Algebraic Aspects of Periodic Operators Matthew Faust, Stephen P. Shipman, and Frank Sottile 26–30 May 2025, Fields Institute Toronto Canada.
		As part of the Fields workshop Commutative algebra and applications, we are organizing a short course and research experience on algebraic aspects of periodic operators.

Reading List

1. Text of Frank's Introductory talk.
2. Algebraic Aspects of Periodic Graph Operators, by Stephen Shipman and Frank Sottile.
   This is a fairly detailed development of algebraic aspects of peroidic operators for a Springer reference/encyclopedia on Operator Theory
3. Toric Geometry and Discrete Periodic Operators, By Frank Sottile.
   Sottile wrote this earlier as an abstract from a talk he gave in 2022 at Oberwolfach. It was an early attempt to describe some of this for algebraic geometers.
4. Generic properties of dispersion relations for discrete periodic operators, Peter Kuchment, Ngoc T. Do, and Frank Sottile. Journal of Mathematical Physics, 61, No. 10, 2020. 19 pp. DOI: 10.1063/5.0018562.
   This was written by, and for, spectral theorists, but it is one of the first papers since the 1990's to explicitly apply algebraic geometry to spectral theory.

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5. Critical points of discrete periodic operators, Matthew Faust and Frank Sottile, Journal of Spectral Theory, 14 (2024), no. 1, pp. 1–35 arXiv.org/2206.13649.
   Ideas from combinatorial and computational algebraic geometry are explicitly used to study features on Bloch varieties.
6. On the density of eigenvalues on periodic graphs, Cosmas Kravaris, SIAM Journal on Applied Algebra and Geometry, 2023.
   In this paper, commutative algebra is used to study the density of states.
7. Defect bound states in the continuum of bilayer electronic materials without symmetry protection, D. Massatt, S.P. Shipman, I. Vekhter, J.H. Wilson, Physical Review B, 2025.
   This paper uses reducibility of the Fermi variety to construct defect states embedded in the spectral continuum. Localization of the defect (p. 3) involves factorization of polynomial matrices B(z) and the vanishing set of polynomial vectors of the form B(z)v(z).