Global theory of one-frequency Schrödinger operators
Sparsity in data analysis and computation
Universality of spectral statistics for random matrices
Random matrices appear in many different branches
of mathematics, with applications ranging from statistics,
combinatorics and communication networks to quantum mechanics
and even number theory. Despite many facets of random matrices,
their spectral statistics exhibit a remarkably universal behavior.
The celebrated Wigner-Gaudin-Mehta-Dyson conjecture asserts that
the local eigenvalue statistics depend only on the symmetry class
of the matrix and is independent of the detailed structure of the
matrix ensemble. We have recently proved this conjecture by embedding
the random matrix into a stochastic flow of matrices
and analyzing the relaxation mechanism of the coupled
stochastic differential equation for the eigenvalues.
This approach has revealed the intrinsic underlying mechanism
behind matrix universality.
The talk will be an overview of these developments
for non-experts, summarizing our recent joint works with P. Bourgade,
A. Knowles, B. Schlein, H.-T. Yau and J. Yin.
Postmodern Fourier Analysis:
Reconsidering Classical Fourier Analysis from a Time-Frequency View-Point
Classical Fourier Analysis has developed from Fourier series
through Fourier transform in one and several variables.
In both situations Lebesgue integration plays a fundamental role,
e.g. for Fourier inversion or the verification of Plancherel's theorem.
The existence of the Haar measure and Pontrjagin's duality theory for LCA groups
have laid the foundation for Harmonic Analysis over LCA groups,
as promoted in the book by A.Weil,
providing the appropriate natural framework for an abstract Fourier transform,
convolution theorems or Plancherel's theorem.
Finally the theory of Schwartz tempered distributions has allowed
to extend the Fourier transform beyond the setting of functions
and has made it a crucial tool for Hörmander's approach
to (pseudo-) differential operators.
A rich variety of fast codes for the DFT,
the discrete Fourier transform (FFT, FFTW, etc.)
plays a crucial role in many electronic devices of our daily life,
allowing the efficient realization of algorithms
for digital signal and image processing.
Modern time-frequency analysis is providing an appropriate framework
for signals, functions, or distributions which cannot be analyzed
by any of the classical tools,
because they assume typically either periodicity of decay at infinity.
The very natural way out of this problem
is the use of a local Fourier transforms,
with a “sliding window”, the so-called STFT (short-time Fourier transform).
As a result the objects under consideration are (continuous) functions
over the so-called time-frequency plane of phase space.
Among the most important windows one finds of course the Gauss function,
due to its optimal TF-concentration and its Fourier invariance.
In this way a very natural link to Fock spaces
(the range of the transform) and to coherent states is given.
It is one of the crucial observations that for any decent window
the STFT can be completely recovered from its samples
over sufficiently dense (in a geometric sense) lattices.
The corresponding theory is Gabor analysis,
referring to Gabor's seminal paper of 1946.
The mathematical analysis of Gabor expansions
is an ongoing enterprize with many interesting and deep mathematical results,
with many interesting links between group representation theory
(mostly of the Heisenberg group),
the theory of function spaces and complex analysis methods.
Robustness of Gabor expansions, the study of Gabor multipliers,
the localization theory of regular Gabor frames and many other results
require typically some mild properties on the windows under consideration,
and not just square integrability (which is OK for the continuous STFT).
The (minimal TF-invariant) Segal algebra $S_0(R^d)$
appears to be the most appropriate universal space for many of these questions.
It is also in the center of a so-called Banach Gelfand triple
which allows to describe mapping properties of classical linear mappings,
e.g. the Fourier transform,
or the correspondence between linear operators and corresponding spreading
or Kohn-Nirenberg symbols
and have by now an important role in the theory of pseudo-differential operators
(cf. the so-called Sjöstrand class).
As we will point out these Banach spaces (of function resp. distributions)
are not only appropriate for the description of questions
arising in Gabor analysis or TF-analysis in general,
but form also a quite appropriate setting for questions of classical Fourier analysis.
