Welcome to the research page of Margaret Beck
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Research Interests

My primary interest is determining the nonlinear stability and large-time behavior of solutions to dissipative PDEs, such as reaction-diffusion equations and viscous conservation laws. This includes studying nonlinear waves such as traveling waves and spatially and/or temporally periodic patterns. I typically view these PDEs as infinite-dimensional dynamical systems, and I analyze them using a variety of mathematical techniques, for example invariant manifolds, similarity variables, geometric singular perturbation theory, exponential dichotomies, and pointwise estimates.

In much of my work, the main mathematical difficulty arises from the fact that the linear operator lacks a so-called spectral gap. For example, the spectrum could be as in the above figure on the left, where a zero eigenvalue is embedded in the continuous spectrum. As a result, one cannot use standard spectral decomposition methods to separate the solutions that decay from those that simply remain bounded. Furthermore, estimates necessary for the related nonlinear analysis can become extremely delicate, as can the properties of associated bifurcations.

One particular topic I've been interested in recently is known as "metastability." Roughly speaking, this refers to long transients in the dynamics. For example, solutions could spend large periods of time near unstable states before settling down to their stable, asymptotic limit. This is interesting mathematically, because there are far fewer techniques available for analyzing transient behaviors than there are for analyzing asymptotic behaviors. Also, in real world systems, the transient time-scale may be so long that one will never actually see the limiting behavior. Thus, metastability is important in applications, as well. The type of behavior arises, for example, in the two-dimensional Navier-Stokes equation.

The above picture on the right is of me in the Catalan town of Vic. It doesn't really have anything to do with my research. However the statue, known as "Merma," is said to come alive during the festival of de Patron and chase the children while carrying a lash. Occaisionally, while working on my research, I feel as if Merma is chasing me around in circles.


Publications and Preprints (or via google scholar)
Lecture Notes, Thesis, Slides for Talks, Poster Presentations, and Miscellany

Lecture notes
PhD Thesis
Slides for talks
Poster presentations
Miscellany