The seminar meets several times a year in UC Berkeley and Stanford University.
The organizers are Semyon Dyatlov (Berkeley), András Vasy (Stanford), and Maciej Zworski (Berkeley).
Related seminars: Analysis&PDE (Berkeley), Analysis&PDE (Stanford), HADES (Berkeley), Geometry (Stanford), Analysis videoseminar (Paris–Berkeley–Bonn–Zurich).
| Date / Location | Time | Speaker | Title / Abstract |
| Wed Apr 15, 2020 Zoom | 3–4 | Maciej Zworski (Berkeley) | Analytic hypoellipticity of Keldysh operators For operators modeled by P=x1(D1)2+(D2)2+aD1 we show that if u is smooth and Pu is analytic then u is analytic. This is motivated by the question of analyticity of quasinormal modes of black holes across event horizons and is a consequence of a general microlocal result. Joint work with J. Galkowski. |
| 4:15–5:15 | Sung-Jin Oh (Berkeley) | On the Cauchy problem for the Hall magnetohydrodynamics In this talk, I will describe a recent series of work with I.-J. Jeong on the Hall MHD equation without resistivity. This PDE, first investigated by the applied mathematician M. J. Lighthill, is a one-fluid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate the ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which not only includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field) but also possibly large perturbations thereof. Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character. With such a viewpoint, the key ill- and well-posedness mechanisms can be understood in terms of the properties of the bicharacteristic flow associated with the appropriate principal symbol. |
| Date / Location | Time | Speaker | Title / Abstract |
| Wed Feb 26, 2020 Stanford | 3–4 383N | Eugenia Malinnikova (Stanford) | Landis conjecture in dimension two In 1960’s Landis conjectured that a nontrivial solution to a time-independent Schr ̈odinger equation with bounded potential cannot decay faster than exponentially. In 1992 the conjecture was disproved by Meshkov, who constructed a counter example and found the optimal rate of decay for complex valued solutions. The conjecture is still open for the case of real valued potentials. We will outline a solution in dimension two. The talk is based on a joint work in progress with A.Logunov, N.Nadirashvili and F.Nazarov. |
| 4:30–5:30 380C | András Vasy (Stanford) | Fredholm theory for the Laplacian near zero energy on asymptotically conic spaces In this talk I will discuss and compare two approaches via Fredholm theory to resolvent estimates for the Laplacian of asymptotically conic spaces (such as appropriate metric perturbations of Euclidean space), including in the zero spectral parameter limit. | |
| Fri Apr 12, 2019 Berkeley 740 Evans | 2:10–3 | Jared Wunsch (Northwestern) | Schrödinger equations with conormal potentials Consider the semiclassical Schrödinger equation (h2Δ+V-E)u=0, where, instead of being smooth, V is allowed to be singular across a hypersurface. The singularity in the potential turns out to have very interesting consequences for the structure of solutions u; in effect, WKB solutions include not just contributions from classical propagation across the interface but also reflected singularities, in what amounts to a quantum diffraction effect (meaning one that is not visible at the level of classical Hamiltonian dynamics). I will discuss the propagation and reflection of semiclassical singularities in this setting, and also its consequences for the existence of quantum resonances in systems where trajectories escape to infinity under classical flow but not under the branched flow where we allow reflections. This is joint work with Oran Gannot. |
| 4:10–5 | Hamid Hezari (UC Irvine) | The inverse spectral problem for strictly convex domains We discuss the recent developments in the inverse length spectral theory of smooth strictly convex domains, including the works of Avila–De Simoi–Kaloshin and Kaloshin–Sorrentino on the Birkhoff conjecture, and De Simoi–Kaloshin–Wei on the length spectral rigidity of nearly circular domains with a reflectional symmetry. In a joint work with Zelditch we explore the inverse Laplace spectral problem for nearly circular ellipses, among all smooth domains without any symmetry or convexity assumption. | |
| Wed Mar 13, 2019 Stanford 384H | 2–3 | Rafe Mazzeo (Stanford) | Analytic aspects of Kapustin–Witten theory Kapustin and Witten introduced a new set of gauge-theoretic equations, which were later proposed by Gaiotto and Witten as a tool to access some old and new manifold invariants. I will describe recent progress on the analytic foundations of this subject. Joint work with Witten and with S. He. |
| 3:20–4:20 | Gabriel Paternain (Cambridge) | Nonlinear detection of connections I will discuss the geometric inverse problem of recovering a connection from the parallel transport along geodesics of a compact Riemannian manifold with strictly convex boundary or along light rays in Minkowski space. This problem is motivated by other geometric inverse problems and is tackled with a range of techniques including energy estimates, regularity results for the transport equation associated with the geodesic flow and microlocal analysis. | |
| Wed Nov 14, 2018 Berkeley | 2:40–3:30 740 Evans | Maciej Zworski (Berkeley) | Internal waves for (linearized) fluids and 0th order pseudodifferental operators Colin de Verdière and Saint-Raymond have recently found a fascinating connection between modeling of internal waves in stratified fluids and spectral theory of 0th order pseudodifferential operators on compact manifolds. The purpose of this talk is to show how a version of their results follows from the now standard radial estimates for pseudodifferential operators and some results about Lagrangian surfaces in classical and wave (quantum) settings. Some numerical simulations and comments about the case of positive viscosity will also be provided. Joint work with S. Dyatlov. (For the brave souls who attended the Harmonic Analysis and Differential Equations Student Seminar on the same topic this talk will provide, after re-introduction of the problem, some technical details avoided then.) |
| 4:10–5 736 Evans | Yiran Wang (Stanford) | Determination of space-time structures from gravitational perturbations We consider inverse problems for the Einstein equations with source fields. The problem we are interested in is to determine space-time structures e.g. topological, differentiable structures of the manifold and the Lorentzian metric, by generating small gravitational perturbations and measuring the responses near a freely falling observer. We discuss some unique determination results for Einstein equations with scalar fields and electromagnetic fields under a microlocal linearization stability condition. A key component of our approach is to analyze the new waves generated from the nonlinear interaction of multiple gravitational waves using microlocal techniques. The talk is based on joint works with M. Lassas and G. Uhlmann. | |
