Download Title: Natural boundary for the susceptibility function of generic

Viviane Baladi:
Title: Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps
Abstract: (joint with S. Marmi and D. Sauzin) We consider the susceptibility function Ψ(z) of a piecewise expanding unimodal interval map. Combining previous results (deduced from spectral properties of
Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a
generic condition) implies that Ψ(z) has a strong natural boundary on the unit circle. The Breuer-Simon
method provides uncountably many candidates for the outer functions of Ψ(z), associated to precritical
orbits. If the perturbation is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit)
implies that the nontangential limit of the Ψ(z) as z tends to 1 exists and coincides with the derivative
of the acim with respect to the map (linear response formula).
David Borthwick:
Title: Distribution of Resonances for Hyperbolic Surfaces
Abstract: We report on recent numerical investigations of resonance distributions for hyperbolic surfaces. These results are based on an effective method for the approximating the Selberg zeta function for
Schottky groups with a small number of generators. The primary purpose of these numerical studies is
to investigate three issues of resonance distribution that have generated great interest lately: the fractal
Weyl law, the spectral gap, and the concentration of decay rates at half the classical escape rate.
Jean-Marc Bouclet:
Title: Low frequency resolvent estimates on asymptotically flat manifolds, and applications.
Abstract: I will discuss both recent results and ongoing projects with Haruya Mizutani and Julien
Royer on resolvent estimates on asymptotically flat manifolds and their applications to global in time
Strichartz estimates.
Yaiza Canzani:
Title: Counting intersections of nodal lines of Laplace eigenfunctions with curves on real analytic
surfaces
Abstract: Let (M, g) be a compact real analytic surface with no boundary, and let H denote a closed
analytic curve in M . Write ϕλ for the eigenfunctions of the positive definite Laplacian ∆g , ∆g ϕλ = λ2 ϕλ .
When M is the two-torus and H has non-vanishing curvature (Bourgain-Rudnick) or when M is an
arithmetic surface and H is a geodesic circle or a closed horocylcle (Jung), it has been shown that H is
a good curve in the sense that kϕλ kL2 (H) ≥ e−Cλ for some C > 0. In these cases it was proved that
#{ϕ−1
λ (0) ∩ H} = O(λ).
(∗)
In this talk we show that the bound (∗) holds on the general surface (M, g) provided H satisfies an
admissibility condition that is weaker than the “goodness” condition. This is Joint work with John Toth.
Tanya Christiansen:
Title: Resonances in even-dimensional Euclidean scattering
Abstract: The resonances of a compactly supported perturbation of the Laplacian on Rd lie in the
complex plane if d is odd. If d is even, they lie on the logarithmic cover of the complex plane. This difference makes the study of resonances in even dimensions more complicated than in the odd-dimensional
case. In this talk we focus on some results about resonances which are different, or which have different
proofs, in the even-dimensional case than in the odd-dimensional case. Some of the results described in
the talk are joint work with P. Hislop.
Hans Christianson:
Title: High frequency resolvent estimates on warped products
Abstract: We consider the cutoff resolvent on asymptotically Euclidean warped product spaces, and
show that the resolvent is either (almost) bounded, or blows up faster than any polynomial. The proof
proceeds by separating variables to obtain a 1-d semiclassical Schroedinger operator, and then classifying
the microlocal resolvent behaviour near each type of critical point.
Yves Colin de Verdière:
Title: Quantum ergodicity and dynamics for discontinuous systems: examples and the case of “quantum graphs”.
Abstract: the work I will present was inspired by the beautiful ArXiv paper by Jakobson, Safarov
and Strohmaier (ArXiv 1301.6783). I plan to give some example of ergodic geodesic Markov process for
discontinuous Riemannian metrics on the 2-sphere. If time permits, I will also show that Quantum ergodicity is never valid for Quantum graphs.
Kiril Datchev:
Title:
Abstract:
Suresh Eswarathasan:
Title: Intersections of fractal sets and Fourier analysis
Abstract: A classical theorem due to Mattila says that if A, B ⊂ Rd of Hausdorff dimension sA , sB ,
respectively, with sA + sB ≥ d, sB > d+1
2 and dimH (A × B) = sA + sB ≥ d, then
dimH (A ∩ (z + B)) ≤ sA + sB − d
for almost every z ∈ Rd , in the sense of Lebesgue measure. We obtain a variable coefficient version of
this result in which we are able to replace the Hausdorff dimension with the upper Minkowski dimension
on the left-hand-side of the first inequality; specifically, we consider a more general family of variable
coefficient transformations other than rotations and translations. We use microlocal analysis, such as
Fourier integral operator bounds, and other techniques of harmonic analysis in our investigation. This is
joint work with Alex Iosevich and Krystal Taylor.
Frédéric Faure
Title: Band structure for the Ruelle spectrum of prequantum Anosov map and contact Anosov flows
Abstract: Using semiclassical analysis, we will present spectral properties of the transfer operator of
prequantum Anosov maps (this is a U(1) extension of a symplectic Anosov diffeomorphism preserving an
associated specific connection) and spectral properties of geodesic flows on negatively curved manifolds
(or contact Anosov flows). In the case of contact Anosov vector field X on a smooth compact manifold M
and if V is a smooth function on the manifold M, it is known that the differential operator A=-X+V has
some discrete spectrum called Ruelle-Pollicott resonances in specific anisotropic Sobolev spaces. We show
that for —Im(z)— large the eigenvalues of A are restricted to vertical bands and in the gaps between
the bands, the resolvent of A is bounded uniformly with respect to —Im(z)—. In each isolated band the
density of eigenvalues is given by the Weyl law. This band spectrum gives an asymptotic expansion for
dynamical correlation functions. This is a work with Masato Tsujii.
