16:15 Mathematical Colloquium
Alexander VOLBERG:
Learning to be a theoretical computer scientist by learning the classical and quantum PAC learning
This is a story how a harmonic analyst may try to penetrate into the area of theoretical computer science (TCS).
PAC = probably approximately correct is notion of TCS where one should determine the object (a function, a matrix, ...) by making a "small" number of trials. One pays the price: trials are random and so the object will be found only with large probability, another price is that it will be found approximately, with a possible small error. PAC learning is a classical part of TCS, where many beautiful results are known. Not long ago a sudden breakthrough in a classical PAC learning was made by my former student Paata Ivanisvili jointly with Alexandros Eskenazis.The quantum analog is PAC learning of big matrices. This will be the main topic of my talk as it leads to an interesting question a lá Remez and Bernstein inequalities.
The talk is based on joint results with Lars Becker, Ohad Klein, Joseph Slote and Haonan Zhang.
Colloquium Poster
09:00 Registration
09:45 OPENING
10:20 Stefanie PETERMICHL: The matrix A2 conjecture
We give an overview of the development of the modern view of sharp weighted estimates in terms of their A2 characteristic as initiated by Sasha and his collaborators. The main object of the talk is on the failure of the expected linear norm estimate for the vector Hilbert Transform in the matrix weighted L^2 space. It is established that the Hilbert Transform has growth exponent 3/2 (Nazarov, P., Treil, Volberg) and that this exponent cannot be improved, showing 3/2>1 (Domelevo, P., Treil, Volberg).
11:50 Vasily VASYUNIN: On two extremal problems in analysis and probability
Two different extremal problems will be considered. The first one concerns finding the extremal value of an integral functional on BMO, while the second is related to martingale transforms. Both problems are solved using the so-called Bellman function technique. It turns out that the Bellman functions associated with these problems are equivalent in a certain sense, and the solutions to the extremal problems coincide, even though the problems are of very different nature.
In the talk, I will briefly describe the formulation of these two problems and explain what the Bellman functions are. Through these two examples, I will illustrate how the Bellman function machinery works. No prior familiarity with the notion of the Bellman function will be assumed. For this reason, the talk will contain many pictures and avoid technical details.
14:40 Zoltán BALOGH: LSI via OMT and curvature
Optimal Mass Transportation (OMT) proved to be an important method in proving functional inequalities in curved spaces. In this talk I will consider the case of the Logarithmic Sobolev Inequality (LSI) and present some recent results in this direction. One of the main goals will be to understand the implications of various concepts of curvature in this setting.
The results are based on joint work with Sebastiano Don, Alexandru Kristály and Francesca Tripaldi.
16:10 Guillermo REY: Cancellative sparse domination
We present a general sparse domination principle which respects the cancellative structure of the functions under study. We obtain sparse domination results in general measure spaces, including general martingale settings in one and two parameters, and in the Euclidean setting. In the one-parameter martingale setting, we obtain a sparse characterization of the H^1 norm. The proofs make critical use of precise level-set estimates for generalized versions of medians. Our results imply new, quantitatively sharp, weighted results for martingales and Calderón–Zygmund operators acting on H^p spaces.
17:15 Guided Tour (Meeting Point: In front of SAZU)
10:00 Xavier TOLSA: Rectifiability, Riesz transforms, and harmonic measure
In recent years, the connection between the Riesz transform and rectifiability has been essential for the solution of some free boundary problems involving harmonic measure. In my talk I will describe the connection between the Riesz transform and harmonic measure and I will survey some joint results with Damian Dabrowski which characterize the measures \mu with L^2(\mu)-bounded Riesz transform in terms of the \beta_2 coefficients of the measure. I will also describe another recent result which can be applied to show that the density of harmonic measure in balls centered at almost all points of vanishing density is not quasi-increasing with respect to the radius.
11:30 Eva GALLARDO-GUTTIÉREZ:
Quasitriangular operators, compact perturbations of self-adjoint operators, and a theorem of Lomonosov
Building on the classical theorem of Aronszajn and Smith [1], Halmos introduced the notion of quasitriangular operators in the 1960s as a key tool to establish the existence of nontrivial closed invariant subspaces for operators on Hilbert spaces. Subsequent significant results by Douglas and Pearcy, as well as those by Apostol, Foiaş, and Voiculescu in the 1970s, showed that the Invariant Subspace Problem can, in fact, be reduced to the class of quasitriangular operators.
In this talk, we will address the study of subclasses of quasitriangular operators — namely, compact perturbations of self-adjoint and normal operators, from the standpoint of the existence of nontrivial closed invariant subspaces. In particular, our approach would yield a new proof of a theorem of Lomonosov on the existence of nontrivial real invariant subspaces for compact perturbations of self-adjoint operators.
REFERENCES:
[1] N. Aronszajn and K. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. 60 (1954).
14:30 Peter YUDITSKII: On point spectrum of Jacobi matrices generated by iterations of quadratic polynomials
In general, the point spectrum of an almost periodic Jacobi matrix can depend on the element of the hull. Jointly with B. Eichinger and M. Lukic, we study the hull of the limit-periodic Jacobi matrix corresponding to the equilibrium measure of the Julia set of the polynomial z^2-\lambda with large enough \lambda; this is the leading model in inverse spectral theory of ergodic operators with zero measure spectrum. We prove that every element of the hull has an empty point spectrum. To prove this, we introduce a matrix version of Ruelle operators.
The problem arose in the context of studying new universality classes associated to fractals.
16:00 Eero SAKSMAN: Chaos and Riemann zeta function on the critical line
We will discuss some results, both old and new (in preparation), on the statistics of the Riemann zeta function on short intervals on the critical line.
The talk is based on collaboration with Adam Harper (Warwick) and Christian Webb (Helsinki).
19:00 Conference Dinner (@ Gostilna na Gradu)
09:30 Guy DAVID:
More two dimensional surfaces in 3-space with a quasisymmetric parameterization, but no bi-Lipschitz one
Given a Muckenhoupt weight $\omega \in A_1(\R^2)$, with small enough constant, we construct a quasi-symmetric parameterization $z\colon \R^2 \to E \subset \R^3$ such that $| Dz(x) \xi |^2$ stays (uniformly) comparable to $\omega(x) |\xi|^2$, where $Dz(x)$ is the differential of $z$ at $x \in \R^2$. When we do this with weights constructed by Bishop that are not equivalent to any Jacobian of a quasiconformal mapping, we obtain sufaces as in the title (I believe Bishop has some already). This lecture belongs to the realm of algorithms for Reifenberg parameterizations.
Work with M. Badger, J. Krandel, S. Ghinassi, R. Schul, T. Toro, to be checked and written.
10:30 Roman BESSONOV: From local to global asymptotic behavior of orthogonal polynomials
A classical result by Máté, Nevai, and Totik (1991) describes the averaged asymptotic behavior of modules of orthogonal polynomials at almost every point z of the unit circle T provided the measure of orthogonality belongs to the Szegő class. I will present a "global" version of this asymptotic relation that works at almost every Stolz angle \Gamma_z. The proof uses three ingredients: Schur functions viewpoint, Khrushchev formula, and the entropy function of a measure. Recently, all of them have found interesting applications in diverse areas. The talk could be considered as an illustration of their use.
Joint work with Artur Nicolau (Universitat Autònoma de Barcelona).
REFERENCES:
[1] A.Máté, P.Nevai, and V.Totik, Szegő's extremum problem on the unit circle, Ann. of Math. 134 (1991).
12:00 Stanislav SMIRNOV: Harmonic measure and quasiconformal mappings
13:00 Closing & Coffee