presented but also the current work on the two dimensional analog will be briefly discussed. We concentrate on the asymptotic scaling which leads to the, so called, coupled mode equations (CMEs) of first order. CMEs have families of solitary waves parametrized by velocity, such that in the original model propagation of localized pulses is possible for a range of velocities at one fixed frequency. The justification proof relies on the Bloch transformation, Sobolev space estimates and the Gronwall inequality. Besides the idea of the proof we present also some numerical examples.

24 May @VU: Klaus Mohnke (HU Berlin), 16:00-17:00, Room S-623

Title: Counting holomorphic curves with jet conditions

Abstract: I will discuss  constraints on higher derivatives of (pseudo)holomorphic curves. The number of such curves seems to be elusive. I will explain why this is not surprising.  The advantage of higher order conditions over simple tangency conditions will be demonstrated on an application to Lagrangian embedding problems.

26 April @VU: Erik Steur (Eindhoven), 16:00-17:00, Room M-655

Title: Partially synchronous oscillations in networks of time-delay coupled systems

Abstract: Synchronization in networks of interacting systems (species, entities, ...) is profound in nature and finds many interesting applications in engineering. Examples include the simultaneous flashing of fireflies, the synchronized release of action potentials in networks of neurons in the brain, orbital locking in solar systems and coordinated motion in groups of robots. Often such networks show a form of incomplete synchronization that is characterized by the asymptotic match of the states of some, but not all of its systems. Necessary for this type of synchronization, which we call partial synchronization, is the existence of partial synchronization manifolds, which are linear invariant manifolds in the state-space of the network of systems. We present a number of conditions for the existence of partial synchronization manifolds for networks of systems that interact via time-delay coupling functions. Next we discuss local and global stability of partial synchronization. We support our findings with numerical simulations of networks of time-delay coupled Hindmarsh-Rose model neurons. This is joint work with Henk Nijmeijer, Sasha Pogromsky and Wim Michiels.


29 March @VU: Maria Westdickenberg (Aachen), 16:00-17:00, Room C-147

Title: Energy methods for existence and evolution

Abstract: For many PDE it is useful to view the phase space as a complex energy landscape. Solutions of static problems may be viewed as local minima or saddle points of the energy. For time-dependent PDE with a gradient flow structure, energy dissipation can be used to understand qualitative and quantitative properties of solutions. We give an overview of some well-known and newer results in this area, including the use of Gamma-limits to show existence of local minima and the use of energy and dissipation to quantify rates of coarsening, relaxation, and metastable evolution.


15 March @VU: Richard Siefring (Bochum), 15:30-16:30, Room M-655

Title: Slice orbits and a dynamical characterization of the 4-ball

Abstract: We give a characterization of symplectic manifolds with boundary which are symplectomorphic to star-shaped regions in (R^4, \omega_0) in terms of topological-dynamical properties of orbits on the boundary.  As a corollary we prove that certain transverse knots cannot appear as periodic orbits of the Reeb vector field for a dynamically convex contact form on tight S^3.  This is joint work with Umberto Hryniewicz and Pedro Salomao.


01 March @VU: Chiara Gallarati (Delft), 16:00-17:00, Room M-655

Title: Maximal L^p-regularity for parabolic  equations with measurable dependence on time.

Abstract:  In this talk I will introduce the concept of maximal L^p-regularity and explain a new approach to maximal L^p-regularity for parabolic PDEs with generator A(t) that depends on time in a measurable way. As an application I will obtain optimal L^p(L^q) estimates, for every p,q\in\(1,\infty), for systems of non-autonomous differential equations of order 2m. This is a joint work with Mark Veraar (TU Delft).


15 February @VU: Oliver Tse (Eindhoven), 16:00-17:00, Room M-655

Title: Equilibration in Wasserstein distance for damped Euler equations with interaction forces

Abstract: This talk describes the techniques used to provide convergence to (global) equilibrium in the 2-Wasserstein distance of partially damped Euler systems under the influence of external and interaction potential forces.


2016


23 November @VU, 15:30-16:15, Björn de Rijk (Stuttgart), Room WN-S655

Title: Stability of periodic pulse solutions in slowly nonlinear reaction-diffusion systems
 
Abstract: In the stability analysis of pattern solutions, the presence of a small parameter can reduce the complexity of the associated eigenvalue problem. This reduction manifests itself through the complex-analytic Evans function, which vanishes on the spectrum of the linearization about the pattern. For specific 'slowly linear' models it has been shown, via geometric arguments, that the Evans function factorizes in accordance with the scale separation. This leads to asymptotic control over the spectrum through simpler, lower-dimensional eigenvalue problems. Recently, the geometric factorization procedure has been generalized to homoclinic pulse solutions in slowly nonlinear reaction-di ffusion systems. In this talk we study periodic pulse solutions in the slowly nonlinear regime. At first sight this seems a straightforward extension of the homoclinic case. However, the geometric factorization method fails. In addition, due to translational invariance of the pulse profile, there is an entire curve of spectrum attached to the origin, whereas for homoclinic pulse solutions there is only a simple eigenvalue residing at the origin. In this talk we develop an alternative, analytic factorization method that does work for periodic structures in the slowly nonlinear regime. We derive explicit formulas for the factors of the Evans function, which yield asymptotic control over the spectrum. Moreover, we obtain a leading-order expression for the critical spectral curve attached to origin. Together these spectral approximation results lead to explicit (diffusive) stability criteria. 


23 November @VU, 16:15-17:15, Thomas Vogel (U München), Room WN-S655

Title: Non-loose unknots in S^3

Abstract: We outline the classification of non loose unknots in S^3 and discuss implications for the contact mapping class group for overtwisted contact structures in S^3.

26 October @UvA, 16:00-17:00, Roman Golovko (UL Brussels), Room F3.20 (KdVI)

Title:On the stable Arnold conjecture

Abstract: We discuss the relative and absolute versions of the stable Arnold conjecture.
In the relative setting, we show that the number of Reeb chords on a Legendrian submanifold, which admits an exact Lagrangian filling satisfying some technical conditions, is bounded from below by the stable Morse number of the filling. In the absolute setting, given a closed symplectically aspherical manifold, we show that the number of fixed points of a generic Hamiltonian diffeomorphism on it is bounded from below by the stable Morse number of this manifold.  
This is joint work with Georgios Dimitroglou Rizell.

28 September @UvA, 16:00-17:00, Berry Baker, (VU), Room F3.20 (KdVI)

Title:A Floer homology approach to travelling waves in reaction-diffusion equations

Abstract: TBA

14 September @VU, 15:30-16:15, Blaz Mramor (Freiburg), Room WN-S623

Title: Minimisers of the Allen-Cahn equation on hyperbolic graphs

Abstract: The Allen-Cahn equation is a second order semilinear elliptic PDE that arises in mathematical models describing phase transitions and is tightly connected to the theory of minimal hypersurfaces. The variational structure of this equation allows us to study energy-minimal phase transitions, which correspond to uniformly bounded non-constant globally minimal solutions. The set of such solutions depends heavily on the geometry of the underlying space. We shall focus on the case where the underlying space is a Gromov-hyperbolic graph. In this case there exists a minimal solution with any “nice enough” asymptotic behaviour prescribed by the two constant states. The set in the graph where the phase transition for such a solution takes place corresponds to a solution of an asymptotic Plateau problem.  

14 September @VU, 16:30-17:15, Olga Trichtchenko (UvA), Room WN-S623

Title: Comparison of stability of solutions to Hamiltonian water wave models

Abstract: The goal of this work is to compare and contrast the stability of solutions to Euler's equations for an inviscid, incompressible and irrotational fluid. We focus on two types of instabilities, a modulational (Benjamin-Feir) instability and high frequency instabilities. It is known that for gravity waves in deep water, the BF instability exists. In this work, we use the reformulation due to Ablowitz, Fokas and Musslimani and analyse for which parameters the modulational instability occurs both asymptotically and numerically under different conditions at the surface such as presence of surface tension or a thin sheet of ice. It is also known that high frequency instabilities exist for nonlinear solutions to Euler's equations describing gravity water waves. We examine how these instabilities change if we add capillarity or other hydroelastic effects. This stability analysis is easily generalisable and we present a method to predict existence of high frequency instabilities in other periodic Hamiltonian systems, building on the theory first proposed by MacKay as well as Mackay and Saman. This allows us to determine which models meet the necessary conditions
for these instabilities to occur and therefore validate their use to model water waves.

22 June @VU, 15:00-16:00, Alexander Fauck (Berlin), Room WN-C147

Title: Fillable exotic contact structures

Abstract: In 1999, I. Ustilovsky first showed that on the standard sphere $S^{4n+1}$ there exist infinitely many fillable contact structures. In my talk, I will discuss how this result can be extended to any odd-dimensional manifold supporting fillable contact structures. To this purpose, I will use Rabinowitz-Floer homology (RFH), a variant of symplectic homology to disdinguish contact structures. Moreover, I will use explicit calculations of RFH on Brieskorn manifolds - an important class of manifolds, among which one finds all exotic differentiable structures on $S^{2n-1}$.  

15 June @VU, 16:00-17:00, Christian Bick (Exeter), Room WN-S655

Title: Dynamics of Symmetric Phase Oscillators with Generalized Coupling

Abstract: In the limit of weak coupling, networks of oscillatory units can be described as a network of phase oscillators. The Kuramoto equations, where the interaction between phases is given by a single harmonic, has received much attention in particular with respect to synchronization. By contrast, we are interested in the influence of higher harmonics—which arise naturally in the reduction to a phase model—on the dynamics in symmetric networks. Moreover, we explore how such generalized coupling can be exploited to construct dynamically invariant sets on which solutions exhibit local frequency synchronization.

18 May @VU, 16:00-17:00, Charlene Kalle (Leiden), Room WN-S655

Title: Matching for certain piecewise linear maps

Abstract: For piecewise linear, expanding interval maps the absolutely continuous invariant density is an infinite sum of indicator functions. There are situations in which this density becomes piecewise smooth, for example when the map has a Markov partition. Matching is another condition on the map that guarantees that this density is piecewise smooth and it holds much more frequently. In this talk I will consider certain piecewise linear, expanding maps that depend on one parameter and show that matching holds prevalently.

06 April @VU, 16:00-17:00, Jagna Wisniewska(VU), Room WN-S655

Title: Rabinowitz Floer homology for non-compact hypersurfaces

Abstract: One of the main interests in symplectic geometry is the analysis of the Hamiltonian systems. Rabinowitz Floer homology is an algebraic invariant, which relates the existence of periodic solutions of Hamiltons equations on a prescribed energy hypersurface to its geometrical and topological properties. Up to now, the construction of Rabinowitz Floer homology has been carried out for compact hypersurfaces. In my research I will show how to extend this construction to include a class of non-compact hypersurfaces.

23 March Benelux Mathematical Congress at CWI Amsterdam

09 March @VU, 16:00-17:00, Vincent Humiliere (Paris), Room WN-S655

Title: A C°-counter example to the Arnold conjecture

Abstract: According to the now established Arnold conjecture, the number of fixed points of a Hamiltonian diffeomorphism is always greater than a certain value that only depends on the topology of the manifold. In any case, this value is at least 2. Does the same hold if we drop the smoothness assumption? After introducing symplectic/Hamiltonian homeomorphisms, I will sketch the construction of a Hamiltonian homeomorphism with only one fixed point on any closed symplectic manifold of dimension at least 4. This is joint work with Lev Buhovsky and Sobhan Seyfaddini.

