The HADES seminar on Tuesday, October 22nd will be given by  Amir Vig in Evans 740 from 3:40 to 5 pm.

Speaker: Amir Vig, UCI & MSRI

Abstract: The wave trace is a distribution on $\mathbb{R}$ given by $\sum_{j = 1}^\infty e^{it \lambda_j}$, where $\lambda_j^2$ are the (positive) eigenvalues of the Laplacian on a compact domain. In general, two linear waves can be superimposed to give another solution to the wave equation. When we add up a bunch of waves at different frequencies, the peak singularities appear at points with substantial constructive interference. On a manifold, the famous “propagation of singularities” tells us that waves propagate along geodesics, so the constructive interference is most pronounced along orbits which are traversed infinitely often (i.e. periodic orbits). On the trace side of things, this phenomenon is reflected in the Poisson relation, which says that the singular support of the wave trace is contained in the length spectrum (the collection of lengths of all periodic orbits). For planar domains, the geodesic flow is replaced by the billiard (or broken bicharacteristic) flow and we see an interesting connection between geometric, dynamical and spectral properties of the domain. In this talk, we introduce some simple cases of wave trace formulas before discussing recent work on explicit formulas for wave invariants associated to periodic orbits of small rotation number in a smooth, strictly convex bounded planar domain. This involves proving a dynamical theorem on the structure of such orbits and then constructing an explicit oscillatory integral representation, which microlocally approximates the wave propagator in the interior.

The HADES seminar on Tuesday, October 8th will be given by  Malo Jézéquel in Evans 740 from 3:40 to 5 pm.

Speaker: Malo Jézéquel, LPSM & MSRI

Abstract: Fine statistical properties of a smooth Anosov flow may be studied through Ruelle resonances. These resonances may be described as the zeroes of a dynamical determinant, i.e. an entire function defined in terms of the periodic data of the flow. We are interested in another relationship between Ruelle resonances and periodic data of the flow: a trace formula conjectured by Dyatlov and Zworski. This formula is known to be true for real-analytic Anosov flows by a result of Fried, we will see how this hypothesis can be weakened.

The HADES seminar on Tuesday, September 24th will be given by  Mohandas Pillai in Evans 740 from 3:40 to 5 pm.

Speaker: Mohandas Pillai, Berkeley

Abstract: This talk will be about the wave maps problem with domain $\mathbb{R}^{2+1}$ and target $\mathbb{S}^{2}$ in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from $\mathbb{R}^{2}$ to $\mathbb{S}^{2}$, with polar angle equal to $Q_{1}(r) = 2 \arctan(r)$. By applying the scaling symmetry of the equation, $Q_{\lambda}(r) = Q_{1}(r \lambda)$ is also a harmonic map, and the family of all such $Q_{\lambda}$ are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps.

In this talk, I will discuss how to construct a collection of infinite time blowup solutions along the $Q_{\lambda}$ family, with a symbol class of possible asymptotic behaviors of $\lambda$.

The HADES seminar on Tuesday, September 17th will be given by  Alix Deleporte in Evans 740 from 3:40 to 5 pm.

Speaker: Alix Deleporte, MSRI

Abstract: Szegö kernels encode information on weighted holomorphic functions, or holomorphic sections. In an appropriate large curvature limit, they enjoy a semiclassical structure. Among other applications, these kernels are used to define an alternative quantization scheme : Berezin-Toeplitz quantization.

This talk will be an opportunity to further motivate attendance to M. Zworski’s course.

The HADES seminar on Tuesday, September 10th will be given by  Stéphane Nonnenmacher in Evans 748 from 3:40 to 5 pm

Speaker: Stéphane Nonnenmacher, Univ. Paris-Sud & MSRI

Abstract: The spectrum of a nonselfadjoint linear operator can be very unstable, that is sensitive to perturbations, an phenomenon usually referred to as the “pseudospectral effect”. In order to quantify this phenomenon, we investigate a simple class of nonselfadjoint 1-dimensional semiclassical (pseudo-)differential operators, submitted to small random perturbations. The spectrum of this randomly perturbed operator is then viewed as a random point process on the complex plane, whose statistical properties we wish to analyze.

Hager & Sjöstrand have shown that, in the semiclassical limit, the randomly perturbed eigenvalues satisfies a probabilistic form of Weyl’s law, at the macroscopic scale. We in turn investigate the statistical distribution of the eigenvalues at the microscopic scale (scale of the distance between nearby eigenvalues). We show that at this scale, the spectral statistics satisfy a partial form of universality: spectral correlations can be expressed in terms of a universal object, the Gaussian Analytic Function (GAF), and a few parameters depending on the initial operator, and of the type of random disorder.

A central tool in our analysis is a well-posed Grushin problem, which turns our spectral problem on \(L^2(R)\) into an effective nonlinear spectral problem on a finite dimensional subspace (“effective Hamiltonian”). This Grushin problem is set up by studying the “classical spectrum” of our initial semiclassical operator (a region in the complex plane), constructing quasimodes of this operator, and analyzing the (complex-)energy-dependence of these quasimodes.

This is joint work with Matin Vogel.

The HADES seminar on Tuesday, August 27th will be given by Haoren Xiong in Evans 748 from 3:40 to 5 pm

Speaker: Haoren Xiong, Berkeley

Abstract: Scattering resonances of Schrödinger operator with a compactly supported potential are defined as the poles of the meromorphic continuation of the resolvent. I will introduce the method of complex scaling which produces a natural family of non-self-adjoint operators whose discrete spectrum consists of resonances. Furthermore, we will see that similar results hold in the case of dilation analytic potentials.