
Date Time 
Location  Speaker 
Title – click for abstract 

02/09 1:50pm 
BLOC 628 
James Kennedy University of Lisbon 
Asymptotically optimal Laplacian eigenvalues and Polya's conjecture
A longstanding problem in spectral geometry is to determine the domain(s) which minimise a given eigenvalue of a differential operator such as the Laplacian with Dirichlet boundary conditions, among all domains of given volume. For example, the Theorem of (Rayleigh) FaberKrahn states that the smallest eigenvalue is minimal when the domain is a ball. Very little to nothing is known about domains minimising the higher eigenvalues, but the Weyl asymptotics suggest that the ball should in a certain sense be asymptotically optimal.
In the first part of this talk, we will sketch a new approach to this problem initiated by a paper of Colbois and El Soufi in 2014, which asks not after the minimising domains themselves but properties of the corresponding sequence of minimal values. This serendipitously also yields a new Ansatz for tackling the more than 50 year old conjecture of Polya that the kth eigenvalue of the Dirichlet Laplacian on any domain always lies above the corresponding first term in the Weyl asymptotics for that eigenvalue. Along the way, we will additionally meet variants of Gauss circle problem.
In a second part, we will present some recent analogous results for the Laplacian with Robin boundary conditions, which are ongoing joint work with Pedro Freitas. 

02/09 2:50pm 
Blocker 605AX 
Yaiza Canzani UNC Chapel Hill 
On the growth of eigenfunctions averages
In this talk we discuss the behavior of Laplace eigenfunctions when restricted to a fixed submanifold by studying the averages given by the integral of the eigenfunctions over the submanifold. In particular, we show that the averages decay to zero when working on a surface with Anosov geodesic flow regardless of the submanifold (curve) that one picks. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages. This is based on joint work with Jeffrey Galkowski. 

02/23 1:50pm 
BLOC 628 
George E. A. Matsas Instituto de Fisica Teorica, Universidade Estadual Paulista 
Overview of the Unruh Effect for Mathematicians
The Unruh effect is interesting to physicists and mathematicians.
Unveiled by a physicist, Bill Unruh, in 1975, it vindicated Steve Fulling's
surprising conclusion that different observers extract, in general,
different particle contents from the same field theory (e.g., inertial
observers in the usual vacuum would freeze to death at 0 K, where observers
accelerated enough may burn into ashes). This seminar is designed for
mathematicians who are not acquainted with quantum field theory but wish to
understand what the Unruh effect means, up to what extent we must trust it,
and why it is so important to our comprehension of some conceptual issues.


03/02 1:50pm 
BLOC 628 
Gregory Berkolaiko TAMU 
Nodal count distribution of graph eigenfunctions
We start by reviewing the notion of “quantum graph”, its eigenfunctions and the problem of counting the number of their zeros. The nodal surplus of the nth eigenfunction is defined as the number of its zeros minus (n1). When the graph is composed of two or more blocks separated by bridges, we propose a way to define a “local nodal surplus” of a given block. Since the eigenfunction index n has no local meaning, the local nodal surplus has to be defined in an indirect way via the nodalmagnetic theorem of Berkolaiko, Colin de Verdière and Weyand.
We will discuss the properties of the local nodal surplus and their consequences. In particular, its symmetry properties allow us to prove the longstanding conjecture that the nodal surplus distribution for graphs with β disjoint loops is binomial with parameters (β,1/2).
The talk is based on joint work with Lior Alon and Ram Band, arXiv:1709.10413 (accepted to CMP). 

03/09 1:50pm 
BLOC 628 
Peter Kuchment Texas A&M 
On LiouvilleRiemannRoch theorems on cocompact abelian coverings
A generalization by Gromov and Shubin [23] of the classical RiemannRoch theorem describes the index of an elliptic operator on a compact manifold with a divisor of prescribed zeros and allowed singularities. On the other hand, Liouville type theorems count the number of solutions of a given polynomial growth of the LaplaceBeltrami (or more general elliptic) equation on a noncompact manifold. The solution of a 1975 Yau's conjecture [6] by Colding and Minicozzi [1] implies in particular, that such dimensions are finite for LaplaceBeltrami equation on a nilpotent cocompact covering. In the case of an abelian covering, much more complete Liouville theorems (including exact formulas for dimensions) have been obtained by Kuchment and Pinchover [45]. One wonders whether such results have a combined generalization that would allow for a divisor that "includes the infinity." Surprisingly, combining the two types of results turns out being rather nontrivial. The talk will present such a result obtained recently in a joint work with Minh Kha (former A&M PhD student, currently postdoc at U. Arizona).
[1] Colding, T. H., Minicozzi, W. P.: Harmonic functions on manifolds, Ann. of Math. 146 (1997), 725–747.
[2] M. Gromov and M. A. Shubin, The RiemannRoch theorem for elliptic operators, I. M. Gel'fand Seminar, 1993, pp. 211241.
[3] "  , The RiemannRoch theorem for elliptic operators and solvability of elliptic equations
with additional conditions on compact subsets, Invent. Math. 117 (1994), no. 1, 165180.
[4] P. Kuchment and Y. Pinchover, Integral representations and Liouville theorems for solutions of periodic elliptic equations, J. Funct. Anal. 181 (2001), no. 2, 402446.
[5] " , Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds, Trans. Amer. Math. Soc. 359 (2007), no. 12, 57775815.
[6] Yau, S. T.: Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math. 28 (1975), 201–228. 