24 February @VU, 16:00-17:00, Francois Genoud (Delft), Room WN-S655

Title: Stable solitons of the cubic-quintic NLS with a delta-function potential

Abstract: This talk is about the one-dimensional nonlinear Schrödinger equation with a combination of cubic focusing and quintic defocusing nonlinearities, and an attractive delta-function potential. Physically, the model comes from nonlinear optics. All standing waves with a positive soliton profile can be determined explicitly in terms of elementary functions. I will prove by a bifurcation and spectral analysis that all these solutions are orbitally stable. A remarkable feature is a regime of bistability, where two stable solitons with same propagation constant coexist.

10 February @VU, 16:00-17:00, Patrick Hafkenscheid (VU), Room WN-S655

Title:Morse homology for braids

Abstract: This talk gives an overview of the topics I will discuss in my thesis. The main interest is in developing a Morse Homology on spaces of braids. A braid in this context can be thought of as a generlisation of periodic function of integer period. This makes it possible to consider simultaneously functions of different periods. An important ingredient in the definition of the Morse Homology is the Discrete Lyapunov-like behavior of zeros of solutions to scalar parabolic PDEs. After it is defined the Morse Homology will fulfill a bridging role between the hard-to-compute Braid Floer Homology and the easy-to-compute Conley Index of discrete braid classes. I will (if time permits) briefly go over the way these three object fit together.

2015

 

04 February @UvA, 16:00-17:00, Tristan van Leeuwen (Utrecht), Room D1.115

Title:Joint parameter and state estimation for inverse problems

Abstract: Inverse problems are ubiquitous in science and engineering. Applications include seismic and medical imaging, non-destructive testing and remote sensing. In many of these application the underlying physical model is a PDE, and the (unknown) parameters appear as coefficients in the PDE. To solve the inverse problem we now need to reconstruct its solution (the state) and the coefficients from partial measurements of the solution. In this talk I will give an overview of existing methods to solve this joint estimation problem and illustrate their properties using a simple toy example. Finally, I will discuss some of my recent work on adapting these methods for large-scale problems and present some numerical results.

18 February @VU, 16:00-17:00, emptyVincent Knibbeler (Newcastle), Room WN-P663

Title:Invariants of Automorphic Lie Algebras

Abstract: Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s in the field of integrable systems. They are obtained by imposing symmetry under a finite group (the reduction group) on a Lie algebra over a ring of rational functions. Since their introduction in 2005 by Lombardo and Mikhailov, mathematicians aimed to classify Automorphic Lie Algebras. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. That is, many Automorphic Lie Algebras with nonisomorphic reduction groups are isomorphic. To explain this phenomenon we look for properties that are independent of the reduction group, called invariants. In this talk we introduce Automorphic Lie Algebras and discuss the invariants that have been found. We will use them to set up a structure theory, and find that this naturally leads to a cohomology theory of root systems. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras significantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity. From a more general perspective, the success of the structure theory and root cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a graded ring.
This is joint work with Sara Lombardo and Jan Sanders.

18 March @VU, 16:00-17:00, Sonja Hohloch (Antwerp), Room WN-P663

Title:From compact semi-toric systems to Hamiltonian S^1-actions and back.

Abstract: Roughly, a semi-toric integrable Hamiltonian system (briefly, a semi-toric system) on a compact 4-dimensional manifold consists of two commuting Hamiltonian flows one of which is periodic. Thus the flow parameters induce an S^1 x R-action on the manifold. Under certain assumptions on the singularities, semi-toric systems have been classified by Pelayo and Vu Ngoc by means of 5 invariants. Every semi-toric system induces a Hamiltonian S^1-action on the manifold by `forgetting' the R-valued flow parameter. Effective Hamiltonian S^1-actions on compact 4-manifolds have been classified by Karshon by means of so-called `labeled directed graphs'. In a joint work with S. Sabatini and D. Sepe, we linked Pelayo and Vu Ngoc classification of semi-toric systems to Karshon's classification of Hamiltonian S^1-actions. More precisely, we show that only 2 of the 5 invariants are necessary to deduce the Karshon graph of the underlying S^1-action. In an ongoing work with S. Sabatini, D. Sepe and M. Symington, we study how to `lift' an effective Hamiltonian S^1-action on a compact 4-manifold to a semi-toric system. In this talk, we give an introduction to semi-toric systems and Hamiltonian S^1-actions and sketch parts of our constructions.

15 April @VU, 16:00-17:00, Jan-David Salchow (VU), Room WN-P663

Title: The polyfold approach to compactification of moduli spaces

Abstract: The solution space of an elliptic PDE with non-compact symmetries will in general not be compact. Even after modding out the symmetries of the solution space, phenomena like breaking of trajectories and bubbling can prohibit compactness. Polyfolds are a new class of function spaces that allow to do two things that are impossible in the classical setting. Namely they allow to take a classical function space, first mod out the symmetries, and secondly put things like ‘broken trajectories’ or ‘bubble trees’ into them. The spaces obtained this way are still nice enough to admit a meaningful functional analysis. In particular the induced PDE might have a compact solution space. In this talk I will give a biased overview of these techniques on the basis of finite dimensional Morse theory.

29 April @UvA, 16:00-17:00, Martina Chirilus-Bruckner (Leiden), Room D1.115

Title: Inverse spectral theory for an efficient use of center manifold reduction

Abstract: Center manifold theory has been traditionally used to simplify the analysis of dynamical systems (such as differential equations) through a reduction of dimension. In particular for partial differential equations, which can be viewed as infinite-dimensional dynamical systems, it is desirable to possibly reduce to finite, low-dimensional systems that are more amenable to analysis. We present a means of extending this method to problems in which, at first sight, such an endeavour seems hopeless, but a reduction becomes possible after solving an inverse spectral problem.

13 May @VU, 16:00-17:00, Jaap Eldering(Imperial), Room WN-P663

Title: Symmetry reduction of fluid(-like) dynamics to "jetlet" particles

Abstract: I will present a particle model for a smoothed version of incompressible fluid dynamics. We follow Arnold's ideas of viewing fluid dynamics as geodesics on the group of diffeomorphisms. The model is obtained through symmetry reduction by subgroups of the diffeomorphism group that fix (the jets of) a set of points. This leads to finite-dimensional systems of "jetlet" particles that are special, but exact solutions of the original system. I will outline the reduction procedure using a dual pair of momentum maps, present the resulting jetlet dynamics (a Hamiltonian system) and possibly say something about the feature of "particle merging" and show some numerical simulations. This is joint work with Colin Cotter, Darryl Holm, Henry Jacobs and David Meier.

09 September @UvA, 16:00-17:00, Daan Crommelin (CWI), Room F3.20 (NIKHEF)

Title: Rare events in stochastic dynamical systems

Abstract: The extreme states of a dynamical system can be of great importance. Extremes ranging from hurricanes to power blackouts are low probability events but they can have a major impact if they occur. Studying such rare events and assessing their probability is challenging, because one often has to rely on Monte Carlo (MC) methods yet standard MC is known to be inefficient for rare events. To improve the efficiency of MC sampling for rare events, various techniques have been developed, for applications in e.g. communication networks, computational chemistry and reliability analysis. I will discuss a technique called multilevel splitting, in which model sample paths are split into multiple copies each time they cross thresholds (or levels) that lead closer to the rare event set.

23 September @VU, 16:00-17:00, Jonathan Jaquette (Rutgers), Room WN-S655

Title: Rigorous Computation of Persistent Homology

Abstract: Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this talk, I will describe a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology. We study the filtration generated by sub-level sets of a function f:X->R, where X is a CW-complex. In the special case where X is a hypercube, I'll discuss implementation of the proposed algorithms as well as a priori and a posteriori bounds of the approximation error introduced by our method.

07 October @VU, Mini-Workshop on Symplectic Geometry, Room HG 10A20 (Hoofdgebouw !)

04 November @UvA, 16:00-17:00, Jasmin Raissy (UvA), Room G3.05

Title: Local methods in complex dynamics (and how to use them for global results)

Abstract: In this talk I shall briefly discuss local holomorphic dynamics in dimension one, focusing on the the normalization and the linearization problems for germs of biholomorphism, and on parabolic bifurcation. Then I will discuss the local dynamics for germs of biholomorphism in several complex variables with an isolated fixed point and in particular I will focus on the dynamics of polynomial skew-products. If time allows I will show how the techniques of parabolic bifurcation can be used to deduce the existence of wandering Fatou components in dimension 2 as done in the joint work with M. Astorg, X. Buff, R. Dujardin and H. Peters.

18 November @VU, 16:00-17:00, Marco Mazzucchelli (Lyon), Room WN-S655

Title:Periodic orbits of magnetic flows on surfaces

Abstract: This talk is about the existence of periodic obits of exact magnetic flows on the cotangent bundle of closed surfaces. The dynamics of these Hamiltonian systems on high energy levels is well known: it is conjugated to a Reeb flow, and actually to a Finsler geodesic flow. In the talk, I will focus on low energies, more precisely on energies below the so-called Mañé critical value of the universal covering. After introducing the setting, I will present a recent result asserting the existence of infinitely many periodic orbits on almost all energy levels in this range. This is a joint work with A. Abbondandolo, L. Macarini, and G. P. Paternain.

