### Inference in "Likelihood-Free" Bayesian Networks

Marco Cusumano-Towner

### Unique ergodicity and measures invariant under permutations of N

Cameron Freer
Consider dense linear orders without endpoints having underlying set the natural numbers. They admit a unique probability measure that is invariant to the logic action of S_\infty on the underlying set, called the Glasner-Weiss measure. In contrast, the Rado graph admits continuum many ergodic invariant measures (among them, the Erdős-Rényi constructions for arbitrary p such that 0 < p < 1). We characterize all isomorphism classes of structures that admit an unique invariant measure, and...

### Spectral instabilities of Schrödinger operators with complex potentials

Petr Siegl
We present an overview of recent results on pseudospectra and basis properties of the eigensystem of one-dimensional Schrödinger operators with unbounded complex potentials. In particular, we address the problem of localizing the transition between spectral (Riesz basis of eigenvectors and "normal" behavior of resolvent norm) and pseudospectral (vast regions in the complex plane where resolvent norm explodes) character of these operators depending on the size of real and imaginary parts of the potential.

### Microstructures of Electricity and Magnetism

Antonio Capella Kort

### Asymptotic geometry of the Hitchin moduli space

Jan Swoboda
In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiss and Frederik Witt on the asymptotics of the natural $L^2$ metric on the Hitchin moduli space of rank-$2$ Higgs bundles. It will be shown that on the regular part of the Hitchin fibration this metric is well-approximated by the so-called semiflat metric coming from the algebraic completely integrable system moduli space is endowed with. This result confirms some aspects of a...

### On a question of Assaf Naor

Krzysztof Oleszkiewicz
For any separable Banach space $(F,\| \cdot \|)$ and independent $F$-valued random vectors $X$ and $Y$ such that ${\mathbf E} \|X\|, {\mathbf E} \|Y\|< \infty$, we have $\inf_{z \in F} ({\mathbf E}\|X-z\|+{\mathbf E}\|Y-z\|) \leq 3 \cdot {\mathbf E}\| X-Y\|.$ Indeed, it suffices to consider $z=({\mathbf E} X+{\mathbf E} Y)/2$ and use Jensen's inequality. Assaf Naor asked whether the constant $3$ in the inequality is optimal. We will discuss this and related problems.

### On a Schrödinger operator with a purely imaginary potential in the semiclassical limit

Yaniv Almog
We consider the operator ${\mathcal A}_h=-h^2\Delta+iV$ in the semi-classical limit $h\rightarrow0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more...

Linc Kesler

### Klaus-Shaw potentials for the Ablowitz-Ladik lattice

Some PDEs and ODEs admit a Lax pair (a pair of linear operators) to be completely solve the equation. One of these operators defines a spectral problem. For some equations (as for the Korteweg-deVries, KdV, equation) this operator is self-adjoint and, consequently, its discrete spectrum is real. However, for some other equations, this operator is non-selfadjoint, such as for the nonlinear Schrödinger (NLS) equation or the Ablowitz-Ladik equation. In 2001, M. Klaus and J. K....

### Kink dynamics in a parametric $\phi^6$ system: a model with controllably many internal modes

Aslihan Demirkaya-Ozkaya
In the present work, we intend to explore a variant of the $\phi^6$ model originally proposed in \textit{Phys. Rev. D} \textbf{12}, 1606 (1975) as a prototypical, so-called, bag'' model where domain walls play the role of quarks within hadrons. We examine the prototypical steady state of the model, namely an apparent bound state of two kink structures. We explore its linearization and find that as a function of a prototypical parameter controlling the curvature of...

### Existence and stability of fronts in inhomogeneous wave equations

Gianne Derks
Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. Travelling waves in such models tend to interact with the inhomogeneity and get trapped, reflected, or slowed down. In this talk, wave equations with finite length inhomogeneities will be...

### Stability of traveling fronts in a model for porous media combustion

Anna Ghazaryan
We consider a model of combustion in hydraulically resistant porous media. There are several reductions of this systems that can be used to understand the evolution of the combustion fronts. One reduction is based on the assumption that the ratio of pressure and molecular diffusivities is close to zero, a different reduction is obtained when the Lewis number chosen in a specific way. Fronts exists in both reduced systems. For the stability analysis of the...

Alexandru Suciu

David Harbater

Eliyahu Matzri

Danny Neftin

### Which drift/diffusion formulas for velocity-jump processes?

This talk examines a class of linear hyperbolic systems which generalizes the Goldstein-Kac model to an arbitrary finite number of speeds with transition rates. Under the basic assumptions that the transition matrix is symmetric and irreducible, and the speed differences generate all the space, the system exhibits a large-time behavior described by a parabolic advection-diffusion equation. The main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of...

### Analyzing Hamiltonian spectral problems via the Krein matrix

Todd Kapitula
The Krein matrix is a matrix-valued function which can be used to study Hamiltonian spectral problems. Akin to the Evans matrix, it has the property that it is singular when evaluated at an eigenvalue. Unlike the Evans matrix, it is not analytic, but is instead meromorphic. I will briefly go over its construction, and then apply it to the study of spectral stability of small periodic waves for a couple of equations.

Alexander Betts

### Overturned traveling interfacial waves

Benjamin Akers
Periodic traveling waves are computed on parameterized interfaces, which are not functions of the horizontal coordinate(s). These overturned traveling waves are computed on one and two-dimensional interfaces, on a classic interface between two fluids as well as on boundary formed by a hydroelastic ice sheet. Numerical continuation procedures are coupled with local and global bifurcation theorems. Extreme wave types and bifurcation surfaces are presented. The prospects for stability of overturned traveling waves are discussed.

### Stability of periodic travelling wave solutions to Korteweg-de Vries and related equations

Olga Trichtchenko
In this talk, we explore the simplest equation that exhibits high frequency instabilities, the fifth-order Korteweg-de Vries equation. We show how to derive the necessary condition for an instability of a perturbation of a small amplitude, periodic travelling wave solutions. We proceed by examining how these unstable perturbations change and grow in time as the underlying solution changes. We conclude by commenting on what happens with a different nonlinearity in the underlying equation.

### The Maslov index and the spectrum of differential operators

Yuri Latushkin
This is a joint work with M. Beck, G. Cox, C. Jones, R. Marangell, K. McQuighan, A. Sukhtayev, and S. Sukhtaiev. In this talk we discuss some recent results on connections between the Maslov and the Morse indices for differential operators. The Morse index is a spectral quantity defined as the number of negative eigenvalues counting multiplicities while the Maslov index is a geometric characteristic defined as the signed number of intersections of a path...

### Stability of periodic waves in Hamiltonian PDEs

Sylvie Benzoni-Gavage
For Hamiltonian systems of PDEs the stability of periodic waves is encoded by the Hessian of an action integral, as shown in earlier work. This talk will deal with two asymptotic regimes, namely for waves of small amplitude and for waves of long wavelength. In both cases stability criteria can be investigated analytically, thanks to the asymptotic expansions of the Hessian of the action and their special structure. The stability results thus obtained apply to...

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