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research.md

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+## Research
+
+Operators that act on an infinite-dimensional space of functions or vectors may exhibit a range of spectral phenomena that have no analog in the finite-dimensional world of matrices. What role do these play in practical problems and how can we compute them? As a computational scientist and numerical analyst, I am fascinated by three broad themes:
+
+**Continuous spectrum.** Operators with continuous spectrum play a key role in resonance phenomena and wave-propagation in electromagnetics, acoustics, quantum mechanics, and various
+ regimes of fluid flow. _How does one capture the continuous spectrum on a computer?_
+
+**Robust eigensolvers.** Discretizations of infinite-dimensional operators may miss eigenvalues,
+ converge to false eigenvalues, and amplify the sensitivity of the spectrum to small perturbations.
+ _Can one avoid the pitfalls of discretization when computing eigenvalues in infinite dimensions?_
+
+**Data-driven design.** Interactions between mathematical models and real data are a key ingredient in engineering analysis and design. Modal decompositions play a key role in constructing
+ models and analyzing data from complex and nonlinear systems, but infinite-dimensional challenges abound. _Can rigorous mode decompositions provide new bridges between models and data?_
+
+Software implementations of efficient and robust algorithms for infinite-dimensional spectral computations are available at [https://github.com/SpecSolve](https://github.com/SpecSolve).