Upcoming REUs

Patterns and PDEs (2025)

Patterns

We are organizing a summer REU program in 2025, to be held at the University of California, Irvine. We are seeking two motivated undergraduate students to spend 8 weeks (June 23 - August 15, 2025) in Irvine, to work on research related to nonlinear waves and pattern formation, using the theory of dynamical systems and partial differential equations, with applications in mathematical biology, ecology, and/or physics.

See the program flyer for additional information.

Faculty mentor: Paul Carter (University of California, Irvine)

Application deadline: February 3, 2025
Apply at: MathPrograms.org

Past REU Projects

Patterns and PDEs (2023): Invasive tumor dynamics

Fingers

In this project, we analyzed interfaces which appear between healthy tissue and tumor tissue in the context of invasive tumor growth. When viewed as a planar interface in two spatial dimensions, the tumor boundary can exhibit instabilities which result in complex pattern formation along the interface, indicative of more aggressive tumors. We explored this behavior in the context of a reaction diffusion model of acid-mediated tumor growth, using geometric singular perturbation theory, asymptotic methods, and numerical simulations.

Collaborators: A. Doelman, P van Heijster, P Maini
REU Students: Daniel Levy, Erin Okey, Paige Yeung

Publications:

Patterns and PDEs (2022): Wrinkling patterns

Wrinkles

In this project during, we explored wrinkling patterns in anisotropic Swift-Hohenberg systems, which have applications in flexible electronics and materials with tunable surface properties. In particular, we explored the effect of parameters which control the amplitude, wavelength, and orientation of the emerging patterns, using computational approaches.

REU Students: Guogen Lan

Patterns and PDEs (2021): Vegetation fronts

Fingers

In this project, we investigated traveling fronts which arise as interfaces between desert and vegetatation in semiarid regions. These fronts propagate as the desert invades the vegetated regions. Transverse instabilities in these fronts can cause fingering patterns to appear, which propagate in the opposite direction, allowing the vegetation to re-invade the desert region, providing a possible mechanism for slowing or reversing desertification. We investivated these instabilities using numerical simulation and continuation in the Klausmeier and Gilad models.

Collaborators: A. Doelman
REU Students: Kaitlynn Lilly, Erin Obermayer, Shreyas Rao

Publications:

Patterns and PDEs (2020): Vegetation spots

Spots

In semiarid regions, limited water resources result in the formation of vegetation patterns, and there is evidence that the pattern structure is influenced by topographical conditions. On flat ground, vegetation patches or spot patterns are commonly observed. In this project, using a variant of the Klausmeier reaction-diffusion model of vegetation pattern formation, we analyzed radially symmetric spot patterns using geometric singular perturbation theory and numerical methods.

Collaborators: A. Doelman
REU Students: Ellie Byrnes, Lily Liu

Publications:

Stellar winds (2019)

Stellar wind

In the setting of viscous stationary spherically symmetric equations of gas dynamics, we are interested in the existence of sub-to-supersonic transitions which describe the phenomena of stellar wind and stellar accretion. These transitions arise as canard solutions in an appropriate fast-slow framework and can be constructed using geometric singular perturbation theory. In this project, we analyzed the effects of viscosity and heat conduction on the structure of stellar winds.

REU Students: Adam Bauer

Publications:

Localized roll patterns (2016)

Moebius

In a project hosted by the Summer@ICERM program in 2016, groups of undergraduate students studied snaking localized roll patterns in the Swift-Hohenberg equation. They analyzed the effect of the underlying geometry/topology of the equation on the structure of the associated snaking curves, as well as implications for higher dimensional patterns.

Collaborators: T. Aougab, M. Beck, J. Bramburger, B. Sandstede
REU Students: Dylan Altschuler, Chloe Avery, Surabhi Desai, Tharathep Sangsawang, Melissa Stadt, Aric Wheeler

Publications:

Traffic flow (2014/2016)

In this REU project, undergraduate students studied the application of data assimilation to models of traffic flow. In particular, a framework was developed to assimilate both Eulerian (fixed sensor) and Lagrangian (GPS) observations in order to predict traffic states and estimate model parameters. In another project, students applied equaton-free modelling techniques to the study of traffic dynamics.

Collaborators: B. Sandstede, L. Slivinski, C. Xia
REU Students: Tracy Chin, Courtney Cochrane, Joseph DeGuire, Bridget Fan, Emma Holmes, Melissa McGuirl, Patrick Murphy, Jenna Palmer, Jacob Ruth, Clayton Sanford, Rebecca Santorella

Publications: