Minisymposia Abstracts
Speaker:  Yaroslav Vorobets Texas A&M University 
Title:  Laplacians isospectral under multiple boundary conditions 
Abstract:  Two domains in a plane are called isospectral if the Dirichlet
Laplacians in these domains have the same spectrum. The isospectral
domains need not be congruent (i.e., in general, one cannot hear
the shape of a drum). The first counterexample of that kind was
constructed by Gordon, Webb, and Wolpert in 1992. The domains were
polygons, and it was noted that the Neumann Laplacians in them
also had the same spectrum. More recently, it was observed that
Laplacians in the same polygons with mixed boundary conditions
(Dirichlet on some sides, Neumann on the others) can be isospectral
as well. Two examples were found by Driscoll and Gottlieb
(numerically) and another one by Band and Parzanchevski. In the talk, all the above examples will be placed into a family of 16 pairs of isospectral Laplacians corresponding to various choices of mixed boundary conditions for the two polygons. Isospectrality is established in a unified way using transplantation of eigenfuctions. The construction extends to other known examples of isospectral polygons. They admit multiple isospectrality too. 
Speaker:  Ruth Gornet University of Texas Arlington 
Title:  The eta invariant on Nilmanifolds 
Abstract: 
In joint work with Ken Richardson of TCU, we discuss recent results on the eta invariant on
twostep nilmanifolds 
Speaker:  Hamid Hezari MIT 
Title:  Spectral rigidity of the ellipse 
Abstract:  This is a joint work with Steve Zelditch. We prove that ellipses are infinitesimally spectrally rigid among smooth domains
with the symmetries of the ellipse. Spectral rigidity of the ellipse has been expected for a long time and is a kind of model
problem in inverse spectral theory. Ellipses are special since their billiard flows and maps are completely integrable.
It was conjectured by G. D. Birkhoff that the ellipse is the only convex smooth plane domain with completely integrable billiards.
Our results are somewhat analogous to the spectral rigidity of flat tori or the sphere in the Riemannian setting.
The main novel step in the proof is the Hadamard variational formula for the wave trace. It is of independent interest and has
applications to spectral rigidity beyond the setting of ellipses. The main advance over prior results is that the domains
are allowed to be smooth rather than real analytic.

Speaker:  Gregory Berkolaiko Texas A&M University 
Title:  Nodal count of eigenfunctions and stability of nodal partitions 
Abstract:  In this talk we address the nodal count (i.e., the number of nodal
domains) of eigenfunctions of Schroedinger operators with Dirichlet
boundary conditions in bounded domains (also called billiards).
According to the classical Sturm theorem, in dimension one the nth
eigenfunction has exactly n nodal domains. In higher dimensions the
situation is less definite: the Nodal Theorem of Courant asserts that,
in any dimension, the number of nodal domains of the nth
eigenfunction cannot exceed n. However, in most cases the ``nodal
deficiency''  the difference between n and the number of nodal
domains of the nth eigenfunction  is nonzero. Moreover, examples
are known of eigenfunctions with an arbitrarily large index n that
have just two nodal domains. One can say that the nature of the nodal
deficiency had not been understood.
We show that, under some genericity conditions, the nodal deficiency is connected to stability of the partition of the billiard by the zeros of the eigenfunction. More precisely, we define an "energy" functional on an infinite dimensional variety of partitions of the billiard. We show that the critical points of the functional correspond exactly to the partitions induced by the eigenfunctions while the Morse indices of the critical points coincide with the nodal deficiencies. 
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