# November Prize Spotlight: David M. Ambrose, Derek Gorthy, Christine Reilly, Marc Thomson, Michael I. Weinstein, and Yufei Zhao

Congratulations to these six members of the SIAM community who were recently awarded the T. Brooke Benjamin Prize in Nonlinear Waves, SIAM Award in the Mathematical Contest in Modeling, Martin Kruskal Lecture Prize, and Dénes König Prize, respectively.

- David M. Ambrose - T. Brooke Benjamin Prize
- Derek Gorthy, Christine Reilly, and Marc Thomson - SIAM Award in the Mathematical Contest in Modeling
- Michael I. Weinstein - Martin Kruskal Lecture Prize
- Yufei Zhao - Dénes König Prize

### David M. Ambrose, T. Brooke Benjamin Prize

The award recognizes Ambrose for his fundamental research in fluid mechanics and nonlinear waves. He has made seminal contributions in several key areas: weak solutions for fluids with free surfaces and with surface tension, time periodic solutions for free surface problems, overturning traveling waves, and equations with degenerate dispersion.

David M. Ambrose is Professor and Associate Department Head of the Department of Mathematics at Drexel University. He received his degrees in mathematics at Carnegie Mellon University and Duke University. He earned his PhD in 2002 from Duke University under the supervision of J. Thomas Beale. Before joining Drexel University in 2008, he was a Courant Instructor at the Courant Institute of New York University and a faculty member at Clemson University. His research interests center on applied analysis and scientific computing for nonlinear systems of partial differential equations, especially free-surface problems in fluid dynamics.

**Q:** *Why are you excited about winning the prize?*

**A:** I never had the good fortune to meet Brooke Benjamin, but our community holds him and his work in high regard; to be recognized with an award named after him is a great honor. Furthermore, I think there are many people in the nonlinear waves community doing excellent research, and to be chosen among them as the recipient of this prize is truly humbling.

**Q:** *Could you tell us a bit about the research that won you the prize and what it means to the public?*

**A:** I have done work over many years on existence theory for problems in nonlinear waves. These waves could arise in the motion of lakes or oceans or in optical fibers used in telecommunications, for instance. Showing that the equations modeling these phenomena have solutions is generally an important baseline result that helps to determine whether the mathematical models are adequately describing the real-world situation we are studying. Sometimes we find that the equations actually are not solvable, and then this leads to a discussion about whether further modeling is necessary to develop more robust equations.

**Q:** *What does being a SIAM member mean to you?*

**A:** I am delighted to be a life member of SIAM. I appreciate how SIAM helps to bring people together through conferences and as a publisher, including support for junior researchers. SIAM also provides very necessary advocacy in the public realm for science policy and funding.

### Derek Gorthy, Christine Reilly, and Marc Thomson, SIAM Award in the Mathematical Contest in Modeling

Derek Gorthy, Christine Reilly, and Marc Thomson received the SIAM Award in the Mathematical Contest in Modeling. The award recognizes them for their outstanding solution to Problem A, the Continuous Problem, in the 2018 Mathematical Contest in Modeling (MCM) administered annually by the Consortium for Mathematics and Its Applications (COMAP). Their solution paper, “Wave Goodbye to Poor Reception,” addressed the 2018 Problem A, “Multi-hop HF Radio Propagation.”

The SIAM Award in the MCM recognizes students for outstanding solutions to real world math problems. It is awarded every year to two of the teams judged “Outstanding” in the MCM. One winning team is chosen for each of the two problems, Problem A, the Continuous Problem, and Problem B, the Discrete Problem.

Professor Anne Dougherty of the University of Colorado Boulder Department of Applied Mathematics acted as faculty advisor for the team. The Department of Applied Mathematics recruits undergraduates across the university, so many teams represent a diversity of fields.

**Q:** *Why are you excited to be receiving this prize?*

**A:** We are thrilled to be recognized by SIAM for our work. Validation by such a well-known mathematical organization encourages us to continue our mathematical work and builds our confidence as we transition from academic to professional life. We hope that this success inspires more students to participate in mathematical competitions.

**Q:** *Could you tell us a bit about the work that won you the prize?*

**A:** Our project involves modeling how high-frequency radio waves travel over long distances. These waves are sent toward the sky at an angle, but interaction with charged particles in the ionosphere causes them to “bounce” back to the Earth. When the waves hit the Earth, they can “bounce” again, repeating the process. The model provides insight into how far these waves can travel and still be received, considering the angle of transmission, time of day, and roughness of the ocean or ground. Accurate modeling of “skywave” propagation allows us to predict communication paths beyond line of sight -- to all points of the globe.