02 December @UvA, 16:00-17:00, Rob Stevenson (UvA), Room F3.20 (NIKHEF)

2014

  • 12 February @VU, 16:00-17:00, Thomas Rot (VU), Room P656
    Title:Rigidity in non-variational systems
    Abstract: Morse theory investigates the relationship between critical points of a function in terms of the topology of its domain. Witten give a compact description of this relationship by counting rigid solutions of the gradient flow, which is now known as Morse homology. The Morse homology turns out to independent of the function and metric, which gives lower bounds on the minimal number and type of critical points of a function in terms of the topology of its domain. In joint work with Rob Vandervorst, we use similar rigid counts to develop a homology theory (Morse-Conley-Floer homology) to study dynamical systems which are not necessarily of gradient type. The homology theory is invariant under continuation, which gives existence results for the minimal number and type of isolated invariant sets of the flow, in terms of the topology of the ambient space.
  • 26 February @UvA, 16:00-17:00, Jan Bouwe van den Berg (VU), Room D1.115
  • 12 March @VU, 16:00-17:00, Joost Hulshof(VU), Room P640
  • 26 March @UvA, 16:00-17:00, Bas Teusink VU), Room B0.209
  • 9 April @VU: Double seminar! 15:00-17:00 Speakers: Milan Tvrdy and Oliver Fabert, Room M616
    Title Milan (Academy of Sciences of the Czech Republic): On singular periodic problems
    Title Oliver (Hamburg): New algebraic structures in Hamiltonian Floer theory
    Abstract Milan: First, I suppose to give a survey of recent results concerning the existence of positive periodic solutions to the problems of the type
    \begin{equation}\tag{P}
    u''+a(t)\,u=f(t,u),\quad u(0)=u(T),\quad u'(0)=u'(T),
    \end{equation}
    where $0<T<\infty,$ $a\in L_1[0,T]$ and $f\,{:}\,[0,T]\times (0,\infty)\to\mathbb R$ is regular in $[0,T]\times (0,\infty),$ but it can have singularity for $x=0.$ Main tools utilized in the proofs are the method of lower and upper functions and the anti-maximum principle (called also inverse nonnegativity). Further, applications to particular real world models, like the Brillouin electron beam focusing equation and the model describing valveless pumping in the one pipe - one tank configuration will be treated.
    Abstract Oliver: Floer cohomology is the most important tool to prove results about periodic orbits of Hamiltonian systems in symplectic geometry and also plays an important role in the mirror symmetry conjecture from superstring theory. In my talk I will show how Eliashberg-Givental-Hofer's symplectic field theory can be used to define new algebraic structures in Hamiltonian Floer theory. Apart from making the information of all rational Gromov-Witten invariants applicable to Hamiltonian dynamics, they are used to formulate the analogon of the classical mirror symmetry conjecture for open Calabi-Yau manifolds.
  • 17 September @VU, 16:00-17:00, Tristan Hands (UvA), Room S-631.
    Title:Bourgain's Entropy Criterion for Pointwise Convergence
    Abstract: In this presentation a sketch of the proof of Bourgain's entropy criterion will be presented. This criterion allows one to show divergence of certain sequences of L2 operators. The significance will be displayed by solving a long standing question from Ergodic theory, posed by Bellow. Other applications to number theory will be stated, including a question raised by Erdős and Khintchine.
  • 1 October @UvA, 16:00-17:00, Todd Young (Ohio), Room G.529.
    Title:Models of Cell Cycle Dynamics and Clustering.
    Abstract: Motivated by experiments and theoretical work on metabolic oscillations in yeast cultures, we study phenomenological ODE models of the cell cycles of large numbers of cells, with cell-cycle dependent feedback. We assume very general forms of the feedback and study the dynamics, particularly the temporal clustering behavior of such systems. Biologists have long observed periodic-like oxygen consumption oscillations in yeast populations under certain conditions. We hypothesized that certain of these oscillations could be caused and/or accompanied by cell cycle clustering. We study models of the cell cycle in which cells in one phase of the cycle may influence the progress of cells in another phase (presumably by production or depletion of diffusible chemical products.) We give proofs of the existence and stability of certain periodic solutions in which cells are clustered. Furthermore, this clustering phenomenon is robust; it occurs for a variety of models, a broad selection of parameter values in those models. Related experiments have shown conclusively that cell cycle clustering occurs in the oscillating cultures.
  • 15 October @VU, 16:00-17:00, Roberto Castelli (VU), Room S-631.
    Title:A method to rigorously compute the tangent bundles of hyperbolic periodic orbits of vector fields.
    Abstract: The first ingredient necessary to parametrize the invariant manifolds of periodic orbits is the tangent bundle, that is the tangent space of the invariant manifold at the orbit. The tangent directions, as like as the stability parameters, result by integrating a non-autonomous system of differential equations with periodic coefficients of the form
    \begin{equation}\label{eq}
    \dot y=A(t) y,\quad A(t)\in \mathbb R^{n\times n}, \tau \ {\rm periodic}
    \end{equation}
    obtained by linearizing the vector field around the periodic orbit. In this talk we combine the Floquet theory and rigorous numerics to compute the Floquet normal form decomposition $\Phi(t)=Q(t)e^{Rt}$ of the fundamental matrix solution of \eqref{eq}. Taking advantage of the periodicity of the function $Q(t)$, the methods aims at computing the Fourier coefficients of $Q(t)$ and the constant matrix $R$ by solving an infinite dimensional algebraic problem in a suitable Banach space. As an application we compute the tangent bundles for orbits of different systems and we relate the dynamical and geometrical properties of the manifolds to the particular form of the Floquet decomposition.
  • 29 October @UvA, 16:00-17:00, Iris Smit (UvA), Room TBA.
    Title:Basins of sequences of attracting holomorphic automorphisms of C^2
    Abstract: Given a sequence of holomorphic automorphisms of C^2 with attracting fixed point zero, we can study its attracting basin. This basin is definitely homeomorphic to R^4. But under what conditions is it also biholomorphic to C^2? In this talk, I will give an overview of old and new results and counterexamples for this question.
  • 2 November @VU, 16:00-17:00, Nena Röttgen (Munster), Room S-631.
    Title:Closed geodesics on complete Riemannian manifolds with convex ends
    Abstract: The study of closed geodesics is a classical topic in Riemannian geometry. However, in the noncompact case little is known about existence of closed geodesics, at least if the dimension is larger than two. In this talk I will give an overview of the subject and present existence results for complete Riemannian manifolds with convex ends.
  • 26 November @UvA, 16:00-17:00, Bob Planque (VU), Room G-210
    Title:Dynamics, stochastics, and optimal steering in systems biology
    Abstract: I'm going to present some recent ongoing work in two areas. First, I will discuss the problem how a cell may regulate gene expression to optimise the flux through a network. After quite some work, we finally now have a concrete system of ODEs to work on, and I will present the ideas behind it and some initial results. Second, I will discuss a nice problem in which many branches of mathematics come together, including ODEs, SDEs, statistics, information theory and others. The question is really simple: suppose you know all about the mechanism by which an external signal is sensed and relayed in the cell nucleus for gene expression, but you also know that numbers of molecules are finite, and hence fluctuate stochastically. Can you then predict the probability of a gene being expressed? We have some initial results, but aren't there yet.
  • 10 December @VU, 16:00-17:00, Konstantinos Efstathiou (Groningen), Room S-631.
    Title:Hamiltonian Monodromy: Overview & Generalizations
    Abstract: Hamiltonian monodromy refers to the monodromy of torus bundles over circles that naturally appear in integrable Hamiltonian systems. The non-triviality of the monodromy is related to the non-existence of global action-angle coordinates and explains features of the joint spectrum of the corresponding quantum system. In this talk I will give an overview of research on Hamiltonian monodromy looking at it from different points of view: some of them Hamiltonian, some of them not. Furthermore, I will discuss generalizations of Hamiltonian monodromy where the space of interest is no longer a torus bundle over a circle but it contains exceptional (singular) fibres.

2013

  • 18 December @UvA, 16:00-17:00, Han Peters (UvA), Room A1.04
    Title: Complex dynamics from a real perspective.
    Abstract: Imagine observing a real world dynamical system. This system will likely depend on so many variables that in order to study its behavior, it is necessary to suppress some of these variables. In doing so, the space becomes more manageable, but information may be lost. In what ways is the new system "similar" to the original dynamical system? In joint work with John Erik Fornaess, we investigate this question in a setting which is very well understood: the iteration of polynomials in the complex plane. We consider an observer who only sees the real parts of the orbits. We focus on ergodic theoretic properties of the observed and the original dynamical systems.
  • 4 December @VU, 15:00-17:00, Georgios Dimitroglou Rizell and Romain Dujardin, Room C624
    Title Georgios: Wrapped Floer homology and Seidel's isomorphism. 
    Abstract: Reeb flows are ubiquitous. Famous examples are the geodesic flow, and the Hamiltonian flow on a given energy hypersurface. Even though the Reeb flow for the standard contact form on the (2n+1)-plane is just a translation, there are interesting questions about its interaction with a Legendrian submanifold. We prove Seidel's isomorphism between the Legendrian contact homology induced by an exact Lagrangian filling and the singular homology of the filling. This implies that the number of integral curves of the Reeb vector-field having endpoints on such a Legendrian submanifold is bounded from below by the rank of the singular homology of the filling.
    Title Romain: Homoclinic tangencies in the complex Hénon family

    Abstract: We explore the stability/bifurcation dichotomy for families of polynomial automorphisms of C^2. Our results are reminiscent both of the classical bifurcation theory of rational mappings of the Riemann sphere (due to Mané-Sad-Sullivan and Lyubich) and of the Palis conjectures on the global dynamical structure of the space of diffeomorphisms. This is joint work with Misha Lyubich. 
    Abstract: TBA
  • 19 November @VU, 11:30-12:30, Doris Hein (IAS), Room P631
    Title: Applications of Morse theory for Cuplength Estimates in Symplectic Geometry
    Abstract: There are many different cuplength estimates in symplectic and contact geometry, e.g. on fixed points of Hamiltonian diffeomorphisms, Hamiltonian chords of Lagrangian submanifolds and solutions to a Dirac-type equation. These results can be obtained by a unified proof method based on Morse theory, which also yields new results for leafwise intersections. The base case is a result for critical points of smooth functions which are a perturbation of a Morse-Bott situation. I will sketch the proof in the Morse case and state the results obtained by applying this proof to the action functionals in symplectic and contact geometry.
  • 20 November @VU, 16:00-17:00, Marcio Gameiro (Sao Carlos), Room P631
    Title: Rigorous continuation of solutions of PDEs 
    Abstract: We present a rigorous numerical method to compute solutions of infinite dimensional nonlinear problems. The method combines classical predictor corrector algorithms, analytic estimates and the uniform contraction principle to prove existence of smooth branches of solutions of nonlinear PDEs. The method is applied to compute equilibria and time periodic orbits for PDEs defined on two- and three-dimensional spatial domains.
  • 6 November @VU, 16:00-17:00, Barney Bramham (Bochum), Room S623
    Title: Hamiltonian surface maps and pseudo-holomorphic curves 
    Abstract: In this talk we will discuss results, questions, and ongoing work concerning area preserving disk maps using a new approach that makes use of foliations by pseudo-holomorphic curves.
  • 23 October @UvA, 16:00-17:00, Tom Kempton (Utrecht), Room A.106
    Title: How to do dynamics on overlapping self similar sets and measures. 
    Abstract: It is easy to associate dynamical systems to non-overlapping self similar sets, such as the middle 1/3 cantor set. One can view the middle 1/3 cantor set K as the union of two scaled down copies of itself, and by applying these contractions in reverse one gets the dynamical system T on K defined by T(x)=3x (mod 1). Using T it is easy to deduce various properties of the Cantor set K (such as its Hausdorff dimension).Attempting to apply this technique directly to self similar sets and measures with overlaps does not work very well, since it gives rise to functions which take multiple values in the overlap region. We discuss ways in which this 'overlapping dynamical system' can be made to make sense and how it gives rise to some results and many conjectures on Beta Expansions and Bernoulli Convolutions.
  • 9 October @VU, 16:00-17:00, Hil Meijer (Twente), Room S623
    Title: Waves in an excitatory neural network with adaptation 
    Abstract: Neural fields provide a mesoscopic description of neuronal activity. In this talk we consider a one-dimensional field with excitatory spatial connections. Inhibition or spike-frequency adaptation provide negative feedback that we model with linear dynamics. Adding this local feedback, yields one of the simplest examples of neural fields that produce wavetrains and travelling pulses and fronts. First we choose a Heaviside step function for the activation function. In this setting we can analyse the existence and stability of travelling pulses and fronts. We find a codimension 2 heteroclinic cycle as organizing centre. It implies the existence of a new anti-pulse solution, which is stable. Second, we consider how these results persist for smooth sigmoidal activation functions. We do this with numerical continuation (with MatCont). We find that the pulses and fronts follow the Heaviside analysis for decreasing slope. For wavetrains, however, the dynamics becomes much richer as their dispersion curves have monotone tails and then develop oscillatory tails. We show that the transition is due to a homoclinic bifurcation where three leading eigenvalues have the same real part. Note: Joint work with S. Coombes (Nottingham)
  • 17 July @ VU, 15:00-17:00, Antonio Ponno (Padova) and Tim Myers (Barcelona). Room P631
    Title Antonio: The Fermi-Pasta-Ulam problem: old questions and new results.
    Title Tim: Mathematics at the nanoscale. 
  • 5 June @ VU, 16:00-17:00, Hermen Jan Hupkes (Leiden). Room M632 
    Title: Travelling around Obstacles in Planar Anistropic Spatial Systems
    Abstract: We study dynamical systems posed on a discrete spatial domain, with a special focus on the behaviour of basic objects such as travelling waves under (potentially large) perturbations of the wave and the underlying spatial lattice.
  • 13 March @ UvA: Erik Fornaess (Michigan)
    Title: Remarks on Complex Analysis
    Abstact: In this seminar talk I will discuss complex dynamics. I will be talking about a recent joint work with Feng Rong on Fatou components.
  • 27 March @ VU, 16:00-17:00, Igor Hoveijn (Groningen). Room P656. 
    Title: Singularities on the boundary of the stability domain near 1:1 resonance
    Abstract: We study the linear differential equation x' = Lx in 1:1 resonance. That is, L is 4 by 4 matrix with eigenvalues (ib,-ib,ib,-ib). We wish to find the stability domain in gl(4,R), the space of 4 by 4 matrices. Moreover we wish to find the singularities of the boundary of the stability domain. The 1:1 resonance turns up in many applications, ranging from fluid dynamics and wave phenomena to rotating mechanical devices. Such systems are frequently considered as perturbed Hamiltonian, reversible or equivariant systems. In many examples the latter turn up at the boundary of the stability domain, especially at the singularities. Therefore determining the stability of perturbations can be delicate. Since a neighborhood of L in gl(4,R) is 16-dimensional we put some effort in reducing the dimension. Here keywords are equivalence classes and transversality. In several steps we are able to reduce to a 3-sphere that contains all information about the neighborhood of L. The boundary of the stability domain is contained in two right conoids. The singularities of this surface are transverse self-intersections, Whitney umbrellas and intersections of self-intersections. A Whitney stratification allows us to describe the neighborhood of $L$ and identify the stability domain.
  • 12 April @ VU, 11:00-12:00am, Margaret Beck (Edinburgh). Room M664
    Title: Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations
    Abstract: Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows where they often emerge on time-scales much shorter than the viscous time scale, and then dominate the dynamics for very long time intervals. We propose a dynamical systems explanation of the metastability of an explicit family of physically relevant quasi-stationary solutions, referred to as bar states, of the two-dimensional incompressible Navier-Stokes equation with small viscosity on the torus. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. This is joint work with C. Eugene Wayne.

