We are pleased to announce a follow-up conference on Waves, Spectral Theory, and Applications.

The conference will be centered on about 10 talks over three days given by mathematicians and scientists at various stages in their careers.  It will also feature a contributed poster session, as well as several break-out sessions each afternoon on particularly important topics.  Break-out session ideas so far include:  PT symmetry, modified energy methods, Maslov indices in stability theory, quantum graphs and more.

This is an NSF funded conference to be held at the University of North Carolina, Chapel Hill – co-organized by Jeremy Marzuola (UNC), Roy Goodman (NJIT) and Panos Kevrekidis (UMASS).

## Registration:

Waves, Spectral Theory, and Applications Registration

Pay for Registration Fee and/or Dinner Online.

## Speaker List:

David Ambrose (Drexel)
Ben Harrop-Griffiths (NYU)
Peter Hintz (MIT)
Peter Hislop (Kentucky)
Dmitry Pelinovsky (McMaster)
Nancy Rodriguez (UNC)
Christof Sparber (UIC)
Atanas Stefanov (Kansas)
Chongchun Zeng (Georgia Tech)

## Breakout Sessions (In Progress):

1. PT Symmetry with Ziad Musslimani
2. Maslov Indices in PDE problems with Graham Cox
3. Metric and Discrete Graphs with Greg Berkolaiko
4. Microlocal methods in Wave Equations
5. Modified Energy methods in Quasilinear PDE with Ben Harrop-Griffiths
6. Random Schrödinger Operators

## Conference Schedule:

Friday, October 20th:

9-9:30 AM, Phillips Hall 330:  Coffee and Welcome, Registration Information
9:30-10:15 AM, Phillips Hall 332: Peter Hislop
10:15-10:45 AM, Phillips Hall 330:  Coffee
10:45 AM-11:30 AM,Phillips Hall 330: Atanas Stefanov
11:30 AM-1:00 PM:  Lunch
1:00-2:30 PM, Phillips Hall 367 and 381:  Breakout Sessions –  Maslov Indices in PDE & Random Schrödinger
2:30-3:00 PM, Phillips Hall 330:  Coffee
3:00-3:45 PM, Phillips Hall 332: Dmitry Pelinovsky
4:00-4:45 PM, Phillips Hall 332:Nancy Rodriguez
6:00-9:00 PM, Venable Bistro in Carrboro:  Conference Dinner

Saturday, October 21st:

9:00-9:45 AM, Phillips Hall 332:  Chongchun Zeng
9:45-10:15 AM, Phillips Hall 330:  Coffee
10:15 AM-11:00 AM, Phillips Hall 332: Christof Sparber
11:00 AM-12:00 PM, Phillips Hall 330:  Poster Session Walk-Throughs
12:00 PM -1:00 PM:  Lunch
1:00-2:30 PM, Phillips Hall 367 and 383:  Breakout Sessions –  Quantum Graphs & PT Symmetry
2:30-3:00 PM, Phillips Hall 330:  Coffee
3:00-3:45 PM, Phillips Hall 332: Benoit Pausader
4:00-4:45 PM, Phillips Hall 332: David Ambrose

Sunday, October 22nd:

9:00-9:45 AM, Phillips Hall 332: Peter Hintz
9:45-10:15 AM, Phillips Hall 330:  Coffee
10:15 AM-11:00 AM, Phillips Hall 332:  Ben Harrop-Griffiths
11:00 AM-12:00 PM, Phillips Hall 330:  Poster Session Walk-Throughs
12:00 PM -1:00 PM:  Lunch
1:00-2:30 PM, Phillips Hall 367 and 381:  Breakout Sessions –  Microlocal Methods in Waves & Modified Energy Methods
2:30-3:00 PM, Phillips Hall 330:  Coffee

### Accommodations:

We have reserved a block of rooms at the Hampton Inn.  To book directly at the conference rate, click here.  Unless you are a graduate student or postdoc for whom it would be a financial hardship, please make your own hotel reservations.  Those who are approved for funding will be reimbursed.

Hotel options include the following places regularly used by the department:

## Funding Information:

Supported by NSF DMS-1536072.

The previous workshop information can be found at

Titles and Abstracts:

David Ambrose:

Title: Well-posedness and ill-posedness for degenerate dispersive equations
Abstract:  Equations with nonlinear/degenerate dispersion can admit solutions with interesting features, such as compactly supported traveling waves, but there tends to be little general existence theory.  In this talk, we will demonstrate an ill-posedness theorem for a degenerate Airy equation, and some well-posedness theorems when the dispersion does not degenerate.  Equations in the family under consideration include Rosenau-Hyman compacton equations and the Harry Dym equation.

Benjamin Harrop-Griffiths:

Title: Degenerate disperisve equations and compactons
Abstract:  We discuss a family of Hamiltonian degenerate dispersive equations that admit compactly supported solitons or “compactons”. We discuss their variational properties and stability. This is joint work with Pierre Germain and Jeremy Marzuola.

