The question of optimizing an eigenvalue of a family of self-adjoint operators that depends on a set of parameters arises in diverse areas of mathematical physics. Among the particular motivations for this talk are the Floquet-Bloch decomposition of the Schrödinger operator on a periodic structure, nodal count statistics of eigenfunctions of quantum graphs, conical points in potential energy surfaces in quantum chemistry and the minimal spectral partitions of domains. In each of these problems one seeks to identify and/or count the critical points of the eigenvalue with a given label (say, the third lowest) over the parameter space which is often known and simple, such as a torus. Classical Morse theory is a set of tools connecting the number of critical points of a smooth function on a manifold to the topological invariants of this manifold. However, the eigenvalues are not smooth due to presence of eigenvalue multiplicities or ``diabolical points''. We rectify this problem for eigenvalues of generic families of finite-dimensional operators. The ``diabolical contribution'' to the ``Morse indices'' of the problematic points turns out to be universal: it depends only on the multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family. Using tools such as Clarke subdifferential and stratified Morse theory of Goresky--MacPherson, we express the ``diabolical contribution'' in terms of homology of Grassmannians of ppropriate dimensions.
Based on a joint work with Igor Zelenko (TAMU). Berkolaiko, G., Zelenko, I. Morse inequalities for ordered eigenvalues of generic self-adjoint families. Invent. math. (2024).
Morse theory for eigenvalues of self-adjoint families
Sep. 11, 2024 4:45pm (MATH 3…
Grad Student Seminar
Daniel Lyness (CU Boulder)
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Poincaré Duality is an extremely powerful tool in Geometry and Topology. In this talk, I will briefly introduce Singular, De Rham and Dolbeault Cohomology and explore the theorems relating these different theories. The I will give and compare the appropriate duality statement in each case. Hopefully there will be time at the end to do some calculations and see these results in action! This talk will be most suited to those with some familiarity with singular homology/cohomology and differential forms, but I intend to present the material in such a way that everyone can get something out of it!
Morse theory for eigenvalues of self-adjoint families