Professor
Department of Mathematics
Massachusetts Institute of Technology
Office: 2-377
E-mail: dyatlov@math.mit.edu
WKB structure in a scalar model of flat bands
In Fall 2025 I am teaching 18.03 (Differential Equations).
I am also co-organizing the 18.S995 (Topics in Analysis) course.
Slides for my talk `Uncertainty Principles and Quantum Chaos' at JMM 2025
CV
We study semiclassical measures on compact quotients of the complex hyperbolic space. We show that the support of each such measure is either the entire cosphere bundle or it contains the cosphere bundle of a compact immersed totally geodesic complex submanifold. We use the strategy of the previous paper and a classification of orbit closures of fast unstable/stable vector fields.
We investigate semiclassical measures for higher dimensional quantum cat maps, which are standard toy models in quantum chaos. If the matrix quantized has a spectral gap (i.e. a simple eigenvalue of largest absolute value) and its characteristic polynomial is irreducible over the rationals, then we show that all semiclassical measures have full support. The spectral gap assumption makes it possible for us to still use the one-dimensional fractal uncertainty principle in this higher dimensional situation. When the characteristic polynomial is reducible over rationals, we obtain a more complicated to state result and compare it with existing counterexamples.
We show lower bounds on mass of eigenfunctions on negatively curved surfaces, and more generally surfaces with Anosov geodesic flow. We also obtain observability/control for the Schrödinger equation and exponential decay for the damped wave equation. The proofs partially use the strategy of the previous work in constant curvature, however the case of variable curvature presents substantial challenges due to which a large part of the proof proceeds in a very different way. [Diagram of dependence of sections]
Using the fractal uncertainty principle proved in a previous paper with Bourgain, we show that every semiclassical defect measure on a compact hyperbolic surface has support equal to the entire cosphere bundle.
We show that fractal uncertainty principle implies an essential spectral gap for convex co-compact hyperbolic surfaces. This partially recovers the result of the paper with Zahl. In contrast to the latter paper, which relied on complicated microlocal techniques, our proof uses transfer operators and is relatively short and self-contained.
For a convex co-compact hyperbolic surface with limit set Λ of dimension δ>0, we show that the Fourier dimension of Λ is at least ε for some ε>0 depending only on δ. The proof uses the discretized sum-product theorem in R and the nonlinear nature of the transformations in the group Γ; in fact, our result would be false for linear Cantor sets. Using fractal uncertainty principle, we deduce an essential spectral gap of size 1/2-δ+ε.
We show that every Ahlfors–David regular subset of R of positive dimension satisfies the fractal uncertainty principle with exponent strictly larger than the one coming from the volume bound. The method of proof is inspired by the works of Dolgopyat, Naud, and Stoyanov in the context of spectral gaps for transfer operators, and can be crudely summarized as follows: triangle inequality in a Hilbert space is rarely sharp. As an application we obtain new spectral gaps for convex co-compact hyperbolic surfaces and open quantum baker's maps.
We show that every convex co-compact hyperbolic surface has an essential spectral gap. The proof uses the approach developed in the paper with Zahl and the key new component is the fractal uncertainty principle for δ-regular sets with all δ<1. Previously such gaps were known only under the relaxed pressure condition δ≤1/2.
We study open quantum baker's maps, which are models in open quantum chaos having Cantor sets as their trapped sets. Using the fractal uncertainty approach developed in the paper with Zahl and the arithmetic structure of Cantor sets, we show that all such systems have a spectral gap, and obtain quantitative bounds on the size of the gap. We also show an improved Weyl bound similar to the one here. Both results are supported by numerical evidence.
We give a Weyl upper bound on the number of scattering resonances on convex co-compact hyperbolic quotients in strips. The exponent in the bound depends on the width of the strip and improves on the standard Weyl upper bound, in particular this exponent is negative until the Patterson–Sullivan gap. The appendix gives numerical evidence for the new Weyl bound.
We obtain an improved spectral gap for convex co-compact hyperbolic surfaces with the dimension of the limit set close to 1/2. Compared to previous works on spectral gaps which rely on Dolgopyat's method, we decouple the combinatorial difficulties of the problem from the analytical ones and obtain an explicit formula for the gap in terms of additive energy of the limit set, which in turn is estimated using its Ahlfors–David regularity.
