The talks of the last decade of the Amsterdam-Leiden-Delft Seminar (previously UVA-VU dynamic analysis seminar):
2019
Tuesday 25 June, 15:00-17:00, MI 402 @Leiden
15:00-15:45: Robbin Bastiaansen
Title: Behaviour of self-organised vegetation patterns in dryland ecosystems
Abstract: Vast, often populated, areas in dryland ecosystems face the dangers of desertification. Loosely speaking, desertification is the process in which a relatively dry region loses its vegetation - typically as an effect of climate change. As an important step in this process, the lack of resources forces the vegetation to organise itself into large-scale patterns. The behaviour of these patterns can be analysed using (conceptual) reaction-(advection)-diffusion models, in which these patterns present themselves as localized structures (e.g. as pulse solution). In this talk, first I will present the results of a comparison between conceptual model and real vegetation pattern characteristics. Subsequently, I will explain how the found multistability leads to novel adaptation mechanisms, which can be understood via a mathematical analysis of the dynamics of (disappearing) semi-strong interacting pulses in an ecosystem model with parameters that (may) vary in time and space.
Coffee Break
16:15-17:00: Yves van Gennip
Title: Variational methods on graphs with applications in imaging and data classification
Abstract: Applications that can be described by variational models profit from all the advantages those models bring along. Both on the functional level as on the level of the associated differential equations, powerful techniques have been developed over the years to study these models. Up until fairly recently, such models were typically formulated in a continuum setting, i.e. as the minimization of a functional over an admissible class of functions whose domains are subsets of Euclidean space or Riemannian manifolds. The field of variational methods and partial differential equations (PDEs) on graphs aims to harness the power of variational methods and PDEs to tackle problems that inherently have a graph (network) structure.In this talk we will encounter the graph Ginzburg--Landau model, which is a paradigmatic example of a variational model on graphs. Just as its continuum forebear is used to model phase separation on a continuum domain ---it assigns to each point of the domain a value from an (approximately) discrete set of values--- the graph Ginzburg--Landau model describes phase separation on the nodes of a graph. This makes it extremely well suited for applications such as data clustering, data classification, community detection in networks, and image segmentation.Theoretically there are also interesting questions to ask, often driven by the properties that have already been established for the continuum Ginzburg--Landau model, such as Gamma-convergence properties of the functional and relationships between its associated differential equations. This presentation will give an overview of some recent developments.
2018
Tuesday, 30 April, 15:00-17:00 @VU Amsterdam
15:00-15:45 Bente Bakker
Title: Conley-Floer theory for waves in lattices
Abstract: The focus of this talk is on lattice differential equations. An important class of solutions are so-called travelling waves, which can be formulated as connecting orbits in a differential equation involving both forward and backward delay terms. In this talk I will present a new existence/forcing theorem for monostable waves. This relies on a novel topological invariant which I call the Conley-Floer index of the system.
Coffee break
16:00-16:45 Jan-David Salchow
Title: The L-infinity structure on symplectic cohomology
Abstract: Symplectic cohomology is a variant of Floer cohomology for symplectic manifolds with boundary. It can be related to the cylindrical contact cohomology of an associated stable hamiltonian structure. The extra information contained in full contact cohomology can be encoded as an L-infinity structure on symplectic cohomology. I will explain what goes into this construction of an L-infinity structure and hint at its relevance.
Tuesday, 26 March, 15:00-17:00, MI403 @Leiden
15:00-15:45 Timothy Fraver
Title: Traveling waves and nanopterons in Fermi-Pasta-Ulam-Tsingou lattices
Abstract: Infinite lattices of nonlinearly coupled oscillators are prototypical models of wave propagation in granular media. By tuning the material parameters of a lattice to certain limits, we can produce different kinds of wave behavior in the lattice. In particular, we can excite exact periodic traveling wave solutions to the lattice equations of motion and may possibly construct homoclinic connections between these periodic ``tails’’ and a particular exponentially localized solution that exists when the material parameter reaches its critical limit. Such waves formed by the superposition of a periodic oscillation and an exponentially localized profile are called nanopterons. We give an overview of recent and ongoing investigations in the existence and properties of nanopterons in the long wave and equal mass limits for diatomic Fermi-Pasta-Ulam-Tsingou lattices and the small mass limit for mass-in-mass lattices.
Coffee Break
16:00-16:45 Thomas Rot
Title: The classification of homotopy classes of proper fredholm maps.
Abstract: Non-linear existence problems can attacked with topological methods. For example a map between closed manifolds of the same dimension is surjective if the degree is non-zero. The degree is invariant under a large class of deformations, which allow one to solve complicated non-linear problems. Framed cobordism is another invariant for maps between manifolds of different dimensions. In this talk I will discuss joint work with Alberto Abbondandolo in which we generalize the theory to an infinite dimensional setting. I will not assume any knowledge of the finite dimensional theory.