Functions from $S_0(R^d)$ are appropriate summability kernels,
the dual space can be used to discuss questions of spectral analysis,
and one can base a relatively elementary theory of generalized stochastic processes
on this frame-work.
Finally this setting is also well suited for a description of
the problem of numerical approximation of continuous questions
(e.g. by sampling and periodization) of continuous problems,
both in Fourier analysis or TF-analysis.
Recent challenges in multifractal analysis
The purpose of multifractal analysis is to describe the pointwise
singularities of functions and measures, and, in particular, to determine
their Hausdorff dimensions. We will present the basic principles of the
subject, and show how it allowed to develop interplays between several
areas of mathematical analysis, such as harmonic and functional analysis,
geometric measure theory, stochastic processes and analytic number
theory. We will also present recent applications in signal and image
processing and expose some mathematical open problems which are
motivated by these applications.
Buffon's needle estimates for rational product Cantor sets
The Favard length of a planar set E is the average length of its
one-dimensional projections. In a joint paper with Bond and Volberg,
we prove new upper bounds on the decay of the Favard length of finite
iterations of 1-dimensional planar Cantor sets with a rational product
structure. This improves on the earlier work of Nazarov-Peres-Volberg,
Bond-Volberg, and Laba-Zhai, and introduces new algebraic and
number-theoretic methods to this area of research. The estimates are
of interest in geometric measure theory, ergodic theory and analytic
Eigenvalues of random normal matrices
I will give an overview of various aspects of the random matrix model.
A special emphasis will be made on recent results
(joint with Seung Yeop Lee)
concerning topology of the “droplet”, the region in the plane
occupied by the eigenvalues when the size of the random matrix is very large.
The structure of the random matrix theory will be illustrated
in a picture where SLE curves, Julia sets, and Kleinian groups
all play a role.
Propagation of chaos and return to equilibrium for Kac's random walks
Kac proposed in 1956 to study the derivation of the (spatially
homogeneous) Boltzmann equation from a many-particle jump processes
with random binary collisions (“Kac's walk”). This limit is closely
connected to the notion of propagation of chaos, i.e. product-like
structure of the many-particle distribution in the many-particle
limit. Several questions were raised: propagation of chaos for the
unbounded collision rates encountered in physics, estimations of rates
of relaxation uniform in the many-particle limit, propagation of
entropic chaos and derivation of the H-theorem. We shall present
answers to these questions, obtained in a joint work with Stephane
Mischler. They are based on a new quantitative stability approach for
many-particle limits, that allows to take advantage of dissipativity
of the level of the limit PDE. If time allows we shall discuss
connexions with the BBGKY hierarchy, and how this method applies to
other many-particle limits.
Singular distributions and symmetry of the spectrum
We discuss the “Fourier symmetry” of measures and distributions
on the circle in relation with the size of their support.
The talk is based on joint work with Gady Kozma.
Spectral gaps and oscillations
How to estimate the size of a gap in the Fourier spectrum of
a measure (function, distribution)?
How fast should a function with a spectral gap osscillage near infinity?
A number of classical problems of Harmonic Analysis,
posted by Beurling, Kolmogorov, Krein, Wiener and
others can be reformulated in these terms.
In my talk I will discuss several of such problems along with their
Complex interpolation and rotation of quasiconformal maps
We introduce interpolation on Lp-spaces with complex exponents,
and apply it to obtain optimal estimates
for rotation of quasiconformal maps.
The talk is based on joint work with K. Astala, T. Iwaniec and I. Prause.
The indicator/interval testing characterization of the two weight inequality for the Hilbert transform
We present joint work with M. Lacey. C.-Y. Shen and I. Uriarte-Tuero
that characterizes when the Hilbert transform H_u maps one
locally finite weighted space $L^2(u)$ to another $L^2(v)$.