**Q:** *What does this research mean to the public?*

**A:** Long-distance radio communication enables people to remain connected in remote locations. Modeling radio transmissions can help make skywave communication more reliable, improving networks in developing regions, disaster relief areas, and the open ocean.

In addition, this sort of work demonstrates how issues in every field of study can be better addressed with mathematics. Our team leverages participation in this contest to encourage other students, even those not studying mathematics, to incorporate math in solving difficult and open-ended problems.

**Q:** *What does participation in SIAM mean to you?*

**A:** Being practicing engineers means tackling complex, real-world problems on a daily basis, and mathematics represents a key tool for approaching these problems. SIAM publications provide us with useful information for implementing mathematical theory in practical applications, both in our schooling and jobs, assisting our engineering work in a variety of disciplines.

### Michael I. Weinstein, Martin Kruskal Lecture Prize

The Martin Kruskal Lecture is awarded every two years to an individual for a notable body of mathematics and contributions in the field of nonlinear waves and coherent structures. Weinstein was recognized for his long-standing contributions to the study of nonlinear wave phenomena. He has brought ideas and techniques of classical applied mathematics into a modern form, and the deep originality of his work has had a lasting impact on both theory and applications. A distinguished researcher and a prolific and inspiring mentor, he represents the best of our applied mathematics community.

Michael I. Weinstein is a Professor of Applied Mathematics and a Professor of Mathematics in the Department of Applied Physics and Applied Mathematics and the Department of Mathematics at Columbia University. He received his PhD from the Courant Institute, New York University in 1982 with George C. Papanicolaou and was a postdoctoral fellow at Stanford University with Joseph B. Keller from 1982 to 1984. Between 1984 and 2000, Weinstein was on the faculty at Princeton University and then the University of Michigan in Ann Arbor. From 1998 to 2004 he was Member of Technical Staff at Bell Laboratories / Lucent Technologies and in 2004 joined the faculty of Columbia University. Weinstein is a SIAM Fellow and a Fellow of the American Mathematical Society. In 2015 he became a Simons Foundation Math + X Investigator.

**Q:** *Why are you excited to be awarded the prize?*

**A:** I feel very honored to be awarded the Martin Kruskal Lecture Prize. I got to know Martin while I was on the faculty at Princeton during the 1980s. I have great admiration for his work in nonlinear waves and asymptotic methods, which has had an enormous impact on both fundamental and applied mathematics, as well as the broader scientific community. I thought of Martin as a friend and he is a continuing inspiration.

**Q:** *Could you tell us a bit about the research that won you the prize?*

**A:** This research relates to contributions to an understanding of the dynamics of coherent structures and applications. As a graduate student at the Courant Institute in the late 1970s and early 1980s, I was inspired by the work of M. D. Kruskal, P. D. Lax, V. E. Zakharov and others, on solitons and integrable nonlinear partial differential equations (PDEs), as well as work on nearly integrable systems (Kolmogorov-Arnold-Moser) to explore the robustness of the coherent structures in non-integrable PDEs. At this time I learned a great deal about linear and nonlinear dispersive waves and applications to nonlinear optics, fluids, and plasmas from G. C. Papanicolaou and D. W. McLaughlin.

A central role in the development of the theory of nonlinear dispersive waves has been played by the time-dependent nonlinear Schroedinger equation (NLS). One prominent area of application of NLS is to the description of an intense nearly monochromatic beam, which is propagating through a medium with which it interacts nonlinearly. An important phenomenon in this setting is the self-focusing of optical beams or lasers. Mathematically this is manifested in the formation of singularities in solutions of NLS (the so-called L^{2} critical case) from smooth initial conditions. This mathematical phenomenon is also called “blow-up.” An important motivation for its exploration is that it signifies a limitation of the NLS model.

Numerical simulations of NLS pointed to the important role played by the nonlinear ground state (the Townes soliton) in the structure of singularity formation. I proved that the optical power of the ground state (its L^{2} norm) provides a sharp threshold between initial conditions giving rise to optical fields which attenuate via diffraction (global existence and scattering) and fields which may develop self-focusing singularities (blow-up). The key was to observe that the above threshold could be characterized in terms of the optimal or “best” constant in a classical analytical interpolation inequality of Gagliardo and Nirenberg. The experimental observation of the threshold was made by G. Fibich and A. Gaeta. The perspective taken in the above analytical work has since been useful in numerous other problems in the analysis of PDEs.

Initially motivated by the question of how the ground state profile plays a role in the solution as it evolves toward its singularity, I developed methods for the rigorous modulational stability analysis of coherent structures. Modulation theory is closely related to the notion of orbital (Lyapunov) stability, and the above investigations also led me to general results on the nonlinear stability of solitary waves.

