# Abstracts

Last update: 31-06-2017

Jean-Philippe Anker (University of Orleans)

Title: The Schrödinger equation on symmetric spaces.

Abstract: My aim in this talk is to give a survey about the linear and the nonlinear Schrödinger equation on symmetric spaces, locally symmetric spaces and homogeneous trees.

Semyon Dyatlov (Massachusetts Institute of Technology) on behalf of Jean Bourgain (Institute for Advanced Study, Princeton)

Title: Harmonic analysis issues related to hyperbolic surfaces.

Abstract: We show that every convex co-compact hyperbolic surface has an essential spectral gap, that is a half-plane $$\{ \mathrm{Re} \, s > 1/2 - \beta \}$$, $$\beta > 0$$, where the Selberg zeta function has only finitely many zeroes. The main tool is a fractal uncertainty principle, stating that no function can be localized close to a fractal set in position and frequency. We explain its proof which is based on the Beurling-Malliavin multiplier theorem. Another application of this uncertainty principle is to control of eigenfunctions of the Laplacian on a compact hyperbolic surface by any nonempty open set.
Coming back to the convex co-compact case, we show Fourier decay for the Patterson-Sullivan measure using the fact that the transformations generating the limit set of the group are sufficiently nonlinear. The key tool is a bound on exponential sums following from the discretized ring theorem. As an application we give an essential spectral gap $$\{\mathrm{Re} \, s > \delta-\varepsilon\}$$, where $$\delta \in (0,1)$$ is the dimension of the limit set and $$\varepsilon > 0$$ depends only on $$\delta$$.

Anthony Carbery (University of Edinburgh )

Title: How to think about endpoint multilinear Kakeya results?

Michael Christ (University of California, Berkeley)

Title: Sharp inequalities, sharpened.

Abstract: The Riesz-Sobolev inequality provides an upper bound for convolutions of indicator functions of sets of specified measures. Under natural hypotheses, equality  is attained only by indicator functions of homothetic ellipsoids. A sharper form of the inequality relates the degree of optimality of a tuple of sets to its degree of approximability by suitable tuples of ellipsoids. The proof involves several distinct elements, including an inequality for spherical harmonics. It provides quantitative insight into the interaction between one-dimensional subgroups of Euclidean groups and their cosets.
Corresponding results for Young's convolution inequality, in its sharp form due to Beckner and to Brascamp-Lieb, and for  the Brascamp-Lieb-Luttinger symmetrization inequality, which generalizes the Riesz-Sobolev iequality, will be discussed if time permits.

Jacek Dziubañski (Uniwersytet Wroc³awski)

Title: Hardy spaces for certain semigroups of linear operators.

Abstract: We shall discuss properties of Hardy spaces associated with semigroups of linear operators generated by Dunkl, Grushin, and Schrödinger operators. We shall concentrate on characterizations of these spaces in particular characterizations by relevant Riesz transforms.

Charles Fefferman (Princeton University)

Title: Interpolation and approximation in several variables.

Abstract: Let $$X$$ be our favorite Banach space of continuous functions on $$R^n$$ (e.g. $$C^m$$, $$C^{m,\alpha}$$, $$W^{m,p}$$). Given a real-valued function $$f$$ defined on an (arbitrary) given set $$E$$ in $$R^n$$, we ask: How can we decide whether $$f$$ extends to a function $$F$$ in $$X$$? If such an $$F$$ exists, then how small can we take its norm? What can we say about the derivatives of $$F$$? Can we take $$F$$ to depend linearly on $$f$$? What if the set $$E$$ is finite? Can we compute an $$F$$ whose norm in $$X$$ has the smallest possible order of magnitude? How many computer operations does it take? What if we ask only that $$F$$ agree approximately with $$f$$ on $$E$$? What if we are allowed to discard a few points of $$E$$ as "outliers"; which points should we discard? A fundamental starting point for the above is the classical Whitney extension theorem, which I learned from Eli.

Alexandru Ionescu (Princeton University)

Title: On the global regularity of the Einstein-Klein-Gordon coupled system.

Abstract: I will discuss some recent work on constructing global solutions of quasilinear evolution systems. In particular, I will focus on the Einstein-Klein-Gordon coupled system in General Relativity, and describe some recent work, joint with Benoit Pausader, on the global regularity of this system and certain simplified models.