| Wed Oct 10, 2018 Stanford 383N | 2:30–3:30 | Peter Hintz (MIT) | Trapping in perturbations of Kerr spacetimes We study the trapped set of spacetimes whose metric decays to a stationary Kerr metric at an inverse polynomial rate. In the first part of the talk, I will focus on the dynamical aspects of this problem and show that the trapped set is a smooth submanifold which converges to that of the stationary metric at the same rate. In the second part, I will explain how to use this to prove microlocal estimates at the trapped set for solutions of wave equations on such spacetimes. |
| 4–5 | Semyon Dyatlov (Berkeley) | Long time propagation and fractal uncertainty principle I will show a frequency-independent lower bound on mass of eigenfunctions on surfaces of variable negative curvature. This was previously done in the case of constant curvature in joint work with Jin, relying on the fractal uncertainty principle proved in joint work with Bourgain. I will focus on the new components needed to handle the case of variable curvature, in particular propagation of quantum observables up to local Ehrenfest time. Joint work in progress with Long Jin and Stéphane Nonnenmacher. | |
| Tue Mar 13, 2018 Berkeley | 2:10–3 732 Evans | Long Jin (Purdue) | Control and stabilization on hyperbolic surfaces In this talk, we discuss some recent results concerning the control and stabilization on a compact hyperbolic surface. In particular, we show that
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| 4:10–5 740 Evans | Xuwen Zhu (Stanford) | Deformation of constant curvature conical metrics In this joint work with Rafe Mazzeo, we aim to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of such conical metrics has a nice compactification as the cone points coalesce. This is a key ingredient of understanding the full moduli space of such metrics with positive curvature and cone angles bigger than 2π. | |
| Tue Jan 16, 2018 Stanford 383N | 2:30–3:30 | Jesse Gell-Redman (U Melbourne) | Dirac type operators on pseudomanifolds We study elliptic differential operators on iterated wedge spaces. These are incomplete Riemannian manifolds on which the metric undergoes iterated conical degeneration; they include cones, cone edges, and, products of cone edges, and they live on pseudomanifolds – topological spaces characterized by an analogous topological notion of iterated conical degeneration. We determine the structure of the generalized inverses and the heat kernels of such operators using the radial blow up program of Melrose. In particular, we extend the edge calculus of Mazzeo to manifolds with corners with iterated fibrations structures (resolutions of pseudomanifolds), for both pseudodifferential and heat kernel type operators. We go on to prove an index theorem for those Dirac type operators on pseudomanifolds associated to (iterated) wedge metrics. Joint with Pierre Albin (UIUC). |
| 4–5 | Martin Vogel (Berkeley) | Spectrum of random non-selfadjoint operators The spectrum of non-selfadjoint operators can be highly unstable even under very small perturbations. This phenomenon is referred to as "pseudospectral effect". Traditionally this pseudospectral effect was considered a drawback since it can be the source of immense numerical errors, as shown for instance in the works of L. N. Trefethen. However, this pseudospectral effect can also be the source of many new insights. A line of works by Hager, Bordeaux–Montrieux, Sjöstrand, Christiansen and Zworski exploits the pseudospectral effect to show that the (discrete) spectrum of a large class of non-selfadjoint pseudo-differential operators subject to a small random perturbation follows a Weyl law with probability close to one. In this talk we will discuss the local statistics of the eigenvalues of such operators (in dimension one). That is the distribution of the eigenvalues on the scale of their average spacing. We will show that the pseudospectral effect leads to a partial form of universality of the local statistics of the eigenvalues. This is joint work with Stéphane Nonnenmacher (Université Paris-Sud). | |
| Mon Sep 25, 2017 Berkeley 740 Evans | 2:40–3:30 | Kiril Datchev (Purdue) | Semiclassical resolvent estimates away from trapping Semiclassical resolvent estimates relate dynamics of a particle scattering problem to regularity and decay of waves in a corresponding wave scattering problem. Roughly speaking, more trapping of particles corresponds to a larger resolvent near the trapping. If the trapping is mild, then propagation estimates imply that the larger norm occurs only there. However, in this talk I will show how the effects of heavy trapping can tunnel over long distances, implying that the resolvent can be very large far away as well. This is joint work with Long Jin. |
| 4:10–5 | Charles Hadfield (Berkeley) | Resonances on asymptotically hyperbolic manifolds: the ambient metric approach On an asymptotically hyperbolic manifold, the Laplacian has essential spectrum. Since work of Mazzeo and Melrose, this essential spectrum has been studied via the theory of resonances; poles of the meromorphic continuation of the resolvent of the Laplacian (with modified spectral parameter). A recent technique of Vasy provides an alternative construction of this meromorphic continuation which dovetails the ambient metric approach to conformal geometry initiated by Fefferman and Graham. I will discuss the ambient geometry present in this construction, use it to define quantum resonances for the Laplacian acting on natural tensor bundles (forms, symmetric tensors), and mention an application showing a correspondence between Ruelle resonances and quantum resonances on convex cocompact hyperbolic manifolds. | |
| Mon May 15, 2017 Stanford 384H | 2:15–3:15 | Peter Hintz (Berkeley) | Resonances for obstacles in hyperbolic space We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound Im λ ≤ -1/2 in odd dimensions and for small obstacles with diameter ρ, we improve this to Im λ ≤ -C/ρ 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. |
| 4–5 | François Monard (UCSC) | X-ray transforms and tensor tomography on surfaces On (M,g) a non-trapping Riemannian surface with boundary, the tensor tomography problem consists of inferring (i) what is reconstructible of a symmetric tensor field from knowledge of its integrals along geodesics through that surface, and (ii) how to reconstruct it. In the Euclidean case and zero-th order tensors (i.e., functions), this is the well-known X-Ray (or Radon) transform and it serves as the theoretical backbone of Computerized Tomography. In a geometric setting, the answer to questions (i) and (ii) depends on the order of tensors considered, the underlying geometry, and what functional setting for the X-ray transform is chosen. In this talk I will review recent results on these aspects, and will discuss reconstruction approaches for functions and tensor fields, some valid in rather general settings, others requiring more Euclidean explicitness. | |
| Mon Mar 20, 2017 Stanford 384I | 2:30–3:30 | Jared Wunsch (Northwestern) | Resonances generated by conic diffraction I will report on recent work, joint with Luc Hillairet, that refines our understanding of the strings of resonances along logarithmic curves generated by multiply diffracted trapped rays. |