Wednesday, 27 February, 15:00-17:00, room WN-M648 @VU
15:00-15:45: Oliver Fabert
Title: Pseudo-holomorphic curve methods for Hamiltonian PDE
Abstract: Many important classes of nonlinear PDEs can be viewed as infinite-dimensional Hamiltonian systems. For finite-dimensional Hamiltonian systems, A. Floer has defined a homological invariant more than 30 years ago, which can be used to establish lower bounds for the number of time-periodic orbits. It uses so-called pseudo-holomorphic curves. In my talk I plan to report on ongoing work on the foundations of infinite-dimensional symplectic geometry and towards generalizing Floer theory in all its flavors from finite to infinite dimensions. As a concrete application I illustrate how the Arnold conjecture for finite-dimensional cotangent bundles generalizes to a lower bound for the number of time-periodic solutions of non-linear wave equations.
Coffee Break
16:15-17:00: Sonja Hohloch
Title: Floer homology for non-primary homoclinic points
Abstract: Floer homology was originally designed for counting the number of intersection points of two `nice' Lagrangian submanifolds in a symplectic manifold. Since the stable and unstable manifold of a hyperbolic fixed point of a symplectomorphism are Lagrangian submanifolds, one may ask if there exists a Floer homology for this particular intersection problem. Since the (un)stable manifolds are usually only injectively immersed, not compact and `very wiggling' the intersection problem is very complicated and in particular not very `analysis-friendly'. Intersection points of the stable and unstable manifold of the same fixed point are called `homoclinic points’. We showed in earlier works that, for symplectomorphims on 2-dimensional symplectic manifolds, it is possible to replace parts of the analysis necessary for the construction of Floer theory by combinatorics and obtain a well-defined Floer homology generated by a special class of homoclinic points, called `primary points’. In this talk, we will recall the construction of Floer homology for primary points and sketch our ideas how to generalize it to non-primary points.
Wednesday, 21 November, 15:00-17:00, WN-P656 @VU Amsterdam
15:00-15:45 Onno van Gaans (LU)
Title: Partially ordered vector spaces by means of embedding
Abstract: Many of the familiar function spaces used in analysis are naturally equipped with a vector space structure and a norm or topology. If we consider real valued functions, these spaces also have a natural partial order, which leads to the notion of a partially ordered vector space. The general theory of partially ordered vector spaces is poor. For vector lattices, which are partially ordered vector spaces in which every set of two elements has a least upper bound, a much richer theory is available. Since the 1990s an approach has been developed of studying partially ordered vector spaces by means of embedding in vector lattices. This approach turns out to be fruitful for spaces that allow an ``order dense'' embedding in a vector lattice. Such spaces are called pre-Riesz spaces. An overview of the theory of pre-Riesz spaces will be given with a focus on the notion of disjointness. Some recent results on disjointness preserving operators will be mentioned as well.
Coffee break
16:15-17:00 Magnus Botnan (VU)
Title: Geometry and topology in neural data
Abstract: Understanding what drives neuronal activity is an active of research. For example, it is well-known that the activity level of a place cell in a rodent is a function of the rodent's spatial position as well as its head direction. However, there are most certainly other, possibly unknown, driving forces. In this talk I will discuss a recent framework, based on algebraic topology, to uncover topological properties of a-priori unknown covariates. This is joint work with Gard Spreemann, Benjamin Dunn and Nils Baas.
Drinks and snacks at The Basket
23 May: Stefanie Sonner (Nijmegen), 16:00-17:00, Room WN-623
Title: Exponential attractors for nonautonomous and random dynamical systems
Abstract: Exponential attractors of infinite dimensional dynamical systems are
compact, semi-invariant
sets of finite fractal dimension that attract all bounded subsets at an
exponential rate. They
contain the global attractor and, due to the exponential rate of
convergence, are generally
more stable under perturbations than global attractors.
In the autonomous setting, exponential attractors have been studied for
several decades and
their existence has been shown for a large variety of dissipative
equations. More recently, the
theory has been extended to non-autonomous and random problems. We
discuss general existence
results for exponential attractors for non-autonomous and random
dynamical systems
in Banach spaces and derive explicit estimates on their fractal
dimension. As an application
semilinear heat and semilinear damped wave equations are considered.
This is joint work with Tomas Caraballo (University of Sevilla) and
Alexandre Carvalho (University of São Paulo).