Namely, when the
strong $A_2$ condition holds together with the pair of dual
indicator/interval testing conditions: the $L^2(v)$ norm of $H_u(1_E)$
is at most a fixed multiple of the $L^2(u)$ norm of $1_I$
for all closed subsets $E$ of a finite interval $I$.
This provides a first real variable
characterization of the two weight inequality, but leaves open the
Nazarov-Treil-Volberg conjecture that it suffices to test over only the
cases $E=I$ is an interval.
Return to equilibrium, non-self-adjointness and symmetries
In this talk we review some results for
non-self-adjoint differential operators, where symmetries play an
important role. In the case of the Kramers–Fokker–Planck operator we
explain how a certain supersymmetry and PT-type symmetry lead to
precise results about the reality and the size of the exponentially
small eigenvalues in the low temperature limit. This also applies
sometimes to coupled oscillators with heat baths, but a recent result
says that when the temperatures of the baths are different, the
supersymmetry will sometimes break down and would have to be replaced
by more direct methods of semi-classical analysis.
The talk is mainly based on joint works with Frédéric Hérau and
Michael Hitrik, but uses results and ideas of Witten, Helffer, Nier,
Héerau, Bismut, Tailleur – Tanase-Nicola – Kurchan and other people.
Commutators, paraproducts and BMO in non-homogeneous martingale harmonic analysis
The talk will be devoted to the basic objects of the martingale harmonic analysis
in the non-homogeneous settings, in particular the paraproducts and the commutators.
These operators appear naturally and play an important role in harmonic analysis.
As it is well known, in the classical homogeneous situation
one can characterize BMO by the boundedness of the paraproducts or commutators.
It turns out that unlike the classical case,
the general situation is much more interesting.
For example, unlike the homogeneous case,
the boundedness of a paraproduct does not mean that the symbol is in BMO:
moreover, the $L^p$-boundedness of the paraproduct does depend on $p$.
However, the so-called extended paraproduct also appears naturally,
and the boundedness of this operator characterizes BMO.
This implies that in the non-homogeneous case the space BMO
also can be characterized by the boundedness of commutators,
provided that the underlying martingale transform satisfies some mixing properties.
We will also discuss the geometry of the martingale difference bases in $L^p$.
Turan and Remez-type inequalities, moments, and Taylor domination for some classes of functions
Let a family of analytic functions
be given, and let $R(\lambda)$ be the radius of convergence of $f_\lambda$. The family $f_\lambda$ possesses a property of an
$(N,C)$-uniform Taylor domination if
with $N$ and $C$ not depending on $\lambda$.
Taylor domination provides, in particular,
a uniform bound on the number of zeroes of $f_\lambda$
in each disk strictly contained in the disk of convergence.
One can hope to use Taylor domination in study of
the Poincaré first return mapping (i.e. in counting limit cycles)
of plane polynomial vector fields.
Taylor coefficients in this case satisfy a certain linear
differential-difference recurrence relation.
We present some results on Taylor domination for simpler,
but still important, recurrence relations.
The starting example is the family $R^d_\lambda$
of all rational functions of degree $d$.
Here uniform Taylor domination follows from the classical Turan lemma.
Equivalently, Taylor domination holds
for solutions of a linear recurrence relation with constant coefficients.
Next, we discuss linear recurrence relations with the coefficients
polynomially depending on the index
(Taylor coefficients of solutions of linear polynomial ODE's).
In this case we prove a weaker version of Taylor domination,
where $C$ in (1) is replaced by $C^k$.
Next we consider Taylor domination for Stieltjes transforms,
i.e. for sequences of the consecutive moments of a given function $f$.
Here a key tool is counting sign changes of $f$
and a validity of a Remez-type inequality for this function.
If time allows, we shall discuss also a remarkable connection,
discovered by Bautin in 1935,
of the uniform Taylor domination property
to the commutative algebra of the Taylor coefficients,
as functions of the parameter $\lambda$.
This is a joint work with D. Batenkov.