2012

  • Wed. 19 Dec @ VU: Jan Sanders (VU), Room M648.
    Title: Coupled cell networks: semigroups, Lie algebras and normal forms
    Abstract: Dynamical systems with a network structure arise in applications that range from statistical mechanics and electrical circuits to neural networks, systems biology, power grids and the world wide web. In this talk I will explain what it means for a coupled cell network to possess the "semigroup(oid) property". Networks with this property form a Lie algebra and we recently developed a method to compute their local normal forms near a dynamical equilibrium. This helped us understand and predict certain seemingly anomalous bifurcations in network systems. This is joint work with Bob Rink.
  • Wed. 5 Dec @ UvA, 15:00-16:00, Room B0.209: Sebastian van Strien (Imperial College) 
    Title: Stochastic stability of expanding circle maps with neutral fixed point
    Abstract: One of the best known dynamical systems with intermittency behaviour is the well-known Pomeau-Manneville circle map. This map has a neutral fixed point at $0$ which causes orbits to linger there for long periods. Nevertheless this map has always a physical measure:   for $\alpha\ge 1$ it is the Dirac measure at $0$ while for $\alpha\in (0,1)$ it is absolutely continuous. It was also  known for quite a while that this map is stochastically stable when $\alpha\ge 1$. In this talk I will discuss a result which implies that this map is also stochastic stable when  $\alpha\in (0,1)$. (joint with Weixiao Shen)
  • Wed. 7 Nov @ VU, 15:00-17:00, Room C624: Andre Vanderbauwhede and Leandro Arosio 
    Title Andre: Branches of periodic orbits in reversible systems.
    Abstract: In the typical reversible systems which appear in many applications (symmetric) periodic orbits appear in one-parameter families (branches). In this survey talk we describe how these branches of periodic orbits originate from equilibria, terminate at homoclinic orbits, and branch from each other in period-doubling bifurcations or higher order subharmonic bifurcations. Adding external parameters allows to study degenerate cases and the transition from degenerate to non-degenerate situations. The talk will be mainly "pictorial", avoiding technicalities as much as possible.
    Title Leandro: Geometric aspects of Loewner theory
    Abstract: Loewner theory is a major tool in geometric function theory, introduced by C. Loewner in 1923 as he was working on the Bieberbach conjecture. The classical theory has been generalized by Bracci, Contreras and Díaz-Madrigal in '08 to complete hyperbolic manifolds in several complex variables. In this talk I will review the theory and present some new results concerning the existence of univalent solutions to the Loewner PDE.
  • Wed. 24 Oct @ VU, 16:00-17:00, Room P640: Sobhan Seyfaddini (ESN Paris)
    Title: C^0 continuity of spectral invariants
    Abstract: After introducing the Oh-Schwarz spectral invariants, I will discuss the relation between these invariants and the C^0 topology on the space of Hamiltonian paths.  I will show that, under certain assumptions, spectral invariants are C^0-continuous.
  • Wed. 12 Sept @ VU, 16:00-17:00, Room M648: Pavel Zorin-Kranich
    Title: IP* sets of integers
    Abstract: According to van der Warden's theorem, for every finite coloring of the integers there exist arbitrarily long monochromatic arithmetic progressions {a,a+b,\dots,a+kb}. Ramsey theory is concerned with results of this type: results that guarantee that some structure can be found within some color class of every finite coloring of a larger structure. A general principle is that whenever it is possible to find some structure, then it in fact occurs often. In this talk I discuss what "often" means in this context.
  • Wed. 25 April @ UvA, 16:00-17:00, Room A1.04: Bert Peletier (Leiden)
    Title: The dynamics of "Target-Mediated Drug Disposition"
    Abstract: Drugs are designed to interact with specific targets in order to produce their desired pharmacological effect. This involves a dynamic interplay between drug and target, each of which is supplied and eliminated, and the drug-target complex which is also eliminated or absorbed. In this talk we discuss the often nontrivial dynamics of this process, ways to recognise it experimentally, and attempts at developing simplified models.
  • Wed. 11 April @ VU, 16:00-17:00, Room M-632: Henk Broer (Groningen)
    Title: Resonance and Fractal Geometry
    Abstract: A number of resonant phenomena is reviewed such as Huygens's synchronizing clocks, the tidal resonances of Moon and certain planets as well a swing. Resonance is an interaction of various oscillations with rationally related frequencies which leads to a compatible periodic behaviour. It is conceived of in terms of parameter dependent dynamics. The resonant zones in parameter space then can consist of tongues that are arranged in a fractal pattern in which a Cantor set plays a role. In and near the Cantor set also other types of dynamics may occur, like quasi-periodic or chaotic. In the talk we discuss several examples.
  • Wed. 28 March @ UvA, 16:00-17:00, Room B0.209: Martijn Zaal (VU)
    Title: Time discretization of the osmotic cell swelling problem
    Abstract: A simple model for cell swelling by osmosis can be formulated as a free boundary problem involving diffusion and mean curvature. This problem can in turn be studied using a time discretization originating from the study of gradient flows in Euclidean space. This point of view relates the physics of the model to the mathematics used to construct solutions. Moreover, it illustrates how ideas from the theory of gradient flows can be used outside of Euclidean or even metric space.
  • Wed. 14 March @ VU, 16:00-17:00, Room M-632: Evgeny Verbitskiy (Leiden/Groningen)
    Title: Periodic and homoclinic points in algebraic dynamics
    Abstract: This will mainly be a gentle introduction to actions generated by a finite number of commuting automorphisms of compact abelian groups, and their fascinating connections with algebra, number theory, analysis, and algebraic geometry. I'll focus on the problem of the growth rate of periodic points for such actions, including recent joint work with Doug Lind and Klaus Schmidt.  The main technical tool relies on construction of suitable homoclinic points of algebraic dynamical systems.
  • Wed. 29 Feb. @ UvA, 16:00-17:00, Room TBA: Yonatan Gutman (Warschau)
    Title: The structure of cubespaces attached to minimal distal dynamical systems
    Abstract: Cubespaces were recently introduced by Camarena and B. Szegedy. These are compact spaces $X$ together with closed collections of "cubes" $C^{n}(X)\subset {2^{n}}$, $n=1,2,\ldots$ verifying some natural axioms. We investigate cubespaces induced by minimal dynamical topological systems $(G,X)$ where $G$ is Abelian. Szegedy-Camarena's Decomposition Theorem furnishes us with a natural family of canonical factors $(G,X_{k})$, $k=1,2,\ldots$. These factors turn out to be multiple principlal bundles.We show that under the assumption that all fibers are Lie groups $(G,X_{k})$ is a nilsystem, i.e. arising from a quotient of a nilpotent Lie group.This enable us to give simplified proofs to some of the results obtained by Host-Kra-Maass in order to characterize nilsequences internally.