My later research on coherent structures and nonlinear waves, initially with A. Soffer and with R. L. Pego and then with other outstanding colleagues, students, and postdocs, was aimed at phenomena such as nonlinear scattering and asymptotic stability of coherent structures. In joint work with Soffer, we studied metastable (very long-lived) coherent structures, which eventually decay radiatively due to resonant coupling to continuum modes. We discovered a general “nonlinear ground state selection” phenomenon, which was subsequently observed experimentally in nonlinear optical experiments.

Most recently I have collaborated with C. L. Fefferman and other colleagues on a class of very robust coherent structures (protected edge states) which are concentrated at interfaces between different periodic media. This robustness is closely connected to spectral characteristics arising from symmetries of the bulk periodic structure.

Examples are the honeycomb structure of graphene and its various artificial analogues. This is related to the exciting and rapidly developing field of topological insulators and its applications to robust energy transfer in many physical systems.

**Q:** *What does your research mean to the public?*

**A:** Coherent structures are ubiquitous. We are passive observers to many, such as water waves which break at sea, hurricanes, and tornados, but an understanding of the mechanisms for their stability and instability leads to strategies for their generation and control. I began working on such problems in the context of optical communications while a researcher at Bell Laboratories. An example was a study of (and patent for) the use of “gap solitons” in periodic structures to slow light down and trap it in designer-defects for the purpose of optical buffering of information. A second example is my recent work with colleagues (mathematicians and applied physicists) on topologically protected defect modes, which suggests strategies for robust energy transfer in photonic and other systems.

**Q:** *What does being a SIAM member mean to you?
*

**A:** I feel very much at home scientifically at the interface between mathematics and its applications. This kind of activity is central to the culture of SIAM, which endures as a dynamic professional society due to its adaptive structure of activity groups.

### Yufei Zhao, Dénes König Prize

The SIAM Activity Group on Discrete Mathematics (SIAG/DM) awards the Dénes König Prize every two years to an individual or individuals in their early career for outstanding research contributions in an area of discrete mathematics within the three calendar years prior to the award year. Each candidate must be a PhD student or, at the time of the award, be within four years after completing their PhD.

The award recognizes Zhao for his paper, “A Relative Szemerédi Theorem,” Geometric and Functional Analysis (2015), co-authored by David Conlon and Jacob Fox. The paper strengthens and simplifies celebrated results of Ben Green and Terence Tao concerning the asymptotic patterns of prime numbers and arithmetic progressions of prime numbers. It contains new techniques to analyze pseudorandom sets of numbers, which may have diverse applications.

Yufei Zhao is Assistant Professor of Mathematics at the Massachusetts Institute of Technology, where he also received his PhD in 2015, under the supervision of Jacob Fox. He received his dual BS degrees in mathematics and computer science from MIT in 2010 and a MASt in mathematics from the University of Cambridge in 2011. Prior to returning to MIT, Zhao was the Esmée Fairbairn Junior Research Fellow in Mathematics at New College, Oxford, as well as a Research Fellow at the Simons Institute for the Theory of Computing at the University of California Berkeley. Zhao was a three-time Putnam Fellow. His research focuses on combinatorics and graph theory, and he is particularly interested in understanding pseudorandomness of discrete systems.

**
Q:** *Why are you excited to win this prize?*

**
A:** It is a great honor for me to join the list of distinguished awardees of the Dénes König Prize. König was one of the pioneers of modern graph theory, and it is great to see that the subject has evolved into a more central mathematical topic over the past century. The prize is a welcome acknowledgement of graph theoretic advances developed by myself in collaboration with David Conlon and my former PhD advisor Jacob Fox.

**
Q:** *Could you tell us a bit about the research that won you the prize?*

**
A:** Our paper, “A Relative Szemerédi Theorem,” coauthored with David Conlon and Jacob Fox, advances our understanding of pseudorandom graphs and completes a key missing step in the development of the regularity method for sparse graphs. As an important application, the work significantly simplifies the proof of the Green-Tao theorem, a celebrated result in number theory showing that the primes contain arbitrarily long arithmetic progressions. The techniques that we introduced in the paper have subsequently led to other advances in graph theory and number theory.

**
Q:** *What does your research mean to the public?*

**
A:** Much of our world can be modeled by graphs and networks, and this is especially true in an age of increasing digital connectivity. Fundamental advances in graph theory pave the way for algorithmic applications in the design and analysis of real world networks, allowing us to run better and faster algorithms with greater confidence and reliability. While our work is theoretical in nature, I believe that the graph theoretic ideas we developed in our research will be useful in future computer science and network applications.

**
Q:** *What does being a SIAM member mean to you?*

A: It is wonderful to be part of a community that values mathematical research and its applications. In particular, SIAM connects me to researchers from other scientific communities, giving me a perspective on how mathematics that one group develops can be useful in surprising and productive ways elsewhere.