This paper presents various non-rigorous heuristics, supported by numerical evidence, on magic angles in the scalar model of twisted bilayer graphene, together with rigorous spectral results and a quantization condition in a toy 1-dimensional WKB model.
We work with a linearized model of fluid in an effectively two-dimensional aquarium. Under the Morse–Smale assumption on the chess billiard map, we show a limiting absorption principle and apply it to get long time asymptotics of the solution to the evolution equation, showing that the singular part concentrates on the attractor of the chess billiard flow similarly to this experimental video from ENS Lyon.
We study a forcing problem for a 0th order pseudodifferential operator on a surface whose reduced Hamiltonian flow is a Morse–Smale flow without fixed points. In particular we explain how some of the results of a recent paper of Colin de Verdière and Saint-Raymond can be obtained using radial estimates.
We give an upper bound on the number of resonances in strips for general manifolds with Euclidean infinite ends, without any assumptions on the trapped set. Our bound depends on the volume of the set of trajectories which are trapped for the Ehrenfest time, and gives a polynomial improvement over the standard bound Rd-1 when the escape rate is positive. We also prove a wave decay result with high probability for random initial data.
We obtain a polynomial lower bound on the norm of the scattering resolvent in the lower half-plane under very mild assumptions on trapping, namely existence of a trajectory which is trapped in the past, but not in the future. The power in the bound, depending on the maximal expansion rate along the trajectory, gives the smallest number of derivatives lost in an exponential decay of local energy estimate for the wave equation.
We show that in the presence of certain elliptic trapping, one-sided bounds on the resolvent with a cutoff supported far away into the infinity are at best exponential. This is in contrast with the known polynomial two-sided bounds.
We show existence of a band of resonances with a Weyl law when the trapped set is r-normally hyperbolic for large r and the normal expansion rates are half-pinched. This dynamical setting is stable under perturbations and is applied the next paper to Kerr–de Sitter black holes. The key tool is a Fourier integral operator microlocally projecting onto resonant states in the band. This operator lets us microlocally represent resonance expansions as Taylor expansions; a variety of methods from microlocal analysis, especially positive commutator estimates, finish the proof. The r-normal hyperbolicity assumption is explicitly needed in the construction of the projector to ensure smoothness of its symbol.
We obtain a fractal upper bound on the number of resonances in disks of fixed size centered at the unitarity axis for a general class of manifolds, including convex co-compact hyperbolic quotients. The exponent in the bound is related to the dimension of the trapped set. We use the techniques of Sjöstrand–Zworski on resonance counting for hyperbolic trapped sets and of Vasy on effective meromorphic continuation of the resolvent for asymptotically hyperbolic infinite ends.
It was proved by Bindel and Zworski in this paper that for scattering on the half-line by a compactly supported potential with a constant bump at the end of its support, the purely imaginary poles of the resolvent become symmetric with respect to the real axis modulo errors exponentially small in the semiclassical limit. Our paper is an extension of this result to a more general class of potentials, allowing any positive bump, and also provides a different explanation of why this phenomenon holds.
We use radial estimates (a version for vector bundles) to investigate what Sobolev regularity is needed for the meromorphic continuation of the Pollicott–Ruelle resolvent for an Anosov flow, in case of an arbitrary lift of the flow to a vector bundle. The resulting condition is similar to the one obtained in the paper of Bonthonneau–Lefeuvre for Hölder–Zygmund regularity.
We study the vanishing order of the Ruelle zeta function at zero for a compact connected oriented negatively curved 3-manifold. We show that if the metric is a small perturbation of the hyperbolic metric, then the order of vanishing is equal to 4-b1 where b1 is the first Betti number. This is in contrast with the hyperbolic case where this order is 4-2b1 and with the case of surfaces where the previous paper with Zworski showed that the order only depends on the genus, not on the metric.