2011

  • Wed 7 Dec @ UvA, 16:00-17:00, Room G0.05: Bob Rink (VU)
    Title: A destruction theorem for generalized Frenkel-Kontorova crystal models
    Abstract: The equilibrium states of the classical Frenkel-Kontorova crystal are also the orbits of the Chirikov standard twist map. In a generalized Frenkel-Kontorova crystal there is no such correspondence, because the atoms in such a crystal interact beyond their nearest neighbors. In this talk, I will present the following converse KAM theorem for generalized Frenkel-Kontorova crystals: if the crystal model admits a continuous family of ground states with an average particle spacing that is "easy to approximate by rational numbers", then this family can be destroyed by an arbitrarily small smooth perturbation of the crystal model. This means that a "typical" crystal will display "forbidden regions" for its atoms. This result of Blaz Mramor and myself generalizes a theorem of Mather for the destruction of Liouville invariant circles of twist maps. Our proof is quite different though and may allow for generalizations to lattice Aubry-Mather theory and elliptic PDEs
  • Wed 9 Nov @ UvA, 16:00-17:00, Room G0.05: Tom Kempton (Utrecht University) 
    Title: Bernoulli convolutions and Beta expansions
    Abstract : In this talk we'll discuss two objects, one from number theory and one
    from measure theory, and link them using some dynamics and ergodic theory.
    The first object is the set of beta-expansions of a real number. Given
    beta>1 and a real number x, a beta expansion of x is a sequence (a_n)
    for which we can write x=\sum a_i\beta^{-i}.
    If we put beta=10 and take a_i in the digit set {0,1,...,9} then beta
    expansions are the familiar decimal expansions. Almost every x has a
    unique decimal expansion, making the set of decimal expansions of a
    given real number rather dull. However, for non-integer beta the set
    of beta expansions of a given real number is typically much richer. We
    are interested in the structure of this set.
    The second object that we'll discuss is the Bernoulli convolution.
    Bernoulli convolutions are perhaps the most simple examples of fractal
    measures, and yet the fundamental question of whether the Bernoulli
    convolution associated to some parameter beta is absolutely continuous
    remains unsolved.
    In this talk we'll link some questions about the structure of the set
    of beta expansions of a typical real number with the question of
    absolute continuity of the Bernoulli convolution.   
  • Wed 26 Oct @ VU, 16:00-17:00, Room F-640:  Pablo Barrientos
    Title: Heteroclinic cycles arising in generic unfoldings of nilpotent singularities
    Abstract : In this seminar we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we show that any nilpotent singularity of codimension four in R4 unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in R3 unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes (Bykov cycles). Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.
  • Wed 12 Oct @ UvA, 16:00-17:00, Room G0.05: Erlend Fornaess Wold (Oslo)
    Title: Symplectic Completion of Jets
    Abstract : We will show how one can use some fairly recent
    developments in Several Complex Variables to complete symplectic
    jets; a problem that comes up in accelerator physics.   
  • Wed 28 Sept @ VU, 16:00-17:00, Room TBA: Jens Rademacher                               Title: Unfolding heteroclinic networks of equilibria and periodic orbits with tangencies
    Abstract : In this talk I present a method to study bifurcations from
    heteroclinic network in ordinary differential equations of arbitrary
    dimension. These networks can consist of equilibria and periodic
    orbits, and the heteroclinic connections can be tangent. The problem
    of bifurcating solutions in a neighborhood of this network is reduced
    to algebraic equations by a kind of Lyapunov-Schmidt reduction,
    closely related to Lin's method. The abstract result
    will be explained through some examples. 
  • Wed 14 Sept @ UvA, 16:00-17:00, Room G0.05: Sara Lombardo (Northumbria University Newcastle and VU)
    Title: Three-wave mixing in quadratic nonlinear crystals
    Abstract : The resonant interaction of three waves (3WRI) is an important process in physics describing the resonant mixing of three quasi-monochromatic waves in weakly nonlinear and dispersive media. It appears in various physical contexts, notably fluid dynamics and optics. In quadratic nonlinear crystals parametric three-wave mixing provides a versatile means of achieving widely tunable frequency conversion of laser light. In this talk I will consider mathematical features of the 3WRI model, in particular a novel solution describing three locked dark pulses (simulton) will be presented. The dark-dark-dark triplet is found using standard Darboux-Dressing Transformations adapted to construct soliton solutions for a class of integrable PDEs with matrix rather than scalar coefficients. The soliton dynamics associated to this class of PDEs (also know as boomeronic-type equations) may be richer than those of standard solitons.
  • Wed 18 May, 16:00-17:00, Room P-624: Frederic Bourgeois (Brussels) 
    Title: S^1-equivariant symplectic homology and families of Hamiltonians
    Abstract : This is joint work with Alexandru Oancea. The aim of this talk is to explain the definition of an S^1-equivariant version of symplectic homology, via a Borel-type construction. As an important ingredient of this construction, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.
  • Tue 3 May, 16:00-17:00, Room M-664: Tanja Eisner (UvA) Note the unusual date.
    Title: On rigidity sequences
    Abstract: We study rigidity sequences for weakly mixing measure preserving dynamical systems and in particular for linear systems in Banach spaces and show the connections to ergodic theory, number theory and functional analysis. This is a joint work with Sophie Grivaux.
  • Wed 20 April, 16:00-17:00, Room P-624: Adrian Muntean (Eindhoven)
    Title: Homogenization of a Locally-Periodic Medium with Areas of Low and High Diffusivity
    Abstract: We aim at understanding reaction and transport in those porous materials that present regions with both high and low diffusivities. For such scenarios, the transport becomes structured (here: micro-macro), while the reaction will be mainly hosted by the micro-structures of the low-diffusivity region. The geometry we have in mind include perforations (pores) arranged in a locally-periodic fashion. We choose a prototypical reaction-diffusion system (of minimal size), discuss its formal homogenization – the heterogenous medium being now assumed to be made of zones with circular areas of low diffusivity of x-varying sizes. We report on two type of results. On one hand, we rely on formal asymptotic homogenization, suitable use of the level sets of the oscillating perforations combined with a boundary unfolding technique to derive the upscaled model equations. On the other hand, we prove the weak solvability of the limit two-scale reaction-diffusion model. A special feature of our analysis is that most of the basic estimates (positivity, boundedness, uniqueness, energy inequality) are obtained in x-dependent Bochner spaces. Finally, the homogenization limit is proven rigorously by means of a suitable corrector estimate (an upper bound on the convergence rate). This is joint work with Tycho van Noorden (University of Erlangen-Nuremberg, Germany).
  • Wed 6 April, 16:00-17:00, Room P-624: Sebastiaan Janssens (Utrecht)
    Title: Discrete-time dynamics on a space of measures
    Abstract: Consider a population of agents (e.g. cells or biological oscillators such as integrate-and-fire neurons) whose individual state (i-state) corresponds to a point moving on the one-dimensional unit circle $S$. Assume furthermore that interaction between agents is indirect, via contribution to and dependence on a so-called \emph{environmental condition}. Then the population state (p-state) at a particular moment is given by a measure $\mu \in M_+(S)$, the cone of finite positive Borel measures on $S$. Depending on the presence of birth and death effects, $\mu$ may or may not be a probability measure. We investigate the existence, stability and bifurcation of Dirac-type periodic solutions of the p-state dynamics by studying the fixed points of an associated return map defined on $M_+(S)$. The underlying interpretation is that a stable (unstable) Dirac fixed point corresponds to a synchronised (de-synchronised) p-state.
  • Wed 9 March, 16:00-17:00, Room P-624: Onno van Gaans (Leiden)
    Title: Stationary solutions of stochastic delay differential equations driven by Levy processes
    Abstract: Under what types of stochastic perturbations will a stable deterministic delay differential equation remain stable? Stability of stochastic equations is here interpreted as existence of a stationary solution. Several approaches and results will be discussed for stochastic perturbations with non-Lipschitz coefficients and processes with jumps.
  • Wed. 23 Feb, 16:00-17:00, Room P-624: Jaap Eldering (Utrecht)
    Title: The Perron method: what, why, and how to apply it to NHIMs.
    Abstract: I will start with a review of the Perron method for proving stable and unstable manifolds of hyperbolic fixed points. This is an alternative to the graph transform method, and I will explain its (dis)advantages. Secondly, I will give an overview of how the Perron method can be generalized to apply it to normally hyperbolic invariant manifolds (NHIMs). This can be done by only linearizing the normal directions.
  • Wed. 9 Feb, 16:00-17:00, Room P-624: Joost Hulshof (VU) 
    Energy concentration for the harmonic map heat flow.
    Abstract: It is well known that the gradient flow of the Dirichlet energy of maps from the disk to the unit sphere may develop singularities. We present a self-contained approach which, in the radially symmetric case, describes that there is only one well-defined scale in which a bubble appears and bubbles off.