| 4–5 | Gunther Uhlmann (U Washington / HKUST) | Travel Time Tomography, Boundary Rigidity and Lens Rigidity We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others. The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform. We consider also the partial data case, where you are making measurements on a subset of the boundary. This is joint work with Plamen Stefanov and András Vasy. | |
| Mon Feb 6, 2017 Berkeley 740 Evans | 2:40–3:30 | Jonathan Luk (Stanford) | Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat data I will present a recent work (joint with Sung-Jin Oh) on the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it was previously proved (by M. Dafermos and I. Rodnianski) that a certain formulation of the strong cosmic censorship conjecture is false, namely, the maximal globally hyperbolic development of a data set in this class is extendible as a Lorentzian manifold with a C0 metric. Our main result is that, nevertheless, a weaker formulation of the conjecture is true for this model, i.e., for a generic (possibly large) data set in this class, the maximal globally hyperbolic development is inextendible as a Lorentzian manifold with a C2 metric. |
| 4:10–5 | Jeffrey Galkowski (Stanford / McGill) | Pointwise bounds for Steklov eigenfunctions Let (Ω,g) be a compact, real-analytic Riemannian manifold with real-analytic boundary ∂Ω. The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary and proving a conjecture of Hislop and Lutzer. The estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations Pu=0 near the characteristic set {σ(P)=0}. This talk is based on joint work with John Toth. | |
| Fri Dec 9, 2016 Stanford 384H | 1:30–2:30 | Semyon Dyatlov (MIT) | Resonances for open quantum maps Quantum maps are a popular model in physics: symplectic relations on tori are quantized to produce families of N×N matrices and the high energy limit corresponds to the large N limit. They share a lot of features with more complicated quantum systems but are easier to study numerically. We consider open quantum baker's maps, whose underlying classical systems have a hole allowing energy escape. The eigenvalues of the resulting matrices lie inside the unit disk and are a model for scattering resonances of more general chaotic quantum systems. However in the setting of quantum maps we obtain results which are far beyond what is known in scattering theory. We establish a spectral gap (that is, the spectral radius of the matrix is separated from 1 as N tends to infinity) for all the systems considered. The proof relies on the notion of fractal uncertainty principle and uses the fine structure of the trapped sets, which in our case are given by Cantor sets, together with simple tools from harmonic analysis, algebra, combinatorics, and number theory. We also obtain a fractal Weyl upper bound for the number of eigenvalues in annuli. These results are illustrated by numerical experiments which also suggest some conjectures. This talk is based on joint work with Long Jin. |
| 2:45–3:45 | Maciej Zworski (Berkeley) | Ruelle zeta function at zero for surfaces of variable curvature For surfaces of constant negative curvature the Selberg trace formula shows that the order of vanishing of the Ruelle zeta function at 0 is given by the absolute value of the Euler characteristic of the surface. Using simple microlocal arguments we prove that this remains true for any negatively curved sufficiently smooth surface. Joint work with S. Dyatlov. | |
| Fri Oct 28, 2016 Berkeley 736 Evans | 2:40–3:30 | Peter Hintz (Berkeley) | Non-linear stability of Kerr–de Sitter black holes In joint work with András Vasy, we prove the stability of the Kerr–de Sitter family of black holes in the context of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will describe the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein's equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges. In particular, I will explain our microlocal proof of a key ingredient of this framework, called `constraint damping,' a device first introduced in numerical relativity. |
| 4:10–5 | Laurent Michel (Nice / Stanford) | Metastability for semiclassical random walk We study the return to equilibrium for a semiclassical random walk associated to a multiple well transition density. We describe accurately the small eigenvalues of the associated operator. The proof is based on a supersymmetric approach. As a preliminary we prove a general factorization result on pseudo differential operators. Joint work with J.-F. Bony and F. Hérau. | |
| Mon Feb 22, 2016 Stanford 384I | 2:30–3:30 | Gilles Lebeau (U Nice) | On the holomorphic extension of the Poisson kernel Let Ω be an open subset of Rd with analytic boundary. The Poisson kernel K(x,y), with x∈ Ω, y∈ ∂Ω, is the solution of the following elliptic boundary value problem, where Δ denotes the usual Laplace operator In this lecture, we are interested in the holomorphic extension in x∈ Cd of K(x,y) near a given point y∈ ∂Ω. Very little is known about this problem. We will first recall old and classical results on the regularity of the Dirichlet problem. Then we will state "conjectures" on the location of the singularities of the holomorphic extension of K and will describe some particular cases where it holds true. Finally, we will explain how this problem is related to propagation of singularities and complex billiard dynamics. |
| 4–5 | Jeffrey Galkowski (Stanford) | A quantum Sabine law for resonances in transmission problems We prove a quantum Sabine law for the location of resonances in transmission problems. In this talk, our main applications are to scattering by strictly convex, smooth, transparent obstacles and highly frequency dependent delta potentials. In each case, we give a sharp characterization of the resonance free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances to the chord lengths and reflectivity coefficients for the ray dynamics and hence give a quantum version of the Sabine law from acoustics. | |
| Fri Jan 29, 2016 Berkeley 740 Evans | 2:40–3:30 | Vadim Kaloshin (U Maryland) | On deformational spectral rigidity of convex symmetric planar domains One can associate to a planar convex domain Ω ⊂ R2 two types of spectra: the Laplace spectrum consisting of eigenvalues of a Dirichlet problem and the length spectrum consisting of perimeters of all periodic orbits of a billiard problem inside Ω. The Laplace and length spectra are closely related, generically the first determines the second. M. Kac asked if the Laplace spectrum determines a domain Ω. There are counterexamples. During the talk we show that a planar axis symmetric domain close to the circle can't be smoothly deformed preserving the length spectrum unless the deformation is a rigid motion. This gives a partial answer to a question of P. Sarnak. This is a joint work with J. De Simoi and Q. Wei. |
| 4:10–5 | András Vasy (Stanford) | The Feynman propagator and its positivity properties In this talk, partially on joint work with Jesse Gell-Redman, Nick Haber and Michal Wrochna, I will explain the properties of the Feynman propagator, i.e. the inverse of the wave operator on `Feynman function spaces', in various settings. I will also explain its positivity properties, and the connection to spectral and scattering theory in Riemannian settings, as well as to the classical parametrix construction of Duistermaat and Hörmander. | |
| Mon Nov 30, 2015 Stanford 384H | 2:30–3:30 | Patrick Gérard (Orsay) | Geometry of the phase space and wave turbulence for the cubic Szegő equation I will review the main properties of the cubic Szegő equation, a toy model for nonlinear wave interaction in one space dimension enjoying some Lax pair structure. Emphasis will be put on the structure of the action-angle variables in the phase space and how they are connected to long time strong transition to high frequencies. This is a joint work with Sandrine Grellier. |
| 4–5 | Alexis Drouot (Berkeley) | Scattering resonances for highly oscillatory potentials We study resonances of compactly supported potentials Vε(x) = W(x,x/ε) where W : Rd × Rd/(2π Z)d → C, d odd. That means that Vε(x) is a sum of a slowly varying potential, W0(x), and one oscillating at frequency 1/ε. For W0 = 0 we prove that Vε has no resonances above the line Im λ = -Aln(1/ε) – except a simple resonance of modulus ∼ ε2 when d=1. In the case W0 ≠ 0 we prove that resonances in fixed strips admit an expansion in powers of ε. We use this result to produce an effective potential converging uniformly to W0 as ε → 0 and whose resonances approach resonances of Vε modulo O(ε4). This work proves a conjecture of Duchêne–Vukićević–Weinstein. | |
| Fri Oct 16, 2015 Berkeley 736 Evans | 2:10–3 | Nicolas Burq (Paris XI / Orsay) | From Strichartz estimates to uniform Lp resolvent estimates We prove uniform Lp resolvent estimates for the stationary damped wave operator. The uniform Lp resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Dos Santos Ferreira–Kenig–Salo and advanced further by Bourgain–Shao–Sogge–Yao. Here we provide an alternative proof relying on the techniques of semi-classical Strichartz estimates. This approach allows us also to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semi-classical spectral analysis framework. Joint with with D.Dos Santos and K. Krupchyk. |
| 3:40–4:30 | Franz Luef (NTNU Trondheim) | Modulation spaces and applications to pseudodifferential operators Modulation spaces are a class of Banach spaces that have turned out to be of great use in harmonic analysis, time-frequency analysis and quanitzation. Many classical function spaces, such as generalized Sobolev spaces or the Sjöstrand class, are modulation spaces. In this talk I present the basic theory of modulation spaces and describe the link to pseudodifferential operators, including the Gabor wave front set introduced by Rodino and Wahlberg. The latter turns out to coincide with the global wave front set of Hörmander. | |
| Fri Apr 17, 2015 Stanford 384I | 3:15–4:15 | Xuwen Zhu (MIT / Stanford) | Resolution of the canonical fiber metrics for a Lefschetz fibration We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the length of the shrinking geodesic. This is joint work with Richard Melrose. |
| 4:30–5:30 | Semyon Dyatlov (MIT) | Spectral gaps via additive combinatorics The spectral gap on a noncompact Riemannian manifold is an asymptotic strip free of resonances (poles of the meromorphic continuation of the resolvent of the Laplacian). The existence of such gap implies exponential decay of linear waves, modulo a finite dimensional space; in a related case of Pollicott–Ruelle resonances, a spectral gap gives an exponential remainder in the prime geodesic theorem. We study spectral gaps in the classical setting of convex co-compact hyperbolic surfaces, where the trapped set is a fractal set of dimension 2δ + 1. We obtain a spectral gap when δ=1/2 (as well as for some more general cases). Using a fractal uncertainty principle, we express the size of this gap via an improved bound on the additive energy of the limit set. This improved bound relies on the fractal structure of the limit set, more precisely on its Ahlfors–David regularity, and makes it possible to calculate the size of the gap for a given surface. Joint work with Joshua Zahl. | |
| Tue Feb 24, 2015 Berkeley 740 Evans | 2:10–3 | Boaz Haberman (Berkeley) | Recovering a gradient term from boundary measurements Given a vector potential A and a scalar potential q defined on some domain U we may define the Schrödinger operator LA,q = (D + A)2 + q. The inverse boundary value problem is to determine A and q from the set of Cauchy data associated to LA,q. We will discuss how this problem is related to unique continuation and Carleman estimates, and illustrate some issues that arise when A is unbounded or q is in a Sobolev space of negative regularity index |
| 3:40–4:30 | Michał Wrochna (Grenoble / Stanford) | Characteristic Cauchy data of positive-frequency solutions of the wave equation For the Klein–Gordon or wave equation, a characteristic Cauchy problem can be formulated by specifying as initial datum the restriction to a lightcone. I will demonstrate how such characteristic Cauchy problem can be solved in the inside of a cone in a globally hyperbolic spacetime for data in adapted Sobolev spaces, generalizing a result of Hörmander. I will then discuss applications in Quantum Field Theory, where one is interested in constructing fundamental solutions with specific wave front set and positivity propeties, corresponding to a Feynman propagator. This is joint work with Christian Gérard (Orsay) | |
| Fri Nov 21, 2014 Berkeley 736 Evans | 2:40–3:30 | Elon Lindenstrauss (Hebrew U Jerusalem / MSRI) | Quantum ergodicity on the sphere and averaging operators The quantum ergodicity theorem of Shnirelman, Colin de Verdière and Zelditch gives that for an orthonormal sequence of eigenfunctions of the laplacian on a compact manifold with ergodic geodesic, outside a density one subsequence, the eigenfunctions equidistribute. The geodesic flow on the sphere is very much not ergodic, and indeed quantum ergodicity (QE) fails on the sphere for the standard sequence of spherical harmonics. On the other hand Zelditch has shown QE holds for a random orthonormal basis in this case. We prove QE for joint eigenfunctions of laplacian and an averaging operator over a finite collection of rotations (with some restrictions). We also give a new approach to a QE theorem on graphs by Anantharaman and Le Masson. Joint work with Shimon Brooks and Etienne Le Masson. |
| 4:10–5 | Nils Dencker (Lund) | The solvability and range of differential equations In the 50's, the consensus was that all linear PDEs were solvable. Therefore it was a great surprise in 1957 when Hans Lewy presented a complex vector field that is not solvable anywhere. Hörmander then proved in 1960 that in fact linear partial differential equations generically are not solvable. For nonsolvable equations the range has infinite codimension, and Hörmander proved in 1963 that nonsolvable complex vector fields are determined by their range, up to right multiplication by functions. We shall generalize this to nonsolvable systems of differential equations of constant characteristics and principal type, including scalar equations. We show that the ranges of these equations determine the Taylor expansions of the coefficients at minimal bicharacteristics, up to right composition by differential equations. The minimal bicharacteristics are the smallest sets on which the equation is not solvable. This is joint work with Jens Wittsten. | |