Alexander Iosevich (University of Rochester)

Title: Rigidity, graphs and Hausdorff dimension.

Abstract: We are going to study point configurations with vertices in compact subsets of the Euclidean space of a given Hausdorff dimension. Some of the distances among these vertices are specified and others are not, with a connected graph describing this set of relationships. Two configurations are defined to be congruent if the corresponding specified distances are the same. This definition of congruence nearly matches the classical one if the configuration is rigid in a suitable sense. The set of configurations described by a given graph naturally embeds in Euclidean space of dimension given by the number of edges of the graph. We are going to prove that for any connected configuration there exists a dimensional threshold such that any set of Hausdorff dimension that exceeds this threshold determines a positive proportion of all possible configurations of this type. Our arguments rely on analytic, combinatorial and topological considerations.

Grzegorz Karch (Uniwersytet Wroc³awski)

Title: Homogeneous Boltzmann equitation and the $$\alpha$$-stable processes.

Abstract: In the talk, infinite energy solutions to the homogeneous Boltzmann equation for Maxwellian-type molecule will be discussed. In particular, similarities between the homogeneous Boltzmann equation and the fractional heat equation will be emphasized. Moreover, I will show how to construct self-similar solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules which correspond to $$\alpha$$-stable densities.
This is a joint work with Marco Cannone and Eleftherios Ntovoris.

Michael Lacey (Georgia Institute of Technology)

Title: Sparse Bounds: A Survey.

Abstract: The notion of a sparse bound for a bilinear form provides an upper bound in terms of a sum of local averages of the input functions. As a corollary, one deduces a range of weighted and vector valued inequalities. The weighted inequalities are completely quantitative, and frequently sharp in the $$A_p$$ characteristic. The arguments that lead to these bounds are natural extensions of the Calderon-Zygmund theory, but their application in this way has only recently been established.

Loredana Lanzani (Syracuse University)

Title: On the symmetrization of Cauchy-like kernels.

Abstract: In this talk I will present new symmetrization identities for a family of Cauchy-like kernels in complex dimension one. Symmetrization identities of this kind were first employed in geometric measure theory by P. Mattila, M. Melnikov, X. Tolsa, J, Verdera et al., to obtain a new proof of $$L^2(\mu)$$ regularity of the Cauchy transform (with $$\mu$$ a positive Radon measure in $$\mathbb C$$), which ultimately led to the a partial resolution of a long-standing open problem known as the Vitushkin’s conjecture. Here we extend this analysis to a class of kernels which are more closely related to the classical kernels in complex function theory.
This is joint work with Malabika Pramanik (U. British Columbia).

Elon Lindenstrauss (The Hebrew University of Jerusalem)

Title: Random walks on skew products and spectral gaps.

Abstract: In my talk I will describe two results, both joint work with P. Varju, regarding random walks on skew products: one regarding random walks by isometries of $$R^d$$, in particular a local central limit theorem valid at very fine scales, and one regarding random walk on Cayley graphs of the group of affine maps on $$F_q^d$$. While seemingly quite different, these results are closely analogous.

Ákos Magyar (University of Georgia)

Title: Geometric configurations in sets of positive density.

Abstract: Geometric Ramsey theory studies the existence of geometric patterns, determined up to translations, rotations and possibly dilations, in sets of positive upper density. We outline a new approach to these type of problems, motivated by developments in additive combinatorics, and discuss some recent results both in the continuous and the discrete setting.
This is joint work with Neil Lyall.

Detlef Müller (Christian-Albrechts-Universität zu Kiel)

Title: On Fourier restriction for a non-quadratic hyperbolic surface.

Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Basically, only the quadric $$z=xy$$ has been studied. In this talk, I shall report on joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques compared to the elliptic case are required to handle more general hyperbolic surfaces.

Alexander Nagel (University of Wisconsin-Madison)

Title: Estimates for Bergman kernels in Reinhardt and tube domains.

Abstract: We obtain geometric estimates of the Bergman kernel and its derivatives in tube domains over bounded convex domains. Then using the Poisson summation formula we obtain estimates in Reinhardt domains which are bounded away from the coordinate axes.

Amos Nevo (Technion-Israel Institute of Technology)

Title: The utility of the Kunze-Stein phenomenon.

