I work on nonlinear partial differential equations and dynamical systems,
often leading to pattern formation. I am interested in developing
mathematical methods for such problems, with an open mind for
applications. Subjects include (formal) asymptotic techniques, (rigorous)
computational methods, the harmonic map heat flow (and other geometric
flows), Hamiltonian systems, fourth order differential equations,
biomathematics, periodically modulated travelling waves, Conley index and
Floer homology. I especially enjoy topological and variational methods.
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.