Master projects for Mats Ehrnström
My general web page is here. You can always send me an E-mail or visit me in my office.
My expertise is in nonlinear differential equations, with or without free boundaries, and typically describing wave motion. I take a particular interest in nonlocal Fourier operators, and Fourier techniques to treat nonlinear equations. Typical questions concern the existence, uniqueness and properties of travelling waves. I have also done work on the existence of solutions to time-dependent problems.
More recently, I take an interest in the connection between differential equations and neural networks, and more generally in connections between different models for capturing laws or principles from data.
Some examples of my research can be found here:
Supervision: I have supervised a number of students, almost all of whom have gone on to pursue a PhD in mathematics. We meet once a week, half an hour (more when needed), and additional support is provided via my group of PhD students and postdocs. The thesis often consists of one part of standard material (not new results, but phrased and collected by the candidate), and one part where the student did work that is in some respect novel and that cannot be found anywhere else. Some of my former students and their theses can be found online (see my CV on this page for the list of students).
Prerequisites: One can read up on most topics, but the most common in a thesis work with me would be integration theory (mainly to know what \(L^2\) and \(L^p\) are), Fourier analysis (to know the Fourier transform, periodic and/or on the line), and the very basic concepts of differential equations. Function spaces are where we look for solutions, so one needs the concepts of Hilbert and Banach spaces.
Themes within ODEs/dynamical systems
Fixed-point theorems and their use in ordinary differential equations
This project aims at understanding the use of (different) fixed-point theorems in the theory of ODEs. Classically, the main existence and uniqueness theorem for ordinary differential equations (the Picard–Lindelöf theorem) is proved using the Banach fixed-point theorem on the space of (bounded and uniformly) continuous functions on a compact interval. This, however, is not the only way forward: other fixed-point theorems, and other assumptions, can be used to produce different results. In this project you will learn some classical fixed-point theorems and how they can be applied to different problems in ODEs; you will be able to solve differential equations with prescribed boundary data 'at infinity' (asymptotic integration); and you will encounter some smaller research problems.
Prerequisites: Ordinary differential equations/Dynamical systems, Linear Methods (or Functional Analysis).
N.b. It is possible to vastly extend the scope of this project by considering more general operators than those associated with ordinary differential equations. For example, general evolution equations (including many PDEs and pseudodifferential equations) can sometimes be successfully posed as an ODE in infinite-dimensional function spaces.
The Hartman-Grobman and Centre Manifold theorems for dynamical systems
Every autonomous system of ODE's describes the evolution of a corresponding dynamical system. Constant solutions correspond to equilibria, or fixed points, for the corresponding evolution operator. Near such a point, any linear autonomous system can be completely described and understood in terms of stable and unstable eigenspaces, corresponding to the real parts of the eigenvalues of the operator; eigenspaces corresponding to negative real parts are asymptotically stable (solutions in these spaces approach the equilibrium), and eigenspaces corresponding to positive real parts are unstable (solutions in these spaces do not remain 'near' the equilibrium); when there are eigenvalues on the complex axis (zero real parts) the situation is also more complex and there exist solutions somewhere in between the two other cases. The Hartman–Grobman theorem states that if the linearization of a nonlinear dynamical system at an equilibrium has no eigenvalues on the complex axis, the system can locally be completely described by its linearization (so that we know from the linear theory how the solutions behave near the equilibrium). The Centre manifold theorem deals with the situation when there are also eigenvalues with zero real part, and describes the solution space in terms of three manifolds (which are solution subspaces for linear dynamical systems): a stable, an unstable, and a centre manifold.
In this project you will study the proofs and possible applications of the classical Hartman–Grobman and and Centre Manifold theorems. The range can be extended to include also centre manifolds for infinite-dimensional dynamical systems (which then requires a somewhat better background in functional analysis) .
Prerequisites: Ordinary differential equations/Dynamical systems, Riemannian or differential geometry (or any subject which helps you know what a \(C^k\)-manifold is).
Bifurcation theory for ordinary differential equations
Themes within PDEs
Prerequisites all PDE-projects: Ordinary Differential Equations, Partial Differential Equations, Linear Methods (or Functional Analysis), Real analysis (basically, some form of integration theory).
Bifurcation theory for partial differential equations (existence-theory for stationary nonlinear waves)
Bifurcation describes a sudden qualitative change in a system, such as the branching of a trivial family of solutions of a parameter-dependent equation into two qualitatively different families of solutions as the parameter passes some critical value. For PDEs—or any equation that is posed in the form of an operator on infinite-dimensional Banach spaces—a main problem is the infinite dimension of solution spaces. In some cases, a so-called Lyapunov–Schmidt reduction can be used to reduce the problem to a finite-dimensional setting. In this project you will learn (basic) bifurcation theory and see how it can be applied to problems in fluid mechanics. Time permitting, ongoing research problems can be approached.
Sobolev spaces, calculus of variations, and solutions of PDE:s as minimizers
Solutions of many problems—such as elliptic boundary-value problems—can be seen as minimizing elements of certain functionals on abstract Hilbert or Banach spaces. In the language of physics, a solution minimizes the energy of a system. This project introduces you to some very nice Hilbert spaces, called Sobolev spaces, and puts them as the solution spaces of the posed problems. You will work your way from equations to functionals, characterize solutions in terms of minimizers, and see how to find them using functional analysis (calculus of variations). Similar methods can be used to obtain (optimal) approximate solutions in finite-dimensional subspaces. The aim of the project is mainly to understand and formulate these methods, but can include (differing levels of) research problems.
Waves and solitons (Linear and nonlinear water waves)
This is my main field.
Distributions, the Schwarz space and solutions of linear PDE:s Linear functionals on the space of continuous functions \(\mathbb R \to \mathbb R\) describe distributions. More generally, distributions are generalised functions that are in the dual space of some function space. They are of great importance in the field of linear partial differential equations: the Fourier transform can be extended to distributions to allow for the transformation of linear differential operators (in phase space) into polynomials (in Fourier space). In particular, in the space of distributions any integrable function is differentiable. This makes the search for solutions much easier: if we accept solutions that are not necessarily functions, finding them—or algorithms/formulas for them—is mitigated.
In this project you will learn tempered distribution and their role in partial and non-local differential equations.
Semigroups and functional-analytic methods for PDE:s
This is a theoretic, but very powerful, approach towards solving parabolic PDE:s.