Every two years, the SIAM Activity Group on Dynamical Systems awards the J.D. Crawford Prize "to an individual for recent outstanding work on a topic in nonlinear science, as evidenced by a publication in English in a peer-reviewed journal within the four calendar years preceding the award year." [1] This year's winner is Dr. Margaret Beck of Boston University, for work using the Maslov index to determine stability of nonlinear waves, as well as for other outstanding contributions on coherent structures and nonlinear waves. The work cited in the prize nomination was "Computing the Maslov index for large systems," Proceedings of the AMS, Vol. 143, no. 5, pages 2159-2173 [2]. Dr. Beck has received many previous awards for her research, including a Sloan Fellowship, the inaugural Joan and Joseph Birman Fellowship from the AMS, many grants from the National Science Foundation, as well as being selected as a Kalvi Fellow by the US National Academy of Sciences. This article gives a broad overview of Beck's research.
Beck's work focuses primarily on infinite-dimensional dynamical systems in a large variety of contexts in applied dynamical systems. These include contributions in the study of nonlinear stability of nonlinear waves for partial differential equations in one or more spatial dimensions. The most difficult type of nonlinear stability result is that of marginal stability. That is, the case in which a test for stability using the linearized equation is inconclusive. In a series of joint papers with Zumbrun, Nguyen, and Sandstede, Beck proved stability of nonlinear waves in the marginal stability case for time-periodic Lax shocks in viscous conservation laws.
Another portion of Beck's work considers metastability, the property of non-equilibrium solutions to remain in a vicinity of a particular special solution for a very long period before converging to a different final limit. Beck (in part with C.E. Wayne) was able to make significant contributions to the understanding of metastability in equations for fluids, such as Burger's and the Navier-Stokes equations. In particular, she was able to describe the large-time behavior of certain types of solutions by using invariant manifolds.
Beck’s work also includes the study of localized roll structures. In many systems, as a parameter in the system varies, these structures grow by adding additional copies of the basic underlying periodic state. As one moves along the bifurcation branch, the parameter moves backward and forward between two limits. Due to the appearance of the bifurcation diagram, this is referred to as snaking. In work with Knobloch, Lloyd, Sandstede, and Wagenknecht, Beck has provided a rigorous foundation for the understanding of this snaking behavior.
Beck's most recent topic of research is the application of a topological invariant from the field of symplectic geometry called the Maslov index. The paper recognized in the Crawford Prize citation is on this topic. Her work applies this index in order to establish stability of time-varying localized structures for partial differential equations. In general in this context, it is relatively straightforward to locate the essential spectrum of a linearized equation, and the key difficulty is to find the location of the point spectrum. In one-dimensional systems, the Evans function allows for the determination of the location of the point spectrum. However, the Evans function does not apply in dimensions above one. Previous work suggests that the Maslov index could be a way to establish stability in higher dimensions, but it is extremely difficult to compute, both analytically and numerically. Beck's paper with Malham gives the first step toward a computational method for computing the Maslov index. The method is applied to the modified KdV partial differential equation. While these results are only the first step, they point towards a new approach for understanding the stability of localized structures in higher dimensional partial differential equations.
For further reading, also see an SIAM News article Beck wrote on her work [3] , a previous DSWeb article on Beck's research when she received the Birman Fellowship [4], and an interview with Beck conducted after she received the prize [5].
Acknowledgements: I would like to thank Gene Wayne for his help in writing this article. As the chair of the Crawford Prize committee, I would like to thank the members of the committee Barbara Gentz, Aric Hagberg, and Martin Wechselberger. The four of us are indebted to the community for the excellent prize nominations - making the committee’s work both very difficult and a real pleasure.
References:
[1] J.D. Crawford Prize, SIAM.
[2] Margaret Beck and Simon J A Malham, "Computing the Maslov index for large systems," Proceedings of the AMS, Vol. 143, no. 5, pages 2159-2173 2015.
[3] Margaret Beck and C. Eugene Wayne, "Analyzing Multiple Time Scales in Two-Dimensional Fluids Using Dynamical Systems," SIAM News, March 2017.
[4] Eugene C. Wayne, "Margaret Beck is the first recipient of the AMS Joan and Joseph Birman Fellowship for Women Scholars," DSWeb Magazine, April 2018.
[5] May Prize Spotlight: Margaret Beck and Philip Holmes, SIAM NEWS BLOG, Awards and Recognition, June 03, 2019.