We show that the meromorphic continuation result for Ruelle zeta function of our previous paper applies to Axiom A flows with orientable stable/unstable spaces, using results of Conley–Easton and Guillarmou–Mazzucchelli–Tzou.
Using previous work with Zworski as well as relatively standard microlocal techniques, we show that the Ruelle zeta function for a negatively curved oriented surface vanishes at zero to the order given by the absolute value of the Euler characteristic. This result, giving a relation between dynamics and topology, was previously known only in constant curvature via the Selberg trace formula.
We define Pollicott–Ruelle resonances for hyperbolic flows on noncompact manifolds (in a setting closely related to basic sets of Axiom A flows). They arise either as poles of Fourier–Laplace transforms of correlations or as poles of meromorphic extensions of dynamical zeta functions. Our construction is based on the microlocal approach of Faure–Sjöstrand and Dyatlov–Zworski.
We show that Pollicott–Ruelle resonances of an Anosov flow are the limits of the L2 eigenvalues of the generator of the flow with an additional damping term. The latter eigenvalues characterize decay of correlations for a stochastic modification of the flow.
We describe the Pollicott–Ruelle resonances for compact hyperbolic quotients in all dimensions. In dimensions greater than 2, this description is much more involved than for surfaces and features in particular the spectrum of the Laplacian on trace-free divergence-free symmetric tensors of all orders.
We prove meromorphic continuation of the Ruelle dynamical zeta function for Anosov flows, recovering recent results of Giulietti, Liverani, and Pollicott. We follow the approach of Faure–Sjöstrand; the key new component is the restriction on the wavefront set of the resolvent, using the methods of Melrose and Vasy, making it possible to take the flat trace.
Using the methods of the previous paper, we obtain sharp upper bounds on the number of Ruelle resonances, defined using the framework of Faure and Sjöstrand.
Motivated by Kerr and Kerr–de Sitter black holes, we show existence of spectral gaps of optimal size for normally hyperbolic trapped sets with codimension 1 smooth stable/unstable foliations. The short proofs, based on semiclassical defect measures, simplify and refine some of the earlier results of Wunsch–Zworski, Nonnenmacher–Zworski (in our special case), and myself.
A note outlining some of the recent work on resonances for rotating black holes.
We apply the methods and results of a previous paper to the setting of Kerr–de Sitter black holes, proving a Weyl law for resonances and an asymptotic decomposition of linear waves for high frequencies. The latter fact, related to resonance expansion, is also proved for Kerr black holes using the work of Vasy–Zworski.
We establish a Bohr–Sommerfeld type quantization condition for quasi-normal modes of a slowly rotating Kerr–de Sitter black hole, observing in particular a Zeeman-like splitting once spherical symmetry is broken. We compute the resulting pseudopoles numerically [click here for MATLAB codes and data] and compare them to those numerically studied by physicists. Finally, we prove a resonance decomposition of linear waves.
The red-shift effect and a parametrix near the event horizons are used to extend the exponential decay proved in the previous paper to the whole space.
This paper constructs an analogue of the scattering resolvent in the case of a slowly rotating Kerr–de Sitter black hole. The poles are proven to form a discrete set; they are the quasi-normal modes of black holes that have been numerically studied by physicists. Using the recent result by Wunsch and Zworski, we prove existence of a resonance free strip and exponential energy decay for the wave equation on a fixed compact set.
Using our previous work, we derive scattering phase asymptotics for manifolds with asymptotically Euclidean ends with remainders that are fractal for hyperbolic flows. We next derive asymptotics with a similar remainder for hyperbolic quotients, and non-sharp lower bounds on the remainder in some cases. Finally, we investigate the relation of scattering phase asymptotics with counting resonances near the unitarity axis, in particular with the fractal Weyl law.
For Riemannian manifolds that are either Euclidean or hyperbolic near infinity, and with trapped set of Liouville measure zero, we show that plane waves, also known as Eisenstein functions, converge to some semiclassical measure, if one averages in frequency in an h-sized window and in the direction of the wave. The speed of convergence is estimated in terms of the classical escape rate and the Ehrenfest time; in many cases, this is a power of h (in contrast with quantum ergodicity on compact manifolds, where the best known result is 1/|log h|). As an application, we derive a local Weyl law for spectral projectors with a fractal remainder.
