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UBC cIRcle BIRS Workshop Lecture Videos Logo
Lower bounds on eigenfunctions and fractal uncertainty principle
Banff International Research Station for Mathematical Innovation and Discovery
Dyatlov, Semyon — 2019-03-13 Let $(M,g)$ be a compact Riemannian manifold and $\Omega\subset M$ a nonempty open set. Take an $L^2$ normalized eigenfunction $u$ of the Laplacian on $M$ with eigenvalue $\lambda^2$. What lower bounds can we get on the mass $m_\Omega(u)=\int_\Omega |u|^2$ There are two well-known bounds for general $M$: (a) $m_\Omega(u)\geq ce^{-C\lambda}$, following from unique continuation estimates, and (b) $m_\Omega(u)\geq c$, where $c>0$ is independent of $\lambda$, assuming that $\Omega$ intersects every sufficiently long geodesic (this is known as the <i> geometric control condition</i>). In general one cannot improve on the bound (a) for arbitrary $\Omega$, as illustrated by Gaussian beams on the round sphere. I will present a recent result which establishes the frequency-independent lower bound (b) for <i> any </i> choice of $\Omega$ when $M$ is a surface of constant negative curvature. This bound has numerous applications, such as control for the Schrödinger equation, exponential decay of damped waves, and the full support property of semiclassical measures. The proof uses the chaotic nature of the geodesic flow on $M$. The key new ingredient is a recently established <i> fractal uncertainty principle</i>, which states that no function can be localized close to a fractal set in both position and frequency. This talk is based on joint works with Jean Bourgain, Long Jin, and Joshua Zahl. Non UBC Unreviewed Author affiliation: University of Berkeley Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Spectral densification for hyperbolic surfaces
Banff International Research Station for Mathematical Innovation and Discovery
Dyatlov, Semyon — 2017-06-15 Let M_t, t\neq 0 be a family of compact hyperbolic surfaces which as t\to 0 degenerates to a surface M_0 with two cusps, via pinching a neck. We show a quantization condition for eigenvalues of the Laplacian on M_t in compact subsets of (1/4, \infty), with the subprincipal term determined from the scattering matrix of M_0. We use the Lefschetz fibration model for the degeneration and its metric resolution due to Melrose-Zhu. This is work in progress joint with Richard Melrose. Non UBC Unreviewed Author affiliation: Massachusetts Institute of Technology Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Lower bounds on eigenfunctions on hyperbolic surfaces.
Banff International Research Station for Mathematical Innovation and Discovery
Dyatlov, Semyon — 2018-09-18 I show that on a compact hyperbolic surface, the mass of an L2- normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassi- cal version, has many applications including control for the Schrdinger equation and the full support property for semiclassical defect mea- sures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain. Non UBC Unreviewed Author affiliation: Massachusetts Institute of Technology Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Ruelle zeta at zero for nearly hyperbolic 3-manifolds
Banff International Research Station for Mathematical Innovation and Discovery
Dyatlov, Semyon — 2023-10-26 For a compact negatively curved Riemannian manifold $(\Sigma,g)$, the Ruelle zeta function $\zeta_{\mathrm R}(\lambda)$ of its geodesic flow is defined for $\Re\lambda\gg 1$ as a convergent product over the periods $T_{\gamma}$ of primitive closed geodesics $$ \zeta_{\mathrm R}(\lambda)=\prod_\gamma(1-e^{-\lambda T_{\gamma}}) $$ and extends meromorphically to the entire complex plane. If $\Sigma$ is hyperbolic (i.e. has sectional curvature $-1$), then the order of vanishing $m_{\mathrm R}(0)$ of $\zeta_{\mathrm R}$ at $\lambda=0$ can be expressed in terms of the Betti numbers $b_j(\Sigma)$. In particular, Fried proved in 1986 that when $\Sigma$ is a hyperbolic 3-manifold, $$ m_{\mathrm R}(0)=4-2b_1(\Sigma). $$ I will present a recent result joint with Mihajlo Ceki\'c, Benjamin K\"uster, and Gabriel Paternain: when $\dim\Sigma=3$ and $g$ is a generic perturbation of the hyperbolic metric, the order of vanishing of the Ruelle zeta function jumps, more precisely $$ m_{\mathrm R}(0)=4-b_1(\Sigma). $$ This is in contrast with dimension~2 where $m_{\mathrm R}(0)=b_1(\Sigma)-2$ for all negatively curved metrics. The proof uses the microlocal approach of expressing $m_{\mathrm R}(0)$ as an alternating sum of the dimensions of the spaces of generalized resonant Pollicott--Ruelle currents and obtains a detailed picture of these spaces both in the hyperbolic case and for its perturbations. Non UBC Unreviewed Author affiliation: MIT Researcher
UBC cIRcle BIRS Workshop Lecture Videos Logo
A microlocal toolbox for hyperbolic dynamics
Banff International Research Station for Mathematical Innovation and Discovery
Dyatlov, Semyon — 2015-05-06 Non UBC Unreviewed Author affiliation: Massachusetts Institute of Technology Postdoctoral http://creativecommons.org/licenses/by-nc-nd/2.5/ca/
UBC cIRcle BIRS Workshop Lecture Videos Logo
Control of eigenfunctions on hyperbolic surfaces
Banff International Research Station for Mathematical Innovation and Discovery
Dyatlov, Semyon — 2019-10-14 Given an $L^2$-normalized eigenfunction with eigenvalue $\lambda^2$ on a Riemannian manifold $(M,g)$ and a nonempty open set $\Omega\subset M$, what lower bound can we prove on the $L^2$-mass of the eigenfunction on $\Omega$ The unique continuation principle gives a bound for any $\Omega$ which is exponentially small as $\lambda\to\infty$. On the other hand, microlocal analysis gives a $\lambda$-independent lower bound if $\Omega$ is large enough, i.e. it satisfies the geometric control condition. This talk presents a $\lambda$-independent lower bound for any set $\Omega$ in the case when $M$ is a hyperbolic surface. The proof uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from harmonic analysis called the Fractal Uncertainty Principle. Applications include control for Schrödinger equation and exponential decay of damped waves. Joint work with Jean Bourgain, Long Jin, and Joshua Zahl. Non UBC Unreviewed Author affiliation: UC Berkeley Researcher http://creativecommons.org/licenses/by-nc-nd/4.0/

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