In particular, I am involved in the following research problems:
Connecting Orbits in Nonlinear Systems
Patterns are all around us: our fingerprints, the red spots typical for measles, the stripes on zebras, and the convection rolls in the atmosphere and oceans which shape our climate. Much is known about how such patterns start to develop from a homogeneous state, but we are still unable to predict or analyze the features of fully developed patterns. To analyse the nonlinear problems that describe such complicated dynamics, we combine detailed computer-assisted calculations with abstract mathematics.This research has two principle components. On the one hand, numerical simulations produce clear and detailed pictures of the dynamics, but the reliability of this information is hard to quantify precisely. The other ingredient is topology, which is a branch of abstract mathematics where much detailed information is ignored in order to focus on robust, global properties. We combine the computer calculations with topology in a mathematically rigorous way through analytic methods, so that we can better understand evolution problems for finite and infinite dimensional dynamical systems, such as pattern formation phenomena.
The goal is to create topologically validated computational machinery for finding the pivotal objects of interest in large amplitude pattern formation: the paths along which dynamical systems change from one state into another. These connections play an organizing role. First, connecting orbits describe localized patterns such as pulses and boundary layers. Second, they act as building blocks from which more complicated, sometimes chaotic, patterns can be constructed. Third, the transitions between different patterns, and their emergence from a homogeneous background, are connections between structured and trivial equilibrium states.
Unified Floer Theory of Braided Dynamics
Braids and knots can be made from pieces of string, shoelaces and hair. Some knots are easy to unravel, some difficult, and for other knots it is not even clear if they can be disentangled at all. In mathematics we describe braids and knots by curves that do not intersect in a three-dimensional space. The obstructions in manipulating and disentangling these curves are characterized mathematically by the "topology" of all types of knots and braids.The topological obstruction of braids and knots affect the behaviour of solutions certain nonlinear (partial) differential equations These equations have their origins in models from physics, chemistry and biology. We use the differential equations to describe the time evolution (dynamics) of curves in a three-dimensional space. The special feature of the differential equations that we study is that in the course of the time they make all braids simpler, never more complicated. Therefore we say that such a differential equation "respects" the topological structure of braids.
We have uncovered such a link between the topology of braids and the analysis of differential equations, two very different branches of mathematics, for three types of differential equations. For each of these types, we have investigated this relationship. It turns out that we must consider braids consisting of strings/curves with two different colors. Mathematically, we can express the relationship between differential equations and braiding in term of so-called invariants, roughly a set of numbers that depends only on the kind of braid and not on the details of the differential equation: they are algebraic-topological invariants of two-color braids.
For the three different types of differential equations we now have three a priori different invariants for the same braid. But we suspect that all three invariants are essentially the same! If we manage to prove that, we have discovered a new intimate relationship between the three types of differential equations on the one hand and the topology of braids with strings of two colors on the other hand.
Computational Conley index and Floer homology
The beautiful and highly complex patterns that dynamical systems generate are often difficult to capture analytically. Numerical calculations can be helpful, even more so when they can be put on a rigorous footing. Such computational methods require a finite dimensional, bounded, discrete setting, whereas dynamical systems can be infinite dimensional, unbounded and continuous. We therefore need a reduction which on the one hand is computationally friendly while on the other hand still captures the essential features of the original system.To that end we use topological tools such as Conley index theory, a powerful extension of Morse theory applicable to dynamical systems in general (maps and flows). Due to its nature as a topological tool it has various stability properties built in such as insensitivity under small perturbations of the system, as well as large deviations under the appropriate isolation criteria. This very powerful technique is based on the fundamental theorem of dynamical systems (Conley's decomposition theorem): any dynamical system can be decomposed in recurrent sets and a gradient flow between the recurrent sets.
To be more concrete, Conley index theory relates information of the dynamics on the boundary of a set to the minimal dynamics in the interior of that set. The link is made with the help of (computational) homology. The next challenge is to make Floer homology, a Morse theory in a truly infinite dimensional setting, amenable to computations.
Variational and topological methods in dynamics
Variational and topological methods can be used to study a wide variety of problems in dynamical systems and (nonlinear) partial differential equations. A prime example is the application of these techniques to low-dimensional dynamics in combination with the topological structure of braids and knots. The maximum principle for elliptic and parabolic equations has far-reaching consequences when interpreted in the context of topological braids. For discretized parabolic equations intricate methods have been developed to obtain forcing results based on Conley index theory. The Conley index assigned to a braid type turns out to be independent of the representation of the braid, and is thus a braid invariant. A typical result is that the presence of certain types of periodic orbits implies that a system is chaotic. On the practical side, algorithms for calculating the Conley index (homology) have been created. While the Conley index results are restricted to positive braids, a Floer homology invariant may be assigned to arbitrary braid classes.A related topic concerns the classical problem under what conditions the dynamics on an energy manifold contain periodic behavior (i.e., there are closed characteristics). For closed manifolds this problem is known as the Weinstein conjecture, and it has been (and still is) being studied in great generality and with much success in the past three decades. The focus in our group is on unbounded energy manifolds, for which the results are few and far between. A variational approach, using homological linking has led to a general existence result for closed orbits on non-compact energy surfaces that meet certain topological requirements. In particular, the top half of the homology groups of such non-compact manifolds play a deciding role. This results has solved the Weinstein conjecture for the non-compact case for a special class of non-compact energy manifolds. Current research is targeted towards applying and extending techniques of symplectic and contact topology to this problem.
Asymptotics and blowup in geometric PDEs
Harmonic map heat flows for vector valued functions in which the target space is a manifold, the unit sphere being the most commonly studied case, belong to a class of problems motivated by applications in geometry, and also appear as a benchmark model for nematic liquid crystals. They may be characterized as the gradient flow of the usual Dirichlet energy. In that sense they generalize the standard heat equation. The interest is in the formation of singularities forced to occur by topological obstructions preventing the usual large time existence and convergence to energy minimizers. Our results rely both on (formal) matched asymptotic expansions and on rigorous techniques.Simultaneously we are studying related questions for the much harder Willmore flow, which concerns the gradient flow of the total curvature of a manifold. Recently, we have also directed our attention on Free Boundary Problems in cell biology, in particular on models in cell biology (tip growth in cell boundaries and cell swelling due to osmosis) in which curvature effects play a role. These models have been proposed in the biological literature. The tip growth problem is strongly related to the inverse mean curvature flow studied in the context of geometrical problems and the cell swelling model leads to gradient flow formulations in which the Wasserstein metric is employed.