| Tue Sep 30, 2014 Stanford 380X | 3:15–4:15 | Jonathan Pfaff (Stanford) | Analytic torsion on locally symmetric spaces We will introduce the analytic torsion on locally symmetric spaces and describe some of its asymptotic properties. The talk will focus on the analytic aspects of the theory. In particular, we will describe the torsion in the non-compact, finite-volume setting as well as a gluing formula for the torsion in that case. |
| 4:30–5:30 | Persi Diaconis (Stanford) | Dirichlet eigenvectors in probability Spectral theory (even microlocal analysis) has been very useful for studying the long time behavior of recurrent Markov chains. The quantitative theory of absorbing Markov chains is in its infancy. Here there is a `quasi-stationary distribution' (the first Dirichlet eigenfunction) and one may ask `How close are we to quasi-stationarity if the process has not been absorbed up to time n?' One can also ask about time to absorption (the top Dirichlet eigenvalue) and the `shape' of the quasi-stationary distribution (McKeen–Vlasov equation). In joint work with Laurent Miclo, we show how the Doob transform produces a recurrent chain with upper and lower bounds for rates of convergence. | |
| Mon May 5, 2014 Stanford | 2:12–3:15 381T | Jeffrey Galkowski (Berkeley) | Distribution of resonances for quantum corrals We consider resonances for the operator -Δ +V⊗ δ∂Ω where Ω⊂Rd is a bounded domain. This operator is a model for quantum corrals as well as other lossy systems. We give a bound on the size of the resonance free region for very general Ω and in the case that ∂Ω is strictly convex, we give a dynamical characterization of the resonance free region that is generically sharp. We describe how this characterization can be thought of as a Sabine Law in certain cases. |
| 4–5 381U | Guy David (Orsay) | A variant of the Alt, Caffarelli, and Friedman free boundary problem motivated by the localization of eigenfunctions I will try to describe joint work with M. Filoche, D. Jerison, and S. Mayboroda. The initial motivation (Filoche–Mayboroda) concerns the localization of eigenfunctions, say, for a Schrödinger operator with a complicated bounded potential, or the Laplacian on a complicated domain. What we do is try to find an automatic decomposition of the domain into small pieces, related to the given operator, and for this we minimize a variant of the Alt, Caffarelli, and Friedman free boundary problem, where we authorize a large number of phases (instead of 2). The results concern the regularity of the minimizers. | |
| Mon Mar 17, 2014 Berkeley 891 Evans | 2:10–3 | Michael Christ (Berkeley) | On an inverse problem concerning Bergman kernels |
| 3:40–4:30 | Austin Ford (Stanford) | The wave trace on manifolds with conic singularities We consider the trace of the (half-)wave group on a compact manifold with conic singularities. The trace of the wave group, which on the one hand equals ∑ e-itλj where λj2 are the eigenvalues of the Laplacian, is on the other hand a distribution in t which is singular at the lengths of closed geodesics. Those closed geodesics that interact with the cone points generically do so "diffractively", carrying singularities into regions of phase space inaccessible to ordinary geodesic flow. We describe a formula for the leading order singularity of the wave trace at the lengths of closed diffractive geodesics, generalizing the formula due of Duistermaat and Guillemin in the smooth setting and that of Hillairet in the setting of flat surfaces with conic singularities. | |
| Mon Jan 27, 2014 Stanford 381U | 3:15–4:15 | Gunther Uhlmann (U Washigton / Stanford) | Seeing through space time We consider inverse problems for the Einstein equation with a time-depending metric on a 4-dimensional globally hyperbolic Lorentzian manifold. We formulate the concept of active measurements for relativistic models. We do this by coupling Einstein equations with equations for scalar fields. The inverse problem we study is the question, do the observations of the solutions of the coupled system in an open subset U of the space-time with the sources supported in U determine the properties of the metric in a larger domain? To study this problem we define the concept of light observation sets and show that these sets determine the conformal class of the metric. This corresponds to passive observations from a distant area of space which is filled by light sources. We will also consider inverse problems for other non-linear hyperbolic equations. This is joint work with Y. Kurylev and M. Lassas. |
| 4:30–5:30 | Maciej Zworski (Berkeley) | Microlocal approach to dynamical zeta functions Dynamical zeta functions of Selberg, Smale and Ruelle are analogous to the Riemann zeta function with the product over primes replaced by products over closed orbits of Anosov flows. In 1967 Smale conjectured that these zeta functions should be meromorphic but admitted "that a positive answer would be a little shocking". Nevertheless the continuation was proved in 2012 by Giulietti–Liverani–Pollicott. In my talk I will present a proof of this result obtained by Dyatlov and myself and inspired by a trace formula of Guillemin and by recent work of Faure–Sjöstrand. It is based on a simple idea involving wave front sets and propagation of singularities: we apply methods of microlocal analysis to the generator of the flow, in particular, propagation of singularities results due to Duistermaat-Hörmander, Melrose and Vasy. | |
| Wed Nov 13, 2013 Berkeley | 2:10–3 891 Evans | Peter Hintz (Stanford) | Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes I will discuss the small data solvability of suitable semilinear wave and Klein-Gordon equations on geometric classes of spaces, which include asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes. Our results are obtained by showing the global Fredholm property, and indeed invertibility, of the underlying linear operator on suitable L2-based function spaces, which also possess appropriate algebra or more complicated multiplicative properties. The linear framework is based on b-analysis, introduced in this context by Vasy to describe the asymptotic behavior of solutions of linear equations. An interesting feature of the analysis is that resonances, namely poles of the inverse of the Mellin transformed b-normal operator, play an important role. Joint work with Andräs Vasy. |
| 4:10–5 736 Evans | Gabriel Durkin, Sergey Knysh (NASA AMES) | Quantum Fisher information for noisy dynamics: a semi-classical approach Parameter estimation of probability distributions is one of the most basic tasks in information theory, and has been generalized to the quantum regime since the description of quantum measurement is essentially probabilistic. The "quantum metrology" prescription is straightforward for closed systems evolving unitarily, becoming more challenging when the system is coupled (more realistically) to an external environment. This produces "noisy" non-unitary dynamics. This noise, or decoherence, degrades the precision in any parameter estimation – precision that is quantified by Quantum Fisher Information. For physically relevant noise models including both phase diffusion and dissipation we investigate the scaling of single parameter precision with the noise amplitude and "resource" N, the number of system dimensions. Using a novel operator approach (rather than WKB), we find new saturable precision bounds in the asymptotic limit of large N in tandem with those quantum states uniquely capable of reaching these bounds. Convergence to asymptotic behaviour can set in quickly for modest N<100, and as such our analysis is relevant to sensing and metrology experiments incorporating ensembles of 10 to 100 particles; potentially atoms, magnetic spins, flux-qubits or photons. | |