2010

  • Wed. 20 Oct. 16:00-17:00, Room P-624. Han Peters (UVA)
    Title: Fatou components in two complex variables.
    Abstract: Roughly speaking, a Fatou component is the largest open connected set for which all orbits behave similarly. As such, Fatou components play an important role in our understanding of holomorphic dynamical systems. For the iteration of polynomials and rational functions in the conplex plane, Fatou components are very well understood. By a result of Sullivan every Fatou component is pre-periodic, and periodic Fatou components are completely classified. In higher dimensions the situation is quite different. Whether every Fatou component is pre-periodic is known only in special cases, and the classification of periodic Fatou components is not nearly complete. In this talk I will describe what is currently known and discuss recent research with M. Lyubich
  • Wed. 6 Oct. 16:00-17:00, Room M-664. Sander Hille (Leiden)
    Title: Reverse engineering of the auxin transport system in Arabidopsis plants.
    Abstract: In plants the growth hormone auxin is transported from the top of the plant downwards through the stem. In various experiments in the Plant BioDynamics Lab (PBDL) at Leiden University the dynamical properties of this transport mechanism at macroscopic level are measured for the model plant  Arabidopsis thaliana. We will discuss these experimental results and the various ways we needed to model the system mathematically in the form of a system of coupled ordinary and partial differential equations in order to fit the available data. Moreover, we will present how well-designed macroscopic measurements and mathematical modeling together, in close collaboration with the experimental biologists at PBDL, are able to draw conclusions on the structure and functioning of the system at mesoscopic -- cellular -- level.
  • Wednesday 22 September, 14:00-17:00, Room 08A05 (in the main building of the VU): Mathematics, Science & Engineering. This symposium is organized to bid farewell to Rein van der Hout, whose job at the VU has ended this summer.
    Programme
    14:00-14:50 Cas van der Avoort (NXP): Glass ceiling for mechanical micro-resonators
    14:50-15:00 Break
    15:00-15:50 Michiel Bertsch (IAC, Rome 2):  Heat flow for director fields
    15:50-16:00 Break
    16:00-16:50: Bert Peletier (emeritus Leiden): Drug delivery to the brain in the presence of high-affinity proteins and lipids  
    16:50-17:00 Closing
    17:00-..... Drinks                                     
  • Wednesday 23 June, 11:00-12:00, Room: S-664.Martina Chirilus-Bruckner (CWI).
    Title: Interaction of Pulses in Nonlinear Wave Equations                                        
    Abstract: The interaction of pulses, i.e. spatially and temporarily oscillating waves modulated by a spatially localized envelope, is described via an extended perturbation approach that provides explicit formulas for interaction effects such as a position shift or a shape deformation of the interacting pulses.
    The analysis involves a reduction to amplitude equations (based on Fourier or Bloch analysis) and a rigorous justification of these (using energy estimates or semigroup theory). The presented method is applicable to a wide class of nonlinear, dispersive equations in 1+1 dimensions. In particular, it can be carried out for nonlinear wave equations with periodic coefficients which arise, e.g., as model equations for light propagation in photonic crystals.
  • Wednesday 9 June, 16:00-17:00, Room F-630.  Sander Hille (Universiteit Leiden).
    This talk has been cancelled.
  • Tuesday 8 June, 16:00-17:00, Room S-664. Lennaert van Veen (Univ. Ontario Inst. Technology).
    Title: The tangled edge of turbulence in bursting Couette flow.
    Abstract: In recent years, the scale of dynamical systems-type computations in turbulence research has increased spectacularly. Equilibrium and periodic solutions have been computed for Couette flow, pipe flow and many other geometries. One of the goals of these computations is to explain the process of turbulent bursting in shear flows. Bursting occurs in the presence of an asymptotically stable laminar flow, so that ordinary bifurcation scenarios do not offer an explanation. Instead, the current focus is on "edge states," i.e. saddle-type equilibria or periodic solutions that live on a boundary between turbulent and laminar behaviour. We should be able to clarify the bursting process if we know the geometry of the (un)stable manifolds of such states. However, the systematic computation of these manifolds is a hard task. We present a recently developed algorithm for the computation of unstable manifolds and its application to turbulent Couette flow. This algorithm uses matrix-free linear solving and comes with a strong convergence result. Initial computations indicate that the (un)stable manifolds of an edge state in turbulent Couette flow form a homoclinic tangle, an observation with far-reaching implications for our understanding of the transition to turbulence.
  • Wednesday 12 May, 16:00-17:00, Room F-630. R. Hindriks (VU).                                     Title: Data-driven dynamical models for spontaneous MEG oscillations  
    Abstract: Large-scale brain activity as measured with magnetoencephalography (MEG) during rest typically shows oscillatory features. The intrinsic dynamics of these oscillations and the way in which spatially separated oscillations are coordinated is currently not well understood. In this talk I will focus on a recent development within the field of time-series analysis for MEG oscillations, namely the construction and estimation of explicit dynamical models in continuous-time. In particular, we focus on deterministic coupled limit-cycle models in the weak coupling regime, and on two-dimensional Fokker-Planck equations. Both techniques will be applied to MEG data-sets recorded from human subjects.
  • Wednesday 28 April, 16:00-17:00, Room F-630. G. Pfander (Jacobs University Bremen) 
    Title: A sampling theory for operators
    Abstract: The classical sampling theorem, attributed to Whittaker, Shannon, Nyquist,
    and Kotelnikov, states that a bandlimited function can be recovered from its
    samples, as long as we use a sufficiently dense sampling grid. Here, we
    review the recent development of an operator sampling theory which allows for
    a widening of the classical sampling theorem. In this realm, bandlimited
    functions are replaced by bandlimited operators, that is, by
    pseudodifferential operators which have bandlimited Kohn-Nirenberg symbols.
    Similar to the Nyquist sampling density condition alluded to above, we
    discuss sufficient and necessary conditions on the bandlimitation of
    pseudodifferential operators to ensure that they can be recovered by their
    action on a single distribution. In fact, we show that an operator with
    Kohn-Nirenberg symbol bandlimited to a Jordan domain of measure less than one
    can be recovered through its action on a distribution defined on a
    appropriately chosen sampling grid. Further, an operator with bandlimitation
    to a Jordan domain of measure larger than one cannot be recovered through its
    action on any tempered distribution whatsoever, pointing towards a
    fundamental difference to the classical sampling theorem where a large
    bandwidth could always be compensated through a sufficiently fine sampling
    grid. The dichotomy depending on the size of the bandlimitation is related to
    Heisenberg's uncertainty principle.
  • Wednesday 14 April: no seminar (NDNS+ workshop in Eindhoven)
  • Wednesday 17 February, 16:00-17:00, Room F-630. Antonios Zagaris (Twente). 
    Title: Phytoplankton-nutrient dynamics in oligotrophic envirnoments
    Abstract: In this talk, we will introduce a system of two coupled reaction-diffusion PDEs modeling phytoplankton-nutrient dynamics in an oceanic environment and for a single phytoplankton species depending on a single nutrient (and light) for its survival. In the
    first half of this talk, we will look closely into the linear stability problem for the trivial steady state (no phytoplankton) and use this asymptotic analysis to identify the emergence of localized structures. These turn out to fall into two categories: deep-chlorophyll maxima, in which the plankton concentration is localized in an interior point of the water column, and benthic
    layers, in which the plankton concentration is localized in the bottom of the column. This first half will close with a brief ecological interpretation of our findings. In the second half, we will look into the weakly nonlinear stability problem for the bifurcating deep-chlorophyll maxima. The peculiarity of this problem is focused around the existence of an infinite number of latent (non-bifurcating) modes which invariably have to be included in the analysis. Explicit asymptotic results may nevertheless be derived, not only close to the bifurcation point, but in a far bigger regime. In fact, the talk will conclude with the deep-chlorophyll maximum disappearing in a saddle-node bifurcation, thus offering its place to a secondary pattern 
  • Wednesday 3 February, 16:00-17:00, Room F-630. Speaker: Matteo Sommacal (La Sapienza, Rome).
    Title: Towards a Theory of Chaos Explained as Travel on Riemann Surfaces
    Abstract: Recently, a mechanism to explain the onset of irregular (chaotic) motions in a dynamical system, in terms of the singularity structure of its solutions, was introduced. The dynamics defined by a certain (paradigmatic) set of three coupled (complex) first-order ODEs, featuring two coupling constants, will be illustrated. It is shown that the system under study can be reduced to quadratures which can be expressed in terms of elementary functions. Despite the integrable character of the model, the general solution is a multiple-valued function of time (considered as a complex variable), and we investigate the position and nature of its branch points. For rational values of the coupling constants, the system is isochronous and explicit formulae for the period of the solutions can be given. For irrational values, the motions are confined but feature aperiodic motion. In this case, an argument is introduced to explain why sensitive dependence on initial conditions is expected. The system shows a rich dynamical behaviour that can be understood in quantitative detail since a global description of the Riemann surface associated with the solutions can be achieved. This toy model is meant to provide a paradigmatic first step towards understanding a certain novel kind of chaotic behaviour. This work has been carried out in collaboration with F. Calogero and P. M. Santini of the Università degli Studi di Roma "La Sapienza" (Italy) and D. Gomez-Ullate Oteiza of the Universidad Complutense de Madrid (Spain).

2009

  • Wednesday 9 December, 16:00-17:00, Room M6.16. Onno Bokhove (Twente). 
    Title: Variational water wave model with accurate dispersion and vertical vorticity
    Abstract: A new water wave model has been derived which is based on variational techniques and combines a depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite element profile with a small number of elements (say), leading to a framework for efficient modelling of the interaction of steepening and breaking waves near the shore with a large- scale horizontal flow. The equations are derived from a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical vorticity. It is shown that the potential flow water wave equations and the shallow-water equations are recovered in the relevant limits.
  • Wednesday 25 November, 16:00-17:00, Room M6.16. Antonios Zagaris (Twente). 
    This talk has been cancelled.
  • Wednesday 11 November, 11:00-12:00, Room P-656. Charlene Kalle (Warwick)
    Title: Beta-expansions and invariant measures

    Abstract: An expression of the form $x = \sum_{n=1}^{\infty} b_n / \beta^n$ with all numbers b_n in some fixed and finite set of real numbers A is called a \beta-expansion of x with digits in A. A \beta-transformation is a map from an interval to itself that can be used to generate beta-expansions of the numbers in its domain. We study the invariant measure that is absolutely continuous wrt Lebesgue of such a transformation. By constructing a version of the natural extension of the transformation we obtain an expression for the density function of this invariant measure.
  • Wednesday 28 October, 16:00-17:00, Room M6.16. Marco Mazzucchelli  (MPI Leipzig)
    Title: The Conley conjecture for Tonelli systems 
    Abstract: On a closed configuration space, a time-periodic Lagrangian is called Tonelli when its restriction to any fiber of T M is superlinear with positive definite Hessian. In this talk I shall prove that, for any Tonelli Lagrangian with global Euler-Lagrange flow, the associated Euler-Lagrange system admits infinitely many periodic solutions. More precisely, I will show that there are infinitely many contractible periodic orbits with a priori bounded mean action and either infinitely many of them are 1-periodic or their basic period is unbounded. This result confirms the Conley conjecture for Tonelli Hamiltonian systems on the cotangent bundle of closed manifolds
  • Wednesday 30 September, 16:00-17:00, Room M6.16. Felix Schlenk (Neuchâtel).  Title: Product Lagrangian tori in  tame symplectic manifolds Abstract: A product torus in standard symplectic space R^2n is the n-fold product of circles in the plane. A product torus in a symplectic manifold is the image of such a torus under a Darboux chart. I'll try to explain how such tori can be classified up to Hamiltonian isotopy. This is work joint with Yura Chekanov.
  • Wednesday 14 October, 16:00-17:00, Room M6.16. Carlota Cuesta (Nottingham).  Title:  Front propagation in a heterogeneous Fisher equation and in a
    pseudo-parabolic equation
    Abstract: We address front propagation into unstable states in two evolution PDEs of
    the types mentioned in the title. That is the mechanisms by which, under an initial perturbation of an unstable state, stable patterns `win' over the unstable state invading its domain.
    The homogeneous Fisher equation has been studied widely in terms of its front propagation properties. The steady state 0 is linearly unstable, whereas 1 is stable. An initial perturbation of 0 grows and propagates with speed that tends to a constant for large time; the solution approximates a travelling wave.
    Travelling wave solutions exist for all wave speeds larger than or equal to 2. In the Fisher case it is well-known that for sufficiently rapidly decaying initial data, the front selects the minimal wave speed, whereas for slow decaying initial conditions speeds faster than the minimal are
    realised. In this talk I will recall the formal approach that leads to the later results before giving the ones for the heterogeneous Fisher and the pseudo-parabolic equation. These can be summarised as follows: in terms of the front speed selection mechanism, the Fisher equation represents a border-line case of the heterogeneous one in a way that will be discussed. In the pseudo-parabolic equation the decay of the initial condition has no effect in front speed selection.
  • Wednesday 16 September, 16:00-17:00, Room M6.16. Gaia Lupo (Perugia).  Title: Initial/Moving Boundary Value Problems for the Nonlinear Heat Equation Abstract: A nonlinear evolution equation of diffusive type will be analysed as a
    mathematical model for heat conduction in high polymer systems and in simple
    monoatomic metals of Storm-type. In such a context, a half-line problem with a
    prescribed time-dependent heat flux at the origin will be presented and the
    construction of the Dirichlet-to-Neumann map on a moving boundary will be
    analysed. Some explicit examples will be also considered.
  • Wednesday 24 June, 14:30-15:30, room P.656: A. Momin (Max Planck Institute, Leipzig)
    Intersections and Contact Homology
    I will describe a version of cylindrical contact homology on the complement of a collection of closed, elliptic Reeb orbits. The given orbits are used as a barrier to restrict the gradient-like cylinders defining the differential using the phenomena of positive intersections of pseudoholomorphic curves.
  • Monday 6 April, 16:00-17:00, room P.656: I. Stamova (Bourgas Free University)
    Stability analysis of impulsive functional differential systems and applications
    Many real world processes and phenomena in nature, science and technology are characterized by the fact that system parameters are subject to short-term perturbations in time. An adequate apparatus for mathematical simulation of such processes and phenomena is the Impulsive Differential Equations. (IDEs) A natural generalization of IDEs is the Impulsive Functional Differential Equations (IDFEs). These equations are an adequate mathematical?model of the processes which are characterized by the change of jumps of their state as well as by the fact that the process under consideration depends on its history at each moment of time. The lecture is dedicated to a development of stability theory for IDFEs. Many applications and some open problems will be discussed.
  • Wednesday 25 March, 16:00-17:00, room P.656: J. van de Leur (Universiteit Utrecht)
    LU- factorization and hierarchies of differential equations
    Using the Birkhoff-factorization of Loop groups, i.e. , some kind of LU-factorization, one obtains in an elementary way a hierarchy of PDE's. I will show this for the loop group of type GL(n). The simplest equations in this hierarchy are the n-wave equations.
  • Wednesday 11 March, 16:00-17:00, room P.656: R. Driesse (UvA)
    Bifurcations from robust homoclinic cycles
  • Thursday 26 February, 16:00-17:00, room S6.48: H. Waalkens (RUG)
    Classical and Quantum Reaction Dynamics in Multidimensional Systems A system displays reaction type dynamics if its phase space possesses bottleneck type structures. Such a system spends a long time in one phase space region (the region of `reactants'), and occasionally finds its way through a bottleneck to another phase space region (the region of `products'), or vice versa. In Hamiltonian systems such bottlenecks are induced by equilibrium points of saddle-center-...-center type ('saddles' for short). The main approach to compute reaction rates is Transition State Theory which has its origin of conception in chemistry where it was invented by Wigner, Eyring and Polanyi in the 1930's. The main idea here is to compute the reaction rate from the flux through a dividing surface placed in the bottleneck (or in chemical terms 'transition state') region. In order not to overestimate the rate the dividing surface needs to have the so-called `no-recrossing' property which means that it is crossed exactly once by reactive trajectories and not crossed at all by nonreactive trajectories. The construction of such a dividing surface has posed a major problem in Transition State Theory since its invention. In the first part of my talk I will discuss in detail the phase space structures which govern the dynamics 'across' saddles, and how they can be computed from a normal form. This implies the construction of a dividing surface without recrossing. In fact, such a dividing surface is 'spanned' by a normally hyperbolic invariant manifold (NHIM) whose stable and unstable manifolds moreover form the phase space conduits for the reaction. The NHIM can be viewed as the mathematical manifestation of the transition state as an unstable invariant subsystem poised between reactants and products.
    In the second part of my talk I will discuss the quantum mechanics of reactions, and the role that the classical phase space structures play for these. This relationship can be studied in terms of a quantum normal form. The two main quantum imprints of the transition state are the quantization of the so-called cumulative reaction probability (the quantum analogue of the classical flux) and quantum resonances which describe the decay of wavepackets initialized on the transition state. The quantum normal form can be formulated as an explicit algorithm which, when implemented on a computer, leads to a very efficient method to compute both cumulative reaction probabilities and quantum resonances. The talk summarizes joint work with Roman Schubert and Stephen Wiggins from Bristol University.
  • Wednesday 11 February, 16:00-17:00, room C6.48: L. Sella (CWI)
    Algorithms for Computation of Symbolic Dynamics
    The theory of symbolic dynamics is a powerful tool to study discrete dynamical systems, in particular it is important to analyse qualitative properties of systems which exhibit chaotic dynamics. In the first part of this talk we present algorithms for the computation of symbolic dynamics and computation of topological entropy - a quantity which characterizes the level of chaos of the system - for one dimensional piecewise-continuous maps. We also show how to study the discrete dynamics of an hybrid system by applying results from symbolic dynamics to the return map of the hybrid system. In the second part of the talk we present algorithms for computation of symbolic dynamics and entropy of piecewise-affine two dimensional maps. For this case we show the implementation of a method based on the Conley index of decomposition of disconnected index pair and we compare it with the already well developed approach based on trellis of fixed point tangle. Finally we mention possible extensions of this first technique to higher dimensions.