04/06 1:50pm 
BLOC 628 
Oran Gannot Northwestern University 
Semiclassical diffraction by conormal potential singularities
I will describe joint work with Jared Wunsch on propagation of singularities for some semiclassical Schrödinger equations, where the potential is conormal to a hypersurface. Semiclassical singularities of a given strength propagate across the hypersurface up to a threshold depending on both the regularity of the potential and the singularities along certainbackwards broken bicharacteristics. 

04/13 1:50pm 
BLOC 628 
Vitaly Moroz University of Swansea 
Asymptotic properties of ground states of a semilinear elliptic problem with a vanishing parameter.
We consider an elliptic problem with a doublewell nonlinearity and a vanishing parameter. The behaviour of solutions depends sensitively on whether a power in the nonlinearity is less, equal or bigger than the critical Sobolev exponent. In the most delicate critical Sobolev regime the asymptotic behaviour of the solutions is given by a particular solution of the critical EmdenFowler equation, whose choice depends on in a nontrivial way on the space dimension.
Joint work with Cyrill Muratov (NJIT).


04/16 1:50pm 
BLOC 220 
Barry Simon Caltech 
Szegő–Widom asymptotics for Chebyshev polynomials on subsets of R
Chebyshev polynomials for a compact subset e ⊂ R are defined to be the monic polynomials with minimal $·_∞ $ over e. In 1969, Widom made a conjecture about the asymptotics of these polynomials when e was a finite gap set. We prove this conjecture and extend it also to those infinite gap sets which obey a Parreau–Widom and a Direct Cauchy Theory condition. This talk will begin with a generalities about Chebyshev Polynomials. This is joint work with Jacob Christiansen and Maxim Zinchenko and partly with Peter Yuditskii. 

04/16 4:00pm 
BLOC 117 
Barry Simon Caltech 
A colloquium talk: Tales of our Forefathers
This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. 

04/20 1:50pm 
BLOC 628 
David Borthwick Emory University 
Distribution of Resonances for Hyperbolic Surfaces
For noncompact hyperbolic surfaces, the appropriate generalization of the eigenvalue spectrum is the resonance set, the set of poles of the resolvent of a meromoprhic continuation of the Laplacian. Hyperbolic surfaces serve as a model case for quantum theory when the underlying classical dynamics is chaotic. In this talk I’ll explain how the resonances are defined and discuss our current understanding of their distribution. I’ll introduce some conjectures inspired by the physics of quantum chaotic systems, and discuss numerical evidence for these conjectures and the partial progress that has been made recently. 

05/04 1:50pm 
BLOC 628 
Junehyuk Jung Texas A&M University 
Boundedness of the number of nodal domains of eigenfunctions
The asymptotic of the number of nodal domains of eigenfunctions on a manifold is closely related to the dynamics of the geodesic flow on the manifold. For instance, if a surface with nonempty boundary has an ergodic geodesic flow, then for any given Dirichlet eigenbasis, one can find a subsequence of density one where the number of nodal domains tends to +\infty.
In this talk, I'm going to discuss what happens to the unit circle bundle over a manifold. When equipped with a metric which makes the Laplacian to commute with the circular action on each fiber, the geodesic flow never is ergodic. Recently I and Steve Zelditch proved that among such metrics the following property is generic:
One can find an eigenbasis that has a subsequence of density 1 where the number of nodal domains is identically 2.
This highlights how underlying dynamics can impact the nodal counting.
I will sketch proof when we are considering a unit tangent bundle of a compact surface with the genus \neq 1. 