| Wed Apr 10, 2013 Stanford | 2:30–3:30 383N | Nicolas Burq (Orsay) | Micro-local analysis of the Dirichlet–Neumann operator It is well known that the Dirichlet–Neumann operator in a smooth domain is a pseudo-differential operator. On the other hand, the definition of this operator requires actually only very low regularity (namely it is defined as soon as the domain is Lipshitz). In this talk I will present some recent results describing the micro-local nature of the Dirichlet Neumann operator in rough domains. Our results depend of course on the level of smoothness we assume on the domain, but the micro-local description that involves para differential operators we get is non trivial as soon as the domain is better than Lipshitz. Furthermore, motivated by the analysis of the water-waves system, we work in the framework of uniformly local Sobolev spaces rather than the usual (L2-based) Sobolev setting. This is a joint work with T. Alazard and C. Zuily. |
| 4–5 380X | Michael Hitrik (UCLA) | Spectra and subelliptic estimates for operators with double characteristics For a class of non-selfadjoint semiclassical operators with double characteristics, we give complete asymptotics for low-lying eigenvalues and establish accurate semiclassical resolvent estimates of subelliptic type in a neighborhood of the origin. The assumptions along the double characteristics generalize those valid for operators of Kramers–Fokker–Planck type. This is joint work with Karel Pravda-Starov. | |
| Fri Feb 22, 2013 Berkeley 736 Evans | 2:10–3 | Israel Michael Sigal (U Toronto) | Asymptotic completeness of Rayleigh scattering Experiments on scattering of photons on atoms (Rayleigh scattering) and on free electrons (Compton scattering) led, in the beginning of 20th century, to our understanding of composition of matter and eventually to creation of quantum mechanics. Though these experiments reproduced central physical phenomena, and though quantum mechanics provided a well defined mathematical framework for describing these processes, their mathematical theory is still missing. (The mathematical framework mentioned is given by the Schroedinger equation of the non-relativistic quantum electro-dynamics.) In recent works, jointly Jean-François Bony and Jeremy Faupin, we succeeded in proving asymptotic completeness of Rayleigh scattering. This proof assumes a bound on the average photon number, which is proven in a special case of finite-dimensional quantum systems. In this talk, I describe recent results. I will not assume a prior knowledge of quantum field theory and will provide all necessary definitions in the talk. |
| 4:10–5 | Benjamin Dodson (Berkeley) | The energy critical NLS in an exterior domain In this talk we will discuss the energy-critical nonlinear Schroedinger equation outside a convex obstacle in four space dimensions (that is, a cubic NLS), with Dirichlet boundary conditions. We will explain how the results of Visan for the defocusing energy critical NLS in four dimensions can be used to study this problem. | |
| Fri Dec 7, 2012 Berkeley 891 Evans | 2:40–3:30 | Gunther Uhlmann (U Washigton / UC Irvine) | Travel time tomography with partial data The travel time tomography problem consists in determining the anisotropic index of refraction or sound speed of a medium by making travel time measurements. We will survey what is known about this problem including some recent results on the partial data case. The latter are joint work with András Vasy. |
| 4:10–5 | Austin Ford (Stanford) | Examples of the structure and dispersion of waves on two-dimensional cones In recent years, there has been much effort to understand the dispersive properties of solutions to the wave and Schrödinger equations in various geometries. I will discuss in this talk the beginnings of extending this program to singular spaces, namely the settings of two-dimensional cones and related spaces. This will begin with the asymptotics of the Schrödinger group, and I will show how these lead to dispersive and Strichartz estimates for solutions to this equation on cones. I will also discuss joint work with Matt Blair, Sebastian Herr, and Jeremy Marzuola extending these estimates to solutions on polygonal domains and surfaces with (exact) conical singularities. The analogous results for solutions to the wave equation on these spaces will also be discussed (joint with Matt Blair and Jeremy Marzuola). Time permitting, I'll also mention current work with Andrew Hassell and Luc Hillairet with the goal of understanding the microlocal structure of these "classical waves" and the various implications knowing this structure would have. | |
| Fri Nov 2, 2012 Stanford 380W | 2:30–3:30 | Michael Christ (Berkeley) | Optimal off-diagonal bounds for Bergman/Szegő kernels associated to positive line bundles with smooth metrics |
| 4–5 | Maciej Zworski (Berkeley) | Exponential decay of correlations in scattering and dynamical problems | |
| Fri Apr 20, 2012 Stanford T195 Herrin | 3–4 | Semyon Dyatlov (Berkeley) | Semiclassical limits of plane waves On a complete noncompact Riemannian manifold which is either Euclidean or hyperbolic near infinity, we study microlocal convergence of distorted plane waves E(z,ξ) as z → ∞. Here z is the spectral parameter and ξ indicates the direction of the wave at infinity. The functions E(z,ξ) are generalized eigenfunctions of the Laplacian, they are also known as Eisenstein functions in the hyperbolic setting. We show that if the trapped set has zero Liouville measure, then plane waves converge to a limiting measure, if we average in ξ and in z∈ [R,R+1]. The rate of convergence is estimated in terms of the maximal expansion rate and classical escape rate of the geodesic flow, giving a negative power of z when the flow is Axiom A. As an application, we obtain expansions of local traces and of the scattering phase with fractal remainders. Joint work with Colin Guillarmou. |
| 4:15–5:15 | Daniel Tataru (Berkeley) | Price's law for electromagnetic waves on Schwarzschild/Kerr backgrounds I will describe recent work, joint with Jason Metcalfe, Jacob Sterbenz and Mihai Tohaneanu, on pointwise decay estimates for solutions to the Maxwell system on black hole asymptotically flat relativistic backgrounds. This is related to the nonlinear black hole stability problem for Einstein's equations. | |