2008

  • Wednesday 17 December, 16:00-17:00: Oleg Makarenkov (Imperial College, London)
    Topological degree approach to study bifurcations of periodic solutions in perturbed planar systems
    Classical conditions for bifurcation of periodic solutions from a cycle x0 in periodically perturbed hamiltonian systems are due to Melnikov [1963]. There were assumed that the cycle x0 is nonsingular and that the corresponding bifurcation function possesses a nonsingular zero θ The singular situation has been considered by Yagasaki [1996] and involves some implicit assumptions. The approach of both authors is based on the Lyapunov-Schmidt reduction. This talk proposes an alternative approach based on the topological degree theory and does not depend on whether the cycle x0 or zero θ is singular or not.
  • Thursday 23 October, 16:00-17:00, room S2.01: Georg Prokert (Technische Universiteit Eindhoven) Justifying the Thin Film approximation: A rigorous limit result for Stokes flow driven by surface tension In the moving boundary problem of Stokes flow driven by surface tension, we pass to the limit of small layer thickness. In an appropriate scaling, the limit evolution is given by the well-known Thin Film equation. While this is straightforward on the level of formal asymptotics, a rigorous analysis has to deal with the degeneracy of the limit which is reflected e.g. in the fact that a first-order evolution equation is replaced by a limit problem of order four. Our main techniques are uniform energy estimates in appropriately scaled Sobolev norms of sufficiently high order, based on parabolicity. This is joint work with M. Günther, Leipzig.
  • Wednesday 24 September, 16:00-17:00, room S2.05: Martijn Zaal/Blaz Mramor (VU University Amsterdam)
    Linear Stability of Osmotic Cell Swelling
    A basic mathematical model for cell swelling by osmosis will be introduced. Some results on the stability of the linearized problem when varying one of the parameters will be presented. 

    Unbounded non-singular strange attractors
    We construct a class of unbounded strange attractors. The construction is based on the so called geometric Lorenz attractor and hints at the existence of unbounded strange attractors, robust under uniform C^2 perturbations.
  • Wednesday 17 September, 14:30-15:30, room S2.03: Ferdinand Verhulst (Utrecht University)
    Resonance and near-resonance in a wave equation
    Normal mode manifolds, also called Lyapunov manifolds, arise naturally in ODEs and PDEs. A basic question is then whether these manifolds can be continued for small ε>0. A second basic question is whether the Lyapunov manifolds persist for increasing ε and other changes of relevant parameters. Possible tools to study these questions are averaging-normalization and numerical bifurcation theory. As we will show, the combination of both techniques is very powerful. We will describe two theorems that can be used in an infinite dimensional setting. The technique of averaging-normalization produces a short-cut to normally hyperbolic manifolds that emerge from the normalized equations because of the dominating presence of slow-fast dynamics. We demonstrate this for a parametrically excited wave equation. Our analysis shows that a complicated bifurcational stucture exists for relatively small values of the small parameter ε.
  • Wednesday 4 June, 16:00-17:00, room S2.01: Alberto Abbondandolo (Università di Pisa)
    On the asymptotic Maslov index
    The asymptotic Maslov index is a real number which can be attached to an invariant measure of a Hamiltonian dynamical system. We shall discuss its main properties, with special emphasis on systems with compact and simply connected configuration space, where the existence of invariant measures with a prescribed asymptotic Maslov index is easy to establish. We shall argue that the asymptotic Maslov index might play the role of the rotation vector in Mather theory.
  • Wednesday 21 May, 16:00-17:00, room S2.09: Peter van Heijster (CWI, Amsterdam)
    Front and pulse dynamics in a three-component system
    We study the dynamics of multi-front solutions of a specific three-component reaction-diffusion system. First, we briefly consider the existence and stability of the stationary patterns -- a 1-pulse or 2-front and a 2-pulse/4-front -- by singular perturbation and Evans functions techniques. Then, we use a renormalization group method to rigorously deduce the system of ODEs that govern the front dynamics. Based on our knowledge of the stationary points of this system, i.e. the stationary patterns, we are able to give an accurate description of the dynamics of N-front patterns (for N not too large).
  • Wednesday 7 May, 16:00-17:00, room S2.09: Jens Rademacher (CWI, Amsterdam)
    The Hyperbolic Continuum Limit of FPU chains: Dispersive and Nonclassical Shocks
    Towards the continuum description of FPU mono-atomic chain models in the hyperbolic space-time scaling, we systematically study discrete approximations to Riemann problems composed of constant states by numerical experiments. In the hyperbolic scaling the expected continuum model is a system of conservation laws so that solutions to Riemann problems provide building blocks for a general solution. The naive continuum limit is the p-system of mass and momentum conservation, but the FPU chain also conserves energy, which rules out the Lax-theory for shocks. For non-convex or non-concave flux the classical hyperbolic theory does not even apply to the p-system, and there exist energy conserving shocks. We rigorously study their occurrence and properties in the p-system, and, as a main new result, show that the macroscopic FPU chain can generate supersonic undercompressive shocks. Moreover, we show how non-classical Riemann solvers of the p-system must be adapted to give macroscopic Riemann solvers of the FPU chain in the case of at most one turning point of the flux.
  • Wednesday 9 April, 16:00-17:00, room S2.01: Jason Frank (CWI Amsterdam)
    Statistical accuracy in the numerical discretization of geophysical fluids
    In numerical weather prediction and climate research numerical simulations are conducted long beyond the point for which global errors have completely saturated the solution. This begs the question of whether any meaningful conclusions can be drawn from the simulation data. We will address this in an idealized setting by deriving equilibrium statistical mechanics theories for three related discretizations of the quasigeostrophic potential vorticity equation (Arakawa 1966), having mutually distinct conservation properties. Numerical experiments agree with the statistics in each case, indicatinging that statistical mechanics is a useful tool for numerical analysis. On the other hand, both the methods considered and the statistical theories resulting from them are known to fall short of the statistics of the continuum. We discuss why this is and propose an improved discretization.
  • Friday 14 March, 16:00-17:00, room F6.64: Sara Lombardo (VU University Amsterdam)
    Accelerated solitons: from integrable systems to nonlinear optics
    Accellerated solitons (also known as Boomerons) are soliton solutions of coupled nonlinear PDEs. Contrary to normal solitons they do not move as free particles but feature instead an accelerated motion. Introduced many years ago, boomerons remained strange waves, rather a mathematical curiosity than of physical interest. It has been recently realised that this is not the case; the breakthrough occurred when it was realised that also the well known 3 Wave Resonant Interaction equations, among other systems, possess boomeronic solutions. This opens the way to applications, particularly in nonlinear optics. In the first part of the talk I will review the state of the art and discuss a few optical processes in quadratic media; in the second part I will discuss solution techniques for a family of PDEs admitting boomerons.
  • Wednesday 13 February, 16:00-17:00, room S2.03: Michał Wojtylak (VU University Amsterdam)
    Krein spaces and definitisable operators
    Let us consider a complex Hilbert space (H,<,>) and a bounded, selfadjoint operator J such that J^2=I. It is not hard to see that J is a difference of two orthogonal projections. On the space H we define a new inner product by [f,g]=< J f,g>. Although this inner product is not positive definite, it allows us to define the adjoint of an operator. We will discuss spectral properties of selfadjoint (in the sense of the new inner product) operators. Moreover, we will introduce a class of definitisable operators and provide a version of a spectral theorem for this class. As an example a Sturm-Liouville operator will be considered.