Sebastien Gouëzel:
Title: Numerical estimates for the spectral radius in surface groups.
Abstract: Describing the analogue of the spectrum of the laplacian in discrete analogues of the hyperbolic plane, i.e., Cayley graphs of fundamental groups of hyperbolic surfaces, is difficult since there is
no handy formula in general. We will explain an elementary approach to get numerical estimates for the
corresponding spectral radius, improving upon the previous results in the literature.
Andrew Hassell:
Title: Equidistribution of eigenvalues of the scattering matrix.
Abstract: I will talk about joint work with Datchev, Gell-Redman and Humphries, as well as more
recent work with Gell-Redman and Zelditch, concerning equidistribution of eigenvalues of the scattering
matrix on the unit circle, for semiclassical potential scattering. If time permits I may also discuss the
case of the Cayley transform of the Dirichlet to Neumann operator on a manifold with boundary.
Dmitry Jakobson:
Title: Averaging over Riemannian metrics.
I will survey several recent results related to averaging over different spaces of Riemannian metrics.
The first result is joint work with Y. Canzani and J. Toth. We study the moments of propagated
perturbed eigenfunctions, evaluated at a fixed point x on a compact manifold, considered as random variables that arise from random pertrubations of a metric. The (finite-dimensional) family of Schrodinger
operators corresponds to perturbations of the reference Riemannian metrics. Assuming the perturbation
family has nontrivial projections onto conformal changes of the metric at x, we establish asymptotics for
the odd moments. Assuming the perturbation family has non-degenerate projection onto the space of
volume-preserving transformations at x, we establish bounds for the variance. The perturbed eigenfunctions arise in the study of Loschmidt echo effect in physics.
The second result is joint work with B. Clarke, N. Kamran, L. Silberman and J. Taylor. We dene
Gaussian measures on manifolds of metrics with the xed volume form. We next compute the moment
generating function for the L2 (Ebin) distance to the reference metric.
Fabricio Macia:
Title: Long-time dynamics and semiclassical measures for some completely integrable Schrödinger
flows.
Abstract: In this talk we shall present some results on the dynamics of solutions to semiclassical
Schrödinger-type equations on the torus at time scales that tend to infinity as the semiclassical parameter tends to zero. We are interested in characterizing the set of (time-dependent) semiclassical measures
associated to sequences of solutions to these equations, and in particular how the structure of this set
depends on the time scale. We show the existence of a threshold, below which the set of semiclassical
measures contains all orbit measures associated to classical dynamics and above which all semiclassical
measures are absolutely continuous in position variable. This is joint work with N Anantharaman and C.
Fermanian-Kammerer.
Dan Mangoubi:
Title: A geometric form of logarithmic convexity for harmonic functions and growth of eigenfunctions.
Abstract: We will discuss a geometric version of Agmon’s logarithmic convexity of harmonic functions
and explain how the Donnelly-Fefferman growth bound on eigenfunctions can be easily deduced from it.
Jeremy Marzuola:
Title: Strichartz estimates for the Schrdinger and wave equations on (and outside) polygons
Abstract: We will review some recent results of the speaker with Matt Blair, Austin Ford and Sebastian Herr establishing local in time Strichartz estimates for the wave and Schrdinger equations on
manifolds with conic singularities and relate them to analysis on wedge and polygonal domains. We will
also discuss a recent extension of such results to domains exterior to polygons domain problems with
Dean Baskin and Jared Wunsch using a new local smoothing estimate proved by the two of them, where
the estimates are sharper.
Shu Nakamura:
Title: Propagation of singularities for Schrödinger equations with long range perturbations
Abstract: The singularities of solutions to Schrödinger equations (with short range perturbations) can
be described using the scattering theory for the corresponding classical mechanics. If the perturbation
is long range type, then we need to use long range scattering technologies, namely, we need to employ
a solutions to the Hamilton-Jacobi equation in the momentum space. If the perturbation is modestly
long-range, then we can use the Dollard type approximate solution, and we can describe the singularities
rather explicitly. (Partly joint work with K. Horie).
Marc Pollicott:
Title: The Schottky-Klein prime function and counting functions
Abstract: We relate the classical Schottky-Klein prime function to a counting function for Fenchel
crosses for geodesics in the three dimensional Poincare upper half-space.
Zeev Rudnick:
Title: Nodal Intersections
Abstract: We study the fine structure of nodal lines for eigenfunctions of the Laplacian on a surface
by examining the number of intersection of the nodal lines with a fixed reference curve. It is expected
that in many cases the number of these intersections is bounded above by the wave number k (the square
root of the eigenvalue). Very little is known concerning lower bounds. For the flat torus, we prove the
expected upper bound of k and give a lower bound of almost the same quality. To do so, we connect
this problem to bounds on the Lp norms of the restriction of the eigenfunctions to the curve, and to a
problem in Number Theory. (joint work with Jean Bourgain).
Alexander Strohmaier:
Title: Quantum ergodicity and dynamics for discontinuous systems
Abstract:The dynamical system underlying that of a Laplace operator corresponding a manifold with
discontinuous metric is not a classical one but has features of a ray-splitting billiard. I will discuss some
aspects of its dynamics and theorems that can be proved about when the system is quantum ergodic.
Henrik Ueberschar:
Title: Quantum Limits for Point Scatterers on Flat Tori
Abstract: The Laplacian with a delta potential on a flat torus is an important model to study the
transition between integrability and chaos in quantum systems. I will present joint work with Par Kurlberg (KTH) on the semiclassical measures which can arise in this system.