Over 370 researchers attended the SIAM conference on Analysis of Partial Differential Equations, which was held at the La Quinta Resort in La Quinta, CA from December 11-14, 2019. The beautiful, warm, and sunny location gave many attendees a short break from the winter cold and dreariness and a hectic finals week. The excellence of the location was mirrored in the lineup of plenary speakers and mini-symposia.
The plenary speakers highlighted the broad range of active research topics in partial differential equations. Several of the plenary speakers focused on developing numerical methods for partial differential equations. John Shadid (Sandia National Laboratory/University of New Mexico) described high-order and scalable numerical methods and preconditioning for nonlinear systems with multiple time scales. Chi-Wang Shu (Brown University) provided a history and review of recent developments in numerical methods for hyperbolic conservation laws and a framework for constructing high-order schemes that obey maximum principles.
Analytical techniques to study solutions were also represented. Manuel del Pino (University of Bath) described singular formation and blow-up in solutions to time evolution parabolic PDEs. Inwon Kim (UCLA) provided global-time existence of solutions to transport problems with applications to crowd motion and fluid interfaces. Irene Gambia (University of Texas at Austin) spoke about mixtures of monatomic gasses primarily from the analytical standpoint of the Boltzmann equation, but also commented on numerical approximation schemes for the multi-particle system.
Finally, Pierre-Emmanuel Jabin (University of Maryland) spoke on tweaks to standard mean field, and Andrew Stuart (CalTech) connected a myriad of topics in describing a methodology for calibrating large-scale computational models.
As always, it seems there were too many overlapping mini-symposia to pick from and I regret that I was unable to attend them all. I was glad to see many mini-symposia composed of a mixture of graduate students, early-career academics, and senior researchers. Biasedly, I particularly enjoyed the presentations with applications to biological and social systems and those that used dynamical systems approaches. On the biological side, Tracy Stepein (University of Florida) used a two population model of migrating and stationary cells to investigate tumor growth in an aggressive form of brain cancer. Additionally, Anna Ghazaryan (Miami University) proved the existence of Fisher-KPP like traveling fronts and dynamics in the diffusive Rosenzweig-MacArthur predator-prey population models.
The spread of animals, people, and information was a popular topic, and included a mini-tutorial on Population Dynamics in Moving Ecosystems by Mark Lewis (University of Alberta). In mini-symposia, Heather Zinn Brooks (UCLA) described the propagation of information and opinions from media sources in social network models. Jasper Weinburd (Harvey Mudd) highlighted some results and ongoing work into traveling waves of locust populations, which stemmed from a summer MRC workshop. Pierre-Emmanuel Jabin (University of Maryland) spoke about the necessity of including both short and long term memory in models of animal migration. Rafael Bailo (Imperial College London) talked about pedestrian dynamics with congestion and highlighted both the model building and accurate numerical schemes. The pedestrian models were inspired by traffic models, but incorporated a unique pseudo-pressure force term.
Several talks also investigated pattern formation. Talks by Montie Avery (University of Minnesota) and Ryan Goh (Boston University) each used spatial dynamics techniques to investigate pattern formation on growing domains. Avery focused on the formation of zig-zag patterns in the Swift-Hohenburg equation and how the growth speed selects wave properties and the emergent instability. At the conference, I was introduced to a new application area of pattern formation - fingerprints. Lisa Maria Kreusser (University of Cambridge) discussed generation of realistic fingerprints for use in training databases. The bio-inspired model used an interacting particle approach coupled with an underlying stress field that adds anisotropy and helps dictate where cells will align.
This year, the SIAM Activity Group prize was awarded to Jacob Bedrossian (University of Maryland) and Nader Masmoudi (Courant) for their recent work on solutions to the nonlinear 2D Euler equations near the Couette flow. In the prize lecture, Jacob Bedrossian clearly presented their work by successfully unpacking technical theorems to the broad audience. The presentation highlighted their results, but also situated their work into the broader and historical picture.
Thank you to the conference organizers and SIAM for hosting an enjoyable and productive conference! I am looking forward to further connecting with colleagues at the upcoming SIAM meetings.