Peter Hintz:

Title: Resonances for obstacles in hyperbolic space
Abstract: We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound $\Im\lambda\leq -1/2$; in odd dimensions and for small obstacles with diameter $\rho$, we improve this to $\Im\lambda < -C/\rho$ for a universal constant $C$. Our proofs largely rely on the classical vector field approach of Morawetz. We also explain how to relate resonances for small obstacles to scattering resonances in Euclidean space. This talk is based on joint work with Maciej Zworski.

Peter Hislop:

Title:  SPECTRAL STATISTICS IN THE LOCALIZATION REGIME FOR RANDOM SCHROEDINGER OPERATORS WITH SINGULAR INTERACTIONS
Abstract:  We prove that the local eigenvalue statistics for Schroedinger operators with random point interactions on Rd, for d = 1; 2; 3 is Poissonian in the localization regime. This is the first example of eigenvalue statistics for multi-dimensional random Schroedinger operators in the continuum. The special structure of the point interactions facilitates the proof of the Minami estimate. This is joint work with W. Kirsch and M. Krishna.

Title: On the Einstein-massive scalar field equation
Abstract: (joint work with A. Ionescu) We consider the problem of the stability of Minkowski space in the presence of a massive scalar field. The difficulty comes from the interaction of two types of dispersive waves: the metric satisfies a quasilinear wave equation, while the scalar field solves a quasilinear Klein-Gordon equation. We show asymptotic stability in harmonic coordinates. This extends work of LeFloch-Ma and Q. Wang who considered the case of solutions which agree with Schwarzschild outside a compact set. We will present the arguments on a simpler model problem.

Dmitry Pelinovsky:

Title: Ground state of the conformal flow on three-sphere
Abstract:  We consider the conformal flow model derived as a normal form for the conformally invariant cubic wave equation on three-sphere. We prove that the energy attains a global constrained maximum at a family of particular stationary solutions which we call the ground state family. Using this fact and spectral properties of the linearized flow (which are interesting on their own due to a supersymmetric structure) we prove nonlinear orbital stability of the ground state family. The main difficulty in the proof is due to the degeneracy of the ground state family as a constrained maximizer of the energy. This is a joint work with P. Bizon and D. Hunik-Kostyra (Krakow).

Nancy Rodriguez:

Title: On the global existence and qualitative behavior of solutions to a model for urban crime
Abstract: We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system which was introduced to describe the dynamics of urban crime.  In bounded intervals I will first discuss the existence of global classical solutions for all reasonably regular non-negative initial data. Next I will address the issue of determining the qualitative behavior of solutions.  Finally, I will conclude with some numerical simulations exploring possible effects that may arise when considering large cross diffusion terms not covered by our qualitative analysis.

Christof Sparber:

Abstract:  We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schrödinger equations in which either the nonlinear coupling constant or, equivalently, the solution is asymptotically small. To this end, a notion of criticality is introduced at which the linear bound states stay adiabatically stable, but nonlinear effects start to show up at leading order in the form of a slowly varying nonlinear phase modulation. In addition, we prove that in the same regime a class of nonlinear bound states also stays adiabatically stable, at least in terms of spectral projections.

Atanas Stefanov:

Title: On the existence and stability of normalized ground states for the Kawahara and  fourth order NLS models
Abstract:  We construct variationally the  normalized ground states for the Kawahara equation, which contains both fifth and third order KdV type dispersion.  We also show the spectral stability of these solitons.  Our results provide a significant extension in parameter space of the current rigorous results. At the same time, we verify and clarify recent numerical simulations tat looked into  the stability of these solitons. We also discuss the related question of the stability of the (multidimensional)  solitons for  fourth order NLS. Of particular interest is a new paradigm that we discover. Namely, all else being equal, the form of the second order derivative (mixed second derivatives vs. pure Laplacian) has implications on the range of existence and stability of the normalized waves. This is joint work with Y. Posukhovskyi.

Chongchun Zeng:

Title:  Instability, index theorems, and exponential dichotomy of Hamiltonian PDEs
Abstract: Motivated by the stability/instability analysis of coherent states (standing waves, traveling waves, etc.) in nonlinear Hamiltonian PDEs such as BBM, GP, and 2-D Euler equations, we consider a general linear Hamiltonian system $u_t = JL u$ in a real Hilbert space $X$ — the energy space. The main assumption is that the energy functional $\frac 12 \langle Lu, u\rangle$ has only finitely many negative dimensions — $n^-(L) < \infty$. Our first result is an $L$-orthogonal decomposition of $X$ into closed subspaces so that $JL$ has a nice structure. Consequently, we obtain an index theorem which relates $n^-(L)$ and the dimensions of subspaces of generalized eigenvectors of some eigenvalues of $JL$, along with some information on such subspaces. Our third result is the linear exponential trichotomy of the group $e^{tJL}$. This includes the nonexistence of exponential growth in the finite co-dimensional invariant center subspace and the optimal bounds on the algebraic growth rate there. Next we consider the robustness of the stability/instability under small Hamiltonian perturbations. In particular, we give a necessary and sufficient condition on whether a purely imaginary eigenvalues may become hyperbolic under small perturbations.  Finally we revisit some nonlinear Hamiltonian PDEs. This is a joint work with Zhiwu Lin.