| Fri Mar 16, 2012 Berkeley 736 Evans | 2:40–3:30 | Nick Haber (Stanford) | Propagation of singularities around a Lagrangian submanifold of radial points In this talk we consider the wavefront set of a solution u to Pu = f, where P is a pseudodifferential operator with real-valued homogeneous principal symbol p. We assume that the Hamilton vector field of p has a certain configuration of 'radial points,' that is, points where the vector field points radially outward in the cotangent fiber. Hörmander's propagation of singularities theorem gives no information at such radial points. Nevertheless, we are able to give additional statements about the regularity of u. In addition, we discuss a further regularity result in this radial setting, in the sense of how close u is to being a Lagrangian distribution. All work presented is joint with András Vasy. |
| 4:10–5 | Rafe Mazzeo (Stanford) | Spectral geometry on the Riemann moduli space I will describe joint work with Ji, Müller and Vasy concerning the analytic properties of the Laplacian for the Weil–Petersson metric on the Riemann moduli space. I also hope to describe new work with Swoboda concerning fine regularities of the Weil–Petersson metric. | |
| Wed Nov 17, 2010 Stanford | 1:15–2:15 380D | Justin Holmer (Brown) | Blow-up solutions on a sphere for the 3d quintic NLS in the energy space Solutions to the focusing nonlinear Schroedinger (NLS) equation i∂tu + Δu + |u|p−1u = 0 for nonlinearities between mass-critical (p = 1 + 4/d) and energy-critical (p = (d + 2)/(d − 2)) can blow-up in finite time. In the mass-critical setting, the blow-up occurs on a discrete (dimension zero) set whereas in the mass-supercritical (p > 1 + 4/d) setting, the blow-up can occur on sets of positive dimension. Using microlocal methods, we first prove that the log-log blow-up solutions studied by Merle–Raphaël (2001–2005) with single blow-up point to the mass-critical equation remain regular in the energy space away from the blow-up point, resolving a conjecture of Raphaël–Szeftel (2008). We are thus able to insert such solutions into higher-dimensional equations under symmetry assumptions; such equations will be mass-supercritical. In particular, we construct a large class of radial solutions that blow-up on a sphere for the three-dimensional energy-critical NLS. This is joint work with Svetlana Roudenko. We also discuss some other recent work in the field. |
| 3–4 383N | Plamen Stefanov (Purdue) | The geodesic X-ray transform in presence of caustics We study geodesic type of X-ray transforms X locally, near a geodesic segment with conjugate points. In the case of the sphere, we can have exact cancellations that X cannot recover. We study the more common case of fold type of singularities of the exponential map. We analyze the microlocal invertibility (or not) of X. We show that cancelations of singularities always happens in 2D, at least of a finite order. In 3D, we give examples of cancellations and examples where we can invert X microlocally. We analyze X*X and show that it is a sum of a pseudodifferential operator of order −1 and an FIO of order −n/2 with a Lagrangian given by the conormal bundle of the conjugate locus. The latter may or may not be of graph type. | |
| Mon May 24, 2010 Berkeley 740 Evans | 1:10–2 | André Martinez (Bologna) | Some new results on the width of quantum resonances |
| 2:10–3 | András Vasy (Stanford) | Wave propagation on asymptotically De Sitter and Anti–de Sitter spaces In this talk I describe the asymptotics of solutions of the wave equation on asymptotically De Sitter and Anti–de Sitter spaces. This is part of a larger program to analyze hyperbolic equations on non-product, non-compact manifolds, similarly to how various types of `ends' have been studied for the Laplacian and other elliptic operators on Riemannian manifolds. The AdS setting is particularly interesting from the point of view of propagation phenomena, since for the conformally related incomplete metric, there are null-geodesics which are tangent to the boundary. | |
| 4:10–5 | Alexander Gamburd (UCSC) | Infinite volume generalization of Selberg's 3/16 Theorem | |
| Wed Mar 10, 2010 Stanford 383N | 2–3 | Galina Perelman (École Polytechnique / CNRS) | Vey theorem in infinite dimensions and its application to the KdV equation We develop an infinite dimensional version of the Vey theorem and apply it to construct the Birkhoff coordinates for the KdV equation in the vicinity of the origin in L02(S1). The obtained integrating transformation has the form "identity plus a 1-smoothing map". This is a joint work with S.Kuksin. |
| 4–5 | Andrew Hassell (ANU) | Quasi-orthogonality of boundary values of eigenfunctions Consider Dirichlet eigenfunctions for a smooth bounded plane domain. The normal derivatives of these eigenfunctions are known, at least heuristically, to be "quasi-orthogonal" when the eigenvalues are sufficiently close. I will discuss a new result – with a remarkably simple proof – expressing this quasi-orthogonality, and apply it to give sharp theoretical bounds on the accuracy of the "method of particular solutions" for numerically computing such eigenfunctions and eigenvalues. This is joint work with Alex Barnett (Dartmouth). | |
| Mon Nov 30, 2009 Stanford | 2:15–3:15 380D | Maciej Zworski (Berkeley) | Probabilistic Weyl laws for quantized tori For the Toeplitz quantization of complex-valued functions on a 2n-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law. In numerical experiments the same Weyl law also holds for "false" eigenvalues created by pseudospectral effects. The talk is based on joint work with TJ Christiansen. |
| 4–5 383N | Daniel Grieser (Oldenburg) | Pseudodifferential calculus for manifolds with multiply fibred cusps We present a pseudodifferential calculus generalizing the 'fibred cusp calculus' introduced by Mazzeo and Melrose. The generalization is in two directions: First, in a direction similar to previous work by Vaillant, the calculus allows to construct parametrices which satisfy a weaker ellipticity requirement than 'full ellipticity', namely we do not require invertibility of the normal operator at the boundary. This is important since many operators of interest, for example the Hodge Laplacian, tend to have non-invertibe normal operator. Second, we allow multiple fibrations of the boundary. Such operators arise for example as Hodge Laplace operator on locally symmetric spaces of Q-rank one. In the case of two fibrations these are locally of the form P(x,y,z,w; x3 ∂x, x2 ∂y, x∂z,∂w). This is joint work with E. Hunsicker. | |
| Wed Oct 21, 2009 Berkeley 736 Evans | 2:10–3 | Michael Christ (Berkeley) | Existence of extremals for a Fourier restriction inequality The Fourier transform maps L2(S2) to L4(R3). We show that there exist functions which extremize the associated inequality, and that any extremizing sequence of nonnegative functions has a convergent subsequence. This was previously known for paraboloids, where all extremizers are Gaussians and vice versa. Complex extremizers and extremizing sequences are related to nonnegative ones in a simple way. All critical points of the associated nonlinear functional are real analytic. Constant functions are local extremizers, but we do not know whether they are global extremizers, nor whether extremizers are unique modulo symmetries of the problem. The proofs involve concentration compactness ideas, inequalities for convolutions, facts about Fourier integral operators, symmetrization, a characterization of approximate characters, a perhaps nonstandard regularity theorem, an idea from additive combinatorics, facts about spherical harmonics and Gegenbauer polynomials, and several explicit computations. Joint work with Shuanglin Shao. |
