Gregory Berkolaiko
Texas A&M University-College Station, TX·Mathematics & Statistics
Many problems in physics and chemistry come down to eigenvalues: special numbers, such as allowed energy levels, that an operator (a mathematical rule acting on wave functions) produces. Knowing where eigenvalues sit and how they shift when a system changes tells you which states exist and which configurations are most stable.
- Fields
Mathematics & Statistics
Physics
- Topics
spectral theory
eigenvalues
topological insulators
- Grants
- 2 known grants, through Jul 2028
What they work on
- Derive explicit formulas for the number of edge-state eigenvalues in spectral gaps of tight-binding Hamiltonians for topological insulators, using the Maslov index and its numerically computable discrete version, the Duistermaat index.
- Use the new index formulas to compute the integrated density of states and connect gap indices with band indices, clarifying the bulk-boundary correspondence in topological insulators.
- Prove a link between the Morse index of an eigenvalue at a critical point and the spectral shift relative to a reference operator, as boundary conditions vary over self-adjoint extensions.
- Apply the Morse index-spectral shift link to bound energy landscapes and certify local minima as globally optimal, including in the spectral minimal partition problem, using Lagrangian Grassmannian and Dirichlet-to-Neumann methods.
More about the research
These two projects develop mathematical tools that turn local, easy-to-compute information about eigenvalues into global conclusions, using ideas from geometry such as the Maslov index, a count of how certain geometric objects twist as parameters change.
Why it matters
Topological insulators could support low-energy electronics because their edges carry current with ultra-low resistance; formulas that count edge states from computable data could help analyze candidate materials. Certifying that a locally found energy minimum is truly global could aid questions such as finding stable molecular configurations in quantum chemistry and optimal patterns in reaction-diffusion systems.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Wei Li
DePaul University, IL·Mathematics & Statistics
Prove the existence of and characterize edge states, the waves that travel along a boundary, on sharp interfaces formed by joining photonic materials with various symmetries, within a partial differential equation framework.
Daniel Stern
Cornell University, NY·Mathematics & Statistics
Use new constructions of extremal metrics for Laplacian and Steklov eigenvalues, the natural vibration frequencies of a shape, to produce many new minimal surfaces of prescribed topology in low-dimensional spheres and balls.
Kenneth Burch
Boston College, MA·Physics
Determine whether the boundary mode observed in FeTeSe superconductors reflects true topological order by systematically varying device configurations and material parameters.
The school
Texas A&M University-College Station: 23 labs on PI Pages, $1.4B a year on research, 768 research doctorates a year.
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