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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.