Abstract: We will explain the path from the Kunze-Stein convolution theorem, to mean ergodic theorems for measure-preserving actions of groups, to the non-Euclidean lattice points counting problem, and to the recent development of intrinsic Diophantine approximation on homogeneous algebraic varieties.

Duong H. Phong (Columbia University)

Title: Anomaly cancellation and geometric flows.

Abstract: The cancellation of anomalies is an essential requirement in the construction of string theories. This results in an equation which involves quadratic expressions in the curvature tensor. In presence of torsion, this equation can be viewed as providing a notion of canonical metric in non-Kähler geometry. We develop a parabolic approach for solving this equation. Perhaps surprisingly, the corresponding flow can be interpreted as a next order correction to the Ricci flow. We show that, on toric fibrations, the flow exists for all time and converges, recovering in this way the well-known solution obtained by J. Fu and S.T. Yau in 2006 by solving a delicate elliptic equation of Monge-Ampere type.
This is joint work with S. Picard and X.W. Zhang.

Fulvio Ricci (Scuola Normale Superiore, Pisa)

Title: A maximal restriction theorem and Lebesgue points of Fourier transforms.

Abstract: This is joint work with D. Müller and J. Wright.
Let $$S$$ be the graph in $$\mathbb R^2$$ of a convex $$C^2$$ function, with affine arclength measure $$d\mu=\kappa^\frac13ds$$, $$\kappa$$ denoting the curvature of $$S$$. We prove that, for $$1 < p < \frac87$$ and $$q=p'/3$$, the following maximal restriction inequality holds: $\big\|(M\hat f)_{|_S}\big\|_{L^q(S,\mu)}\le C\big\|f\|_{L^p(\mathbb R^n)}\ ,$ where $$M$$ is the strong maximal function. This implies that, given $$f\in L^p(\mathbb R^2)$$ with $$p<\frac87$$ and $$S$$ as above, $$\mu$$-almost every point of $$S$$ is a Lebesgue point of $$\hat f$$. Moreover, the regularized values of $$\hat f$$ at these points coincide $$\mu$$-a.e. with the values of the image $$\mathcal R f$$ of $$f$$ under the restriction operator $$\mathcal R$$.

Andreas Seeger (University of Wisconsin-Madison)

Title: Spherical maximal functions on the Heisenberg groups.

Abstract: Let $$V$$ be a hyperplane in the Heisenberg group $$\mathbb H^n$$ and $$\mu$$ the surface measure of a sphere in $$V$$. We discuss some old and new results on $$L^p(\mathbb H^n)$$ boundedness for the maximal function $$\sup_t |f*\mu_t|$$ generated by the automorphic dilations.

Brian Street (University of Wisconsin-Madison)

Title: Convenient Coordinates.

Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let $$X_1,...,X_q$$ be either real or complex $$C^1$$ vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., $$C^m$$, or $$C^\infty$$, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".

Terence Tao (University of California, Los Angeles)

Title: Failure of the pointwise ergodic theorem on the free group at the $$L^1$$ endpoint.

Abstract: A well known theorem of Nevo and Stein shows that the pointwise ergodic theorem on the free group holds in $$L^p$$ for every $$p>1$$. We give a construction that shows that the $$L^1$$ endpoint of this theorem is false.

Christoph Thiele (Universität Bonn)

Title: Some results on directional operators.

Abstract: We present some joint work with Francesco DiPlinio, Shaoming Guo, and Pavel Zorin-Kranich. First we prove a Littlewood-Paley diagonalization result for bi-Lipschitz perturbations of the identity map on the real line. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane. Second, we prove a square function estimate for a single scale directional operator. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane.

François Treves (Rutgers University)

Title: On a class of integrable structures on smooth manifolds and the local exactness of the associated differential complexes.

Abstract: The lecture will describe classes of integrable structures on smooth manifolds, the di§erential complexes they give rise to, and what can be said about the local exactness of such complexes. Main points will be the relation with local solvability of linear PDE and the Weierstrass-type approximation formulas, as well as the role and validity of the approximate Poincaré lemma.

James Wright (University of Edinburgh)

Title: A problem of Kahane in higher dimensions.

Abstract: The famous Beurling-Helson theorem characterises mappings of the circle with preserve absolutely convergent fourier series. This has been extended to higher dimensional tori. Kahane asked what happens when we relax absolute convergence to uniform convergence. Much work has been done on this problem for mappings of the circle. We examine what happens on higher dimensional tori.