Organizing Committee
Abstract

Discrete periodic Schrodinger operators describe the behavior of individual electrons in "ideal" crystals in the tight-binding model of solid state physics. Spectra of such operators have the usual band-gap structure, and the corresponding dispersion relations are algebraic varieties. In the 1990's Gieseker, Knorrer, and Trubowitz used toroidal compactifications to solve questions such as irreducibility of Bloch and Fermi varieties and the density of states, for a class of mono-atomic models. Their work showed that while spectral theory is focused on the real part of the Bloch variety, the study of complex singularities and compactifications is crucial for describing formation of bands and gaps.

After a gap of 30 years, spectral theory is again interacting with algebraic geometry. Recently, W. Liu gave an algebraic method to obtain more general proofs of irreducibility for Fermi surfaces, Kravaris used free resolutions to study density of states, and Kuchment and coauthors used toric varieties and numerical nonlinear algebra to study spectral edges. These and other developments have led to the realization that many open questions in spectral theory may be studied using modern tools in algebraic geometry. These include the relation of reducibility/irreducibility with physical symmetries of the crystals, the structure of spectral band edges, and Dirac cones. These methods can also be applied to quantum graphs, such as graphene-type structures, and to obtain existence and non-existence theorems for embedded eigenvalues. The goal of this workshop is to bring together experts from spectral theory, mathematical physics, and algebraic geometry to understand, apply, and advance these new methods and interactions.

A significant aspect of these reemerging interactions involves computation. A key recent development in algebraic geometry is computation, both symbolic and numerical. Computation and experimentation using tools from algebraic geometry have already been important in two recent papers by Sottile on this subject, and we expect it to become a useful tool for studying periodic operators. Speakers in this workshop are firmly rooted in the uses of computation and experimentation in applications of Algebraic Geometry, and many are familiar with exploiting the atmosphere and facilities of the ICERM to initiate collaboration.

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Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

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Friday, February 24, 2023
  • 8:50 - 9:00 am EST
    Welcome
    11th Floor Lecture Hall
    • Session Chair
    • Brendan Hassett, ICERM/Brown University
  • 9:00 - 9:45 am EST
    From spectral theory to algebraic geometry through discrete periodic operators
    11th Floor Lecture Hall
    • Speaker
    • Stephen Shipman, Louisiana State University
    • Session Chair
    • Frank Sottile, Texas A&M University
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Some relevant algebraic geometry and toric varieties
    11th Floor Lecture Hall
    • Speaker
    • Frank Sottile, Texas A&M University
    • Session Chair
    • Stephen Shipman, Louisiana State University
  • 11:30 am - 12:30 pm EST
    Group Work
  • 12:30 - 2:00 pm EST
    Lunch/Free Time
  • 2:00 - 2:45 pm EST
    Irreducibility of varieties associated with periodic Schr\"odinger operators
    11th Floor Lecture Hall
    • Speaker
    • Jake Fillman, Texas State University
    • Session Chair
    • Frank Sottile, Texas A&M University
    Abstract
    We will discuss Bloch and Fermi varieties associated with Schr\"odinger operators and some works concerning their irreducibility.
  • 3:00 - 3:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 5:00 pm EST
    Group Work
  • 5:00 - 6:30 pm EST
    Reception
    11th Floor Collaborative Space
Saturday, February 25, 2023
  • 9:00 - 9:45 am EST
    Finite quantum graphs and algebraic geometry
    11th Floor Lecture Hall
    • Speaker
    • Lior Alon, MIT
    • Session Chair
    • Stephen Shipman, Louisiana State University
    Abstract
    Let G be a finite graph of N edges. A finite standard quantum graph (G,L), with L=(L_1,…,L_N), is a collection of N intervals e_j=[0,L_j] glued at their endpoints, corresponding to the edges of G, equipped with the one-dimensional Laplacian (2ed derivative edgewise) that acts on functions that satisfy standard vertex conditions. We will be interested in this operator's spectral properties (eigenvalues and eigenfunctions) and their dependence on G and L. For a fixed G, its ``Secular Manifold’’ S_G is the set of points (e^{ikL_1},…, e^{ikL_N}) in the N-dimensional torus, such that k^2 is an eigenvalue of (G,L). The eigenvalues and eigenfunctions of (G,L), for any L, are determined by intersections of a curve depending on L with S_G. This allows to decouple spectral properties into G and L dependence and provides an algebraic toolkit for spectral geometry on quantum graphs. After providing the background and defining the secular manifold, I will review previous results: spectral gap distribution, nodal count distribution, and the arithmetic structure of the spectrum. I will discuss the role of the secular manifold in those results, and some open conjectures that may benefit from investigating the algebraic structure and Morse structure of this variety.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Density of States for discrete Schrodinger operators: homological techniques and free resolutions.
    11th Floor Lecture Hall
    • Speaker
    • Hal Schenck, Auburn University
    • Session Chair
    • Frank Sottile, Texas A&M University
    Abstract
    In recent work, Kravaris uses computes the Density of States for discrete Schrodinger operators as the (normalized) rank of a certain module over the ring of Laurent polynomials. I will discuss some of the computational and homological tools used in his result.
  • 11:30 am - 12:30 pm EST
    Group Work
  • 12:30 - 2:00 pm EST
    Lunch/Free Time
  • 2:00 - 3:30 pm EST
    Group Work
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    Spectral band edges of discrete and continuous periodic operators via analytic methods.
    11th Floor Lecture Hall
    • Speaker
    • Ilya Kachkovskiy, Michigan State University
    • Session Chair
    • Stephen Shipman, Louisiana State University
    Abstract
    I will discuss analytic methods of showing that the level sets of spectral band functions at the edges of spectral bands have dimension at most $d-2$, where $d$ is the dimension of the lattice of periods. The approach works in the continuum for $d=2$ and for a large class of discrete Schrodinger operators in all dimensions. I will also summarize some current open questions in the area. The results in the talk are based on joint works with N. Filonov.
Sunday, February 26, 2023
  • 9:00 - 9:45 am EST
    Algebraic and analytic geometry problems in spectral theory of periodic media
    11th Floor Lecture Hall
    • Virtual Speaker
    • Peter Kuchment, Texas A & M University
    • Session Chair
    • Frank Sottile, Texas A&M University
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 am - 12:00 pm EST
    Group Work
  • 12:00 - 2:00 pm EST
    Lunch/Free Time
  • 2:00 - 3:30 pm EST
    Group Work
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 5:00 pm EST
    Group Presentations
    11th Floor Lecture Hall

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