SMRI Seminar: Berkolaiko -- Morse theory for eigenvalues of self-adjoint families

SMRI Seminar: ’Morse theory for eigenvalues of self-adjoint families’ 
Greg Berkolaiko (Texas A&M University) 

Date and time: Thursday 23 May, 13:00-14:00 AEST 
Location: SMRI Seminar Room, A12-03-301
***NOTE NEW LOCATION***IN-PERSON ONLY THIS WEEK*** 

Abstract: 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 Schroedinger 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 appropriate dimensions.  


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Please join us after the seminar for SMRI afternoon tea, 2:00-2:45pm every Thursday on
the SMRI Terrace (accessed through A14-04-L4.36) 

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