09 May: Kees Vuik (TU Delft), 16:00-17:00, Room F3.20 (KdV, UvA)
Title: Scalability and Accuracy of Helmholtz solvers
Abstract: In the 20 years of research on the Helmholtz problem, the focus has either been on the accuracy of the numerical solution (pollution) or the acceleration of the convergence of a Krylov-based solver (scalability).While it is widely recognized that the convergence properties can be investigated by studying the eigenvalues, information from the eigenvalues is not used in studying the pollution error. Our aim is to bring the topics of accuracy and scalability together; instead of approaching the pollution error in the conventional sense of being the result of a discrepancy between the exact and numerical wave number, we show that the pollution error can also be decomposed in terms of the eigenvalues.Recent research efforts aimed at iteratively solving the Helmholtz equation has focused on incorporating deflation techniques for GMRES-convergence accelerating purposes. The requisite for these efforts lies in the fact that the widely used and well acknowledged Complex Shifted Laplacian Preconditioner (CSLP) shifts the eigenvalues of the preconditioned system towards the origin as the wave number increases. The two-level-deflation preconditioner combined with CSLP (ADEF) showed encouraging results in moderating the rate at which the eigenvalues approach the origin. However, for large wave numbers the initial problem resurfaces and the near-zero eigenvalues reappear.Our findings reveal that the reappearance of these near-zero eigenvalues occurs if the near-singular eigenmodes of the fine-grid operator and the coarse-grid operator are not properly aligned. This misalignment is caused by accumulating approximation errors during the inter-grid transfer operations. We propose the use of higher-order approximation schemes to construct the deflation vectors. The results from Rigorous Fourier Analysis (RFA) and numerical experiments confirm that our newly proposed scheme outperforms any deflation-based preconditioner for the Helmholtz problem.
25 April: Konstantinos Efstathiou (Groningen), 16:00-17:00, Room F3.20 (KdV, UvA)
Title: Monodromy and Circle Actions
Abstract: Standard Hamiltonian monodromy was introduced by Duistermaat as an obstruction to the existence of global action-angle coordinates in integrable Hamiltonian systems. It refers to the monodromy of torus bundles that typically exist in such systems. Fractional Hamiltonian monodromy, introduced by Nekhoroshev, Sadovskií, and Zhilinskií, generalizes standard monodromy by considering not only torus bundles but also more general fibrations with singular fibers. In this talk I present results concerning both standard and fractional monodromy that were recently obtained in collaboration with Nikolay Martynchuk. It turns out that, in integrable Hamiltonian systems with a Hamiltonian circle action, both standard and fractional monodromy can be solely determined through a careful study of the fixed points of the circle action and their weights. A basic ingredient of this approach is the definition of generalized parallel transport of homology cycles. These results will be demonstrated in several examples of integrable Hamiltonian systems.
28 March: Jason Frank (Utrecht), 16:00-17:00, Room WN-P640
Title: Tangent-space splittings for data assimilation
Abstract: Data assimilation methods are used for marrying instrumental observations of a physical system to numerical prediction models. There are many flavors, depending on whether one takes a probabilistic/statistical, control theoretic, or dynamical systems point of view. Furthermore there are variational methods that consider a whole time window and sequential methods that proceed step-by-step. In this talk I will consider the relationship between the observation operator and the decomposition of the model tangent space in terms of Lyapunov exponents/vectors. The main conclusion is that the observations should constrain the unstable tangent space. Using this point of view we construct two methods, one variational and one sequential and discuss their convergence behavior. Along the way I will mention some other structural considerations in data assimilation.
14 March: Heinz Hanßmann (Utrecht), 16:00-17:00, Room WN-S623
Title: Bifurcations and Monodromy of the Axially Symmetric 1:1:-2 Resonance
Abstract: We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:-2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational symmetry about the vertical axis. Introducing a detuning parameter we find a rich bifurcation diagram, containing three parabolas of Hamiltonian Hopf bifurcations that join at the origin. We describe the monodromy of the resulting ramified 3-torus bundle as variation of the detuning parameter lets the system pass through 1:1:-2 resonance.
28 February: Alef Sterk (Groningen), 16:00-17:00, Room F3.20 (KdV, UvA)
Title: Extreme value laws for dynamical systems
Abstract: Extreme value theory for chaotic, deterministic dynamical systems is a rapidly expanding area of research. Given a dynamical system and a real-valued observable defined on its state space, extreme value theory studies the limit probabilistic laws for asymptotically large values attained by the observable along orbits of the system. Under suitable mixing conditions the extreme value laws are the same as those for stochastic processes of i.i.d. random variables. In this talk I will discuss the classical results for i.i.d. processes, some recently obtained results for dynamical systems, and promising directions for future research.
14 February: Fabian Ziltener (Utrecht), 16:00-17:00, Room WN-S623
Title:Coisotropic submanifolds of symplectic manifolds and leafwise fixed points for C^0-small Hamiltonian flows
Abstract: Consider a symplectic manifold (M,\omega), a closed coisotropic submanifold N of M, and a Hamiltonian diffeomorphism \phi on M. A leafwise fixed point for \phi is a point x\in N that under \phi is mapped to its isotropic leaf. These points generalize fixed points and Lagrangian intersection points. In classical mechanics leafwise fixed points correspond to trajectories that are changed only by a time-shift, when an autonomous mechanical system is perturbed in a time-dependent way. J. Moser posed the following problem: Find conditions under which leafwise fixed points exist. A special case of this problem is V.I. Arnold's conjecture about fixed points of Hamiltonian diffeomorphisms. In my talk I will provide the following solution to Moser's problem. Namely, leafwise fixed points exist, provided that the Hamiltonian diffeomorphism is the time-1-map of a Hamiltonian flow whose restriction to N stays C^0-close to the inclusion N\to M.
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,
Vincent 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.