Jeremy Marzuola
University of North Carolina at Chapel Hill, NC·Mathematics & Statistics
Many physical systems behave very differently once you introduce a defect: a crystal with an impurity traps light in unexpected ways, a simulated ocean needs an artificial edge that absorbs waves without reflecting them, and a network can be cut into meaningful groups only if you understand how its shape constrains the cut. This project pursues three such problem areas in applied mathematics and argues they share machinery: tools from harmonic analysis, which decomposes functions into waves, together with optimization and the theory of elliptic partial differential equations.
- Fields
Mathematics & Statistics
Physics
- Topics
spectral theory
harmonic analysis
microlocal analysis
- Grants
- 1 known grant, through Jun 2027
What they work on
- Establish rigorous constructions of fundamental modes and resonances for model operators from condensed matter and topological physics, including light propagating in crystalline structures that contain localized defects.
- Extend domain decomposition results for the Helmholtz equation, which describes steady waves, to give quantitative error bounds for numerical schemes treating scattering by obstacles, in parallel with scattering by inhomogeneous media.
- Analyze the damping terms used to absorb outgoing waves in bounded simulations of open-ocean propagation by recasting them as quantum-mechanical operators with complex potentials and applying microlocal analysis.
- Determine how the geometry of a domain or graph constrains methods for partitioning it into meaningful pieces, such as detecting communities or assigning labels like identifying cancer cells from genetic markers.
More about the research
The first area concerns model operators from condensed matter and topological physics, including the behavior of light waves in a crystalline structure containing defects, with attention to rigorous constructions of the fundamental modes and resonances of physically important examples. The second concerns the accuracy of numerical algorithms for wave propagation, electromagnetic or fluid. Earlier work with collaborators on solving the Helmholtz equation, which describes steady waves, by domain decomposition, splitting a region into pieces solved separately, will be extended to give quantitative bounds for scattering off obstacles, paralleling the case of scattering by inhomogeneous media. A related question comes from simulating open-ocean waves on a necessarily finite computational domain: one adds a damping term to absorb outgoing waves, but this perturbs the system substantially. After a suitable transformation, the effect of that damping becomes a question about operators from quantum mechanics with complex potentials, where recent microlocal analysis, a refined way of tracking where and in what direction a wave carries energy, supplies sharp bounds. The third area studies how the geometry of a domain or graph affects how it can be partitioned into useful pieces, for instance identifying communities or labels such as distinguishing cancer cells from genetic markers. Across all three, the emphasis is on quantifiable error estimates, developed with undergraduates, graduate students, and postdocs, and in collaboration with physicists and with numerical analysts simulating each application area.
Why it matters
Error estimates for numerical solutions of the Helmholtz equation matter wherever waves are used to see inside something, including medical imaging and sonar detection, since inverse problems there depend on trusting the simulated wave field. Rigorous understanding of damping layers could make open-ocean wave simulations more reliable, and the partitioning results could inform methods that sort data into groups, such as labeling cells from genetic markers.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Francois Monard
University of California-Santa Cruz, CA·Mathematics & Statistics
Unify three complex-analytic methods for X-ray transforms, including transport twistor space, to derive exact reconstruction formulas in curved (Riemannian and non-Euclidean) geometries.
Avraham Soffer
Rutgers University-New Brunswick, NJ·Mathematics & Statistics
Characterize the large-time asymptotic behavior of solutions for a general class of nonlinear Schrödinger equations, the equations governing quantum waves and light in nonlinear media, and identify all possible asymptotic states.
Mark Ablowitz
University of Colorado Boulder, CO·Mathematics & Statistics
Extend the Inverse Scattering Transform, which turns nonlinear wave equations into solvable linear scattering problems, to new classes of equations including fractional nonlinear wave equations with non-integer-order derivatives.
The school
University of North Carolina at Chapel Hill: 72 labs on PI Pages, $1.6B a year on research, 523 research doctorates a year.
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