Genericity results for metric graphs and the trace space.

If you choose a random symmetric matrix, would it have two equal eigenvalues? Probably not, since symmetric matrices with degenerate eigenvalues capture a "very small portion" of the entire space of symmetric matrices. This is an example of a genericty result. A more complicated and famous result in spectral geometry is Uhlenbeck's genericity theorem ('72) which states that given any manifold, for a Baire-generic choice of Riemannian metric, all eigenvalues of the associated Laplalacian are simple, with eigenfunctions that are Morse and does not vanish at their critical point. A similar result for metric graphs was given by Friedlander ('05) and Berkolaiko and Liu ('17). Given any fixed graph structure, for a Baire-generic choice of edge lengths, all eigenvalues of the Laplacian (with standard vertex conditions) are simple, with eigenfunctions that are Morse and do not vanish on vertices. 

However, apriori, the Baire-generic set of ``good'' edge lengths can have zero Lebesgue measure (and so random sampling would probably miss it). 

In this talk, I will introduce a stronger notion of genericity, that implies full Lebesgue measure. I will give an independent proof of the previous results, showing that they are in fact strongly generic. To do so, I will define the ``Trace Space'', a moduli space of all eigenpairs associated with a graph structure. I will also show that any non-trivial homogenous relation on the traces must fail generically. This proof is algebraic in nature, uses ergodic theory, and is based on a conjecture of Colin de Verdiere which was recently proved by Kurasov and Sarnak.

I will not assume any prior knowledge in spectral geometry or metric graphs.

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