| 4:10–5 | Laurent Michel (Nice) | Semiclassical analysis of the Metropolis algorithm on bounded domains We consider the semclassical Metropolis operator on a bounded domain. We obtain a precise description of its spectrum that give useful bounds on rates of convergence for the Metropolis algorithm. As an example, we treat the random placement of N hard discs in the unit square, the original application of the Metropolis algorithm. Joint work with P. Diaconis and G. Lebeau. | |
| Mon Apr 13, 2009 Berkeley 740 Evans | 3:10–4 | Hart Smith (U Washington) | Strichartz estimates for Wave and Schrödinger equations on manifolds with boundary I will discuss recent work with Blair and Sogge establishing Lp estimates for solutions to the wave and Schrodinger equations in the setting of manifolds with boundary, along with some applications to corresponding nonlinear equations. At a point of convexity on the boundary an example of Ivanovici shows that, at least for the wave equation, the full range of Strichartz estimates cannot hold. Nevertheless, we can use microlocal parametrix constructions to obtain a range of important estimates, including ones used to establish well-posedness for energy critical nonlinear equations. |
| 4:40–5:30 | Laurent Demanet (Stanford) | From canonical relations to numerical computations Locally, coordinates can be selected such that a canonical relation is prescribed by the gradient of a phase. I will discuss which choices of coordinates give the most interesting realizations of certain Fourier integral operators as oscillatory integrals. Combined with old almost-orthogonality ideas and new matrix factorization tools, we will see that these considerations go a long way towards solving the important practical problems of optimal-complexity computation of linear hyperbolic propagators and seismic imaging operators. | |
| Fri Mar 13, 2009 Stanford 380F | 2:15–3:15 | Nalini Anantharaman (École Polytechnique / Berkeley) | Spectral deviations for the damped wave equation I will present some results about the Weyl asymptotics for the damped wave equation on a negatively curved manifold. I will give a fractional Weyl upper bound for the number of eigenvalues with given imaginary part. It is notoriously difficult to prove a lower bound – in fact, it is already difficult to prove existence of infinitely many eigenvalues in a given horizontal strip. I will show a very particular model (twisted Laplacian on an arithmetic surface) where it is possible. |
| 4:15–5:15 | Michael Hitrik (UCLA) | Invariant tori, phase space tunneling, and spectra for non-selfadjoint operators We would like to present some recent work together with Johannes Sjöstrand, dealing with the spectral analysis of non-selfadjoint perturbations of selfadjoint semiclassical operators in dimension 2. Specifically, assuming that the classical flow of the unperturbed part is completely integrable, we analyze spectral contributions coming from both Diophantine and rational invariant Lagrangian tori. Estimating the tunnel effect between the two types of tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the strength of the non-selfadjoint perturbation is not too large. We also hope to talk about the ongoing work, where we study the global distribution of the imaginary parts of the eigenvalues in the entire spectral band. | |
| Fri Apr 18, 2008 Stanford 380C | 2:15–3:15 | Maciej Zworski (Berkeley) | Ruelle resonances for Anosov diffeomorphisms (after Faure, Roy, and Sjöstrand) I will report on a recent paper by Faure, Roy, and Sjöstrand who used microlocal methods to give new (and to some of us simple) proofs of many known facts about Anosov diffeomorphisms on compact manifolds: mixing and the existence of Sinai–Ruelle–Bowen measures for (Lebesgue) measure preserving maps, the spectral properties of the Koopman operator, and the decay of correlations in terms of Ruelle resonances (originally established by Anosov, Ruelle, Baladi, Liverani...). |
| 4:15–5:15 | Persi Diaconis (Stanford) | Methods of sampling from a manifold In a variety of applied problems, one is given a (reasonably nice) compact submanifold embedded in Euclidian space and required to choose a sample of points from the area measure. For example, the set of positive n-tuples with a fixed sum and product. I will motivate this class of problems, give several examples and algorithms, and study the analysis problems posed by the algorithms. This is joint work with Susan Holmes and Mehrdad Shahshahani. | |
| Fri Feb 29, 2008 Stanford 381T/380C | 2:15–3:15 | Michael Christ (Berkeley) | Magnetic Schrödinger operators, the d-bar Neumann problem, and the Aharonov–Bohm effect |
| 3:15–4:15 | Gunther Uhlmann (U Washington) | On Calderón's Inverse Problem We will discuss some recent results on Calderón's inverse problem of determining the electrical conductivity of a medium by making voltage and current measurements at the boundary. In particular we will consider the problem of determining the conductivity from partial data and the case of an anisotropic conductor. | |
| 4:15–5:15 | Kiril Datchev (Berkeley) | Local smoothing for scattering manifolds with hyperbolic trapped sets | |
| Mon Jan 28, 2008 Berkeley | 3:10–4 939 Evans | Jared Wunsch (Northwestern) | Semiclassical second microlocal propagation of regularity and integrable systems I will discuss semiclassical second microlocalization at a Lagrangian submanifold of T*X, a precise way of measuring the failure of a distribution to be a Lagrangian distribution in the sense of Hörmander (translated into the semiclassical setting). One application is to the propagation of local Lagrangian regularity on invariant tori of systems with classically integrable hamiltonian. Joint with András Vasy. |
| 4:10–5 891 Evans | Richard Melrose (MIT) | Semiclassical quantization and index maps | |
| Wed Dec 5, 2007 Berkeley 105 Stanley | 1:40–2:30 | Colin Guillarmou (Nice) | Strichartz estimates for a class of hyperbolic systems We discuss the problem of having (or not) a loss in Strichartz estimates in the setting of manifolds with non-empty hyperbolic trapped set. This is joint work with N. Burq and A. Hassell. |
| 3:40–4:30 | Rafe Mazzeo (Stanford) | Polyhedra, iterated edges and the Stoker conjecture I will discuss recent progress on some analytic problems on spaces with iterated edge singularities, with a focus on geometric applications, especially to an old conjecture by J.J. Stoker on convex hyperbolic polyhedra. | |
| Mon Oct 29, 2007 Stanford | 3–4 383N | Daniel Tataru (Berkeley) | Global energy solutions for the KP-I equation The KP-I equation is an asymptotic model for the propagation of long small amplitude dispersive surface waves with weak transversal effects and large surface tension. Using a multiscale analysis we prove that this equation is globally well-posed in the energy space. This is joint work with Alexandru Ionescu and Carlos Kenig. |
| 4:10–5:10 381T | Francis Nier (Rennes) | LLL functional for Bose-Einstein condensates |
