Paul Carter
University of California-Irvine, CA·Mathematics & Statistics·Early-career
Patterns appear across nature: bands of vegetation in drylands, separating mixtures and waves in chemical reactions. Mathematicians want to know which universal mechanisms create such patterns and how to predict their shape and behavior.
- Fields
Mathematics & Statistics
- Topics
pattern formation
partial differential equations
singular perturbation
- Grants
- 1 known grant, through May 2028
What they work on
- Analyze when bistable planar interfaces, boundaries between two stable states, become unstable and form patterns in multi-component reaction-diffusion systems, motivated by desertification fronts in drylands.
- Build a rigorous theory of temporal pulse replication, where a single traveling pulse self-replicates in a canard-explosion-like way, using geometric singular perturbation methods.
- Characterize far-from-onset patterns formed behind invasion fronts into unstable steady states, using spatial dynamical systems tools for existence, stability and bifurcation analysis.
More about the research
Many of these systems are described by partial differential equations (PDEs), equations for quantities that change in space and time. When parts of a system work on very different length or time scales, the equations are 'singularly perturbed,' and standard tools often fail.
Why it matters
This is fundamental mathematics. It adds rigorous understanding of how complex patterns form in systems with widely different scales. That understanding could eventually inform models of desertification fronts in dryland ecosystems and of wave phenomena in biology and neuroscience.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Chongchun Zeng
Georgia Institute of Technology, GA·Mathematics & Statistics
Analyze two-dimensional water waves linearized at shear flows, where speed varies with depth, establishing the bifurcation of instability and linear inviscid damping for gravity water waves.
Dionyssios Mantzavinos
University of Kansas, KS·Mathematics & Statistics
Develop analytical tools to establish existence, uniqueness and stability of solutions to nonlinear partial differential equations posed in bounded domains in two or more spatial dimensions.
Oleksandr Misiats
Virginia Commonwealth University, VA·Mathematics & Statistics
Build and analyze energy-minimizing models of martensitic phase transitions — the crystal rearrangements behind the shape memory effect — to explain bending-induced effects reported experimentally in shape memory alloys.
The school
University of California-Irvine: 46 labs on PI Pages, $645M a year on research, 524 research doctorates a year.
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