2007


  • Wednesday 5 December, 13:30-14:30, room F2.53: Florian Wagener (University of Amsterdam)
    A stochastic bifurcation theory for discrete time stochastic systems
    We propose a bifurcation theory of smooth stochastic dynamical systems that are governed by everywhere positive transition densities. The local dependence structure of the unique strictly stationary evolution of such a system can be expressed by the ratio of joint and marginal probability densities; this 'dependence ratio' is a geometric invariant of the system. By introducing a weak equivalence notion of these dependence ratios we arrive at a bifurcation theory for which in the compact case, the set of stable (non-bifurcating) systems is open and dense.
  • Wednesday 21 November, 16:00-17:00, room S2.03: Chris Stolk (University of Twente)
    Reflection seismology in presence of multipathing
    Reflection seismology is a branch of seismolgy that uses reflected seismic waves to produce images of the Earth's subsurface. The method employs sources at the surface, that can emit a short pulse of seismic waves. The reflected signals are recorded using a large number of receivers at the surface. From these data, both the position of the reflectors, and the velocity function between the reflectors and the surface must be estimated. For this, signals from a large number of sources and receivers are combined. In this talk we discuss complications that occur when the waves can travel along different raypaths from a subsurface point to the surface, due to e.g. focussing of waves.
  • Wednesday 7 November, 16:00-17:00, room S2.01: Leo Maas (NIOZ)
    Internal Waves and Wave Attractors
    The abstract for this talk can be found here
  • Wednesday 24 October, 16:00-17:00, room S2.03: Jean-Philippe Lessard (Rutgers University and VU University Amsterdam)
    Toward a Computer-assisted Proof of an Old Conjecture in Delay Equations
    The scalar delay equation y'(t)=-ay(t-1)[1+y(t)] , (a>0) often called the Wright's equation is arguably one of the most studied equation in the field of nonlinear functional differential delay equations since the mid 50's. A conjecture that was made by Jones in 1962 is that the Wright's equation has a unique slowly oscillating periodic solution (SOPS), for every fixed a>pi/2. Partial progress in trying to prove the conjecture was made by Xie and Nussbaum in the 90's, but they could prove the result only for a>5.67. In this talk, we give a brief history of the problem and show how the use of rigorous numerical continuation techniques can help in getting significant progress toward the proof of the conjecture.
  • Wednesday 10 October, 16:00-17:00, room S2.03: Martin Bootsma (Utrecht University)
    Modeling of the transmission dynamics of Methicillin-resistant Staphylococcus aureus (MRSA)
    Antibiotic-resistance is primarily a hospital problem although increasingly, antibiotic-resistant bacteria are present in the animal and human reservoirs. In this talk, I will discuss a model (a branching process) to estimate the spreading capacity of a hospital clone of MRSA and a clone present among pigs and calves farmers. Furthermore, I will discuss a model, which mimics the so-called "Search & Destroy" policy of Dutch hospitals to prevent transmission of MRSA. I will also discuss what type of interventions are effective according to these models and what the potential benefits are of recently developed tests which can detect MRSA in microbiological samples of patients within hours instead of within 3 days.
  • Wednesday 26 September, 16:00-17:00, room S2.03: Hermen Jan Hupkes (Leiden University)
    Invariant Manifolds and Applications for Functional Differential-Algebraic Equations of Mixed Type
    Recently, differential equations involving both delayed and advanced arguments have appeared in an increasing number of models, originating from a wide variety of scientific disciplines. We present recent results concerning the existence of center manifolds and the occurrence of Hopf bifurcations for various types of such equations. In particular, we focus on a differential-algebraic equation with mixed arguments that arises from an economic life-cycle model and exhibit, both analytically and numerically, the presence of periodic cycles in the economy under consideration.
  • Wednesday 12 September, 16:00-17:00, room F1.31: Rein van der Hout (VU University Amsterdam)
    Discrete precipitation phenomena in a reaction-diffusion system
    Some reaction-diffusion systems exhibit strikingly regular precipitation-patterns, which have first been observed by R.E. Liesegang in 1896. We discuss a mathematical model, due to Keller and Rubinov. This model contains a nonlocal, discontinuous term which makes the analysis rather hard. We show that this model indeed predicts discrete precipitation bands, but many problems remain unsolved. This is joint work with M. Mimura (Meiji University, Tokyo), D. Hilhorst (Paris-Sud) and I. Ohnishi (Hiroshima).
  • Wednesday 13 June, 16:00-17:00, room S2.03: Peter van der Kamp (La Trobe University, Melbourne) Multi-sums of products
  • Wednesday 16 May, 16:00-17:00, room S2.01: Konstantin Mischaikow (Rutgers University)
    Modeling Transcriptional Control
  • Wednesday 2 May, 16:00-17:00, room C6.24: Ale Jan Homburg (University of Amsterdam)
    Randomly perturbed diffeomorphisms
    Using randomly perturbed circle diffeomorphisms as guide, I will review the dynamics and bifurcations of diffeomorphisms with bounded noise. Of particular interest are bifurcations where the support of a stationary measure explodes. I will explain this scenario and discuss quantitative characteristics. This is joint work with Hicham Zmarrou.
  • Wednesday 18 April, 16:00-17:00, room C6.38: Heinz Hanßmann (Utrecht University)
    On the destruction of resonant Lagrangean tori in Hamiltonian systems
    Poincaré's fundamental problem of dynamics concerns the behaviour of an integrable Hamiltonian system under a (small) non-integrable perturbation. Under rather weak conditions K(olmogorov)A(rnol'd)M(oser) theory settles this question for the majority of initial values. The perturbed motion is (again) quasi-periodic, the number of frequencies equals the number of degrees of freedom. KAM theory proves such Lagrangean tori to persist provided that the frequencies are bounded away from resonances by means of Diophantine inequalities.
    How do Lagrangean tori with resonant frequencies behave under perturbation? We concentrate on a single resonance, whence many n-parameter families of n-tori are expected to be generated by the perturbation; here n+1 is the number of degrees of freedom. For non-degenerate systems we explain the pattern how these families of lower-dimensional tori come into existence, and then discuss what happens in the presence of degeneracies.
  • Wednesday 7 March, 16:00-17:00, room R2.24: Eric Séré (Université de Paris - Dauphine)
    Periodic orbits of singular Hamiltonian systems
    In this talk, I will present a joint work with C. Carminati (University of Pisa) and K. Tanaka (Waseda University, Tokyo), JDE 230 (2006), 362-377.
    We consider a noncompact hypersurface H in R2N which is the energy level of a singular Hamiltonian of ``strong force" type. Under global geometric assumptions on H, we prove that it carries a closed characteristic, as a consequence of a result by Hofer and Viterbo on the Weinstein conjecture in cotangent bundles of compact manifolds. Our theorem contains, as particular cases, earlier results on the fixed energy problem for singular Lagrangian systems of strong force type.
  • Wednesday 7 February, 15:30-17:15, room TBA: Otto van Koert (ULB, Brussel)
    Introduction to contact homology
    In this talk, we will give an informal introduction to contact homology, a powerful invariant of contact manifolds. It is based on studying the dynamics of the Reeb field and can also be used to detect the existence of closed Reeb orbits. It works, roughly speaking, as follows.
    We set up an action functional whose critical points are closed Reeb orbits. Then we do Morse homology on this action functional. In other words, we study a suitable gradient flow of the functional. This gives rise to a homology theory which turns out to be an invariant.
  • Wednesday 24 January, 16:00-17:00, room F3.01: Yves van Gennip (TU Eindhoven)
    Blending polymers with mathematics
    Pattern formation is a common phenomenon in nature, ranging from the stripes on fishes and patterns in the desert sand to magnetisation of iron and convection in hot water. A prime example of pattern formation is given by diblock copolymers, two mutually repelling polymers chemically bonded together. During my talk we will consider a variational model for diblock copolymer/homopolymer blends. Due to competing influences on different length scales complicated patterns arise on intermediate scales. We will clearly see these competing influences in our model. In one dimension the model is understood and we will see some results in this case. The close relation between this model and a model for lipid bilayers by Peletier-Roeger will be examined and inspired by properties of this latter model we will look for the possibility of 'partial localisation' in the polymer model, i.e. the tendency of the polymers to aggregate on lower dimensional sets, e.g. surfaces in three dimensional space.

2006


  • Wednesday 13 December, 14:00-15:00, room F6.64: Jens Rademacher (CWI Amsterdam)
    The saddle-node of nearly homogeneous wave trains in reaction- diffusion systems
    In joint work with Arnd Scheel (University of Minnesota) we study the saddle-node bifurcation of a spatially homogeneous oscillation in a reaction-diffusion system posed on the real line. We use a novel Liapunov-Schmidt reduction applicable to certain singularly perturbed situations and investigate existence and stability of wave trains with large wavelength that accompany the homogeneous oscillation. We find two different scenarios of possible bifurcation diagrams which we refer to as elliptic and hyperbolic. In both cases, we find all bifurcating wave trains and determine their stability on the unbounded real line. We confirm that the accompanying wave trains undergo a saddle-node bifurcation parallel to the saddle-node of the homogeneous oscillation, and we also show that the wave trains necessarily undergo sideband instabilities prior to the saddle-node.
  • Monday 4 December, 16:00-17:00, room C6.24: Chris Wendl (MIT, Boston)
    Holomorphic foliations and Reeb dynamics in dimension 3
    The Weinstein conjecture asserts that periodic orbits always exist for a certain class of (Reeb) vector fields on odd-dimensional (contact) manifolds. These vector fields arise naturally, e.g. as Hamiltonian systems restricted to star-shaped energy hypersurfaces in phase space. We will focus on two recent results related to this conjecture in the 3-dimensional case: the theorem of Hofer, Wysocki and Zehnder, that generic star-shaped energy surfaces always admit either 2 or infinitely many periodic orbits, and another due to Abbas, Cieliebak and Hofer, that the conjecture holds whenever the vector field is determined by a so-called "planar" contact form. Both are obtained using some distinctly low-dimensional features of Gromov's theory of pseudoholomorphic curves, which can sometimes be used to construct singular foliations of a contact 3-manifold. These foliations have some remarkable compactness properties, and suggest a program for solving the Weinstein conjecture in dimension 3.
  • Wednesday 15 November, 14:00-15:00, room H3.58: Guido Carlet (VU Amsterdam)
    Integrable systems and Toda hierarchy
  • Wednesday 1 November, 14:00-15:00, room C6.38: Pieter Eendebak (Utrecht)
    Contact structures and projection methods
    In my talk I will present to two themes in my disseration: contact structures and projection methods. I will assume the listener has a basic notion of a partial differential equation and knows the basics of differential geometry. In particular the words manifold, vector field, the Lie bracket of vector fields and vector bundles should ring a bell.
    Contact structures are used to study the geometry of partial differential equations. Instead of looking at functions and equations we look at integral manifolds and geometric structures. Symmetry methods for (partial) differential equations have been introduces by Sophus Lie in the 19th century. We formulate the symmetry method as a projection method. Then we show that another method, the method of Darboux, is an example of this projection method as well.
  • Wednesday 18 October, 14:00-15:00, room F6.30: Daan Crommelin (CWI Amsterdam)
    Reconstruction of effective stochastic dynamics from data
    Construction of stochastic models that describe the effective dynamics of observables of interest is an useful instrument in various fields of application, such as physics, climate science, and finance. I will discuss a new technique for the construction of such effective models from timeseries. The approach centers on the minimization of an object function that measures the difference between the eigenspectrum of the generator of the stochastic process (for example, the Fokker-Planck operator) and a reference eigenspectrum obtained from the data.
  • Monday 16 October, 16:00-17:00, room F6.54: Klaus Niederkrüger (Université Libre de Bruxelles)
    Contact manifolds that are not energy hypersurfaces
    The natural paradigm for contact manifolds are certain energy level sets of Hamiltonian systems. Not all contact manifolds are of that type though. We describe a powerful obstruction to realizing a given manifold as such a level set.
  • Wednesday 4 October, 14:00-15:00, room F6.30: Konstantinos Efstathiou (Groningen)
    Fractional monodromy
    The monodromy of the Lagrangian fibre bundles that appear in the study of integrable Hamiltonian systems has attracted a lot of attention, since it is the coarsest obstruction to the existence of global action-angle variables and it was not known classically. Recently, a generalization of the standard notion of monodromy, called fractional monodromy, was introduced by Nekhoroshev, Sadovskii and Zhilinskii. Fractional monodromy appears in certain integrable Hamiltonian systems for which the integral map has a family of singular fibers that correspond to hyperbolic periodic orbits with reflection. We describe fractional monodromy in a system in 1:-2 resonance by constructing a concrete basis of the homology group of the fibers and 'passing' part of the basis through the family of singular fibers. In this construction we use extensively the dynamics of the system.