The Amsterdam-Leiden Seminar is organized jointly by VU University Amsterdam and Leiden University.

The seminar (usually) takes place once per month on Wednesday afternoon.

For more information, please contact the organizers: Rob van der Vorst (VU) and Vivi Rottschäfer (LU)

A database of earlier years' seminars (including the former VU-UvA dynamical analysis seminar) can be found here.

And a route description can be found here.

## Upcoming talks in 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.

## Previous talks in 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 Frave r
**

*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.

**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:***Ge****ometry 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.