We show that on a compact Riemannian manifold with ergodic geodesic flow, restrictions of the eigenfunctions of the Laplacian to any hypersurface satisfying a simple geometric condition are equidistributed in phase space. This work generalizes a paper of John Toth and Steve Zelditch using the methods of semiclassical analysis, and provides a shorter proof.
We consider a Riemannian surface with cusp ends and show that the Eisenstein functions in the upper half-plane, away from the real line, converge to a certain canonical measure. This statement is similar to quantum unique ergodicity(QUE); however, being away from the real line considerably simplifies the problem. In particular, no global dynamical properties of the flow are used. As an application, we prove that the scattering matrix converges to zero in any strip away from the real line.
[ First version, using a more direct analysis in the cusp ]We prove certain weighted L-infinity estimates for eigenfunctions on a strictly convex surface of revolution. Our application lies in the area of compressed sensing — these estimates give an improvement on how many random sampling points (chosen with respect to the measure related to the weight) are enough to recover, with high probability, a function whose expansion in spherical harmonics is sparse.
This was my undergraduate project.
An article on quantum chaos for a general mathematical audience.
In this paper, Alex Cohen proved the fractal uncertainty principle in higher dimensions for line porous sets. This expository note gives a different point of view on some parts of the proof.
A review of various old and new results on weak limits and semiclassical measures in Quantum Chaos.
A review of Quantum Ergodicity and related results, including a sketch of the proof of Quantum Ergodicity.
This is a broad review of recent developments related to the fractal uncertainty principle (unlike the previous review which focused in more detail on a single result).
An expository article for the proceedings of Journées EDP (Roscoff, June 2017) describing the results and the proofs of previous works with Bourgain and Jin on fractal uncertainty principle and its application to control of eigenfunctions.
For a series of lectures at MSRI. A quick introduction to b-calculus techniques and an outline of the proof of the Fredholm property of Laplacians on asymptotically cylindrical manifolds on weighted b-Sobolev spaces.
For the course 18.155 at MIT. Starts with definition of distributions and ends with Hodge's Theorem.
For the minicourse taught at Geometry and Analysis Seminar for Boston Area Graduate Students in Fall 2022.
For the minicourse taught at the Summer Northwestern Analysis Program in Summer 2019.
These notes give a self-contained proof of the Stable/Unstable Manifold Theorem (also known as the Hadamard–Perron Theorem) in hyperbolic dynamics. They also give two examples of hyperbolic systems: geodesic flows on negatively curved surfaces and dispersing billiards. The proof of the Stable/Unstable Manifold Theorem starts with a basic model case which however retains the essence of the general case. There are many figures throughout the text, and it is indended as a (somewhat) gentle introduction to some techniques in hyperbolic dynamics.
Lecture notes for lectures given at the Third Symposium on Scattering and Spectral Theory in Florianopolis, Brazil, July 2017 (work in progress).
Lecture notes for lectures given in Tsinghua University in Summer 2016.
Lecture notes for part of the course 18.158 given at MIT in Spring 2016.
Some (rather sloppy) notes for a (rather informal) talk on semiclassical Lagrangian distributions, Fourier integral operators, and their applications to quantization conditions given in March 2012.
This is a short introduction to nontrapping estimates in scattering theory. We discuss (1) how one can obtain exponential decay in obstacle scattering from a nontrapping estimate via the contour deformation argument (2) how to prove the semiclassical propagation of singularities estimate in the presence of a complex absorbing potential via Hörmander's positive commutator method (3) how propagation of singularities and complex scaling lead to a nontrapping estimate in the one-dimensional model case.
An article that served as the final project in Michael Hutchings' course on symplectic geometry in Spring 2009.
For the minicourse taught at the Summer Northwestern Analysis Program.
A few movies demonstrating decay of correlations and equidistribution of velocities for billiards.
A demonstration of concentration in phase space (measured using the FBI transform) of various solutions to the damped wave equation on the circle with a potential.
Some of these have websites which are on Stellar/Canvas and not publicly available.
