Schedule of the Workshop on Real Analysis

Monday, July 14

9:00 - 10:00 Guy David: A variant with many phases of the Alt, Caffarelli, and Friedman free boundary problem
10:00 - 11:00 Rowan Killip: tba
11:00 - 11:30 Coffee break
11:30 - 12:30 Patrick Gerard: Singular value dynamics and nonlinear Fourier transform for Hankel operators on the circle
12:30 - 14:00 Lunch break
14:00 - 15:00 Monica Visan: tba
15:00 - 16:00 Yvan Martel: Dynamics of the critical generalized KdV equation
16:00 - 16:30 Tea and cake

Wednesday, July 16

9:00 - 10:00 Elias Stein: Singular integrals: the product theory and its outgrowth
10:00 - 11:00 Po Lam Yung: A new twist on the Carleson operator
11:00 - 11:30 Coffee break
11:30 - 12:30 James Wright: Maximal functions along lines and along curves
12:30 - 14:00 Lunch break
*14:00 - 15:00 Trevor Wooley: Efficient congruencing and a Diophantine inequality of Bourgain and Demeter
*15:00 - 16:00 Ben Green: Bob Hough's solution of Erdős's covering congruences conjecture
*16:00 - 17:00 Coffee break
*17:00 - 18:00 Michael Christ: A Sharpened Hausdorff-Young Inequality

* The talks on Wednesday and Thursday afternoon are joint sessions with the Workshop "Analytic Number Theory" and take place in the big lecture hall (Großer Hörsaal), Wegelerstr. 10.

Friday, July 18

9:00 - 10:00 Vedran Sohinger: The Gross-Pitaevskii hierarchy on the three-dimensional torus
10:00 - 11:00 Ignacio Uriarte-Tuero: Two weight norm inequalities for singular and fractional integral operators in RN
11:00 - 11:30 Coffee break
11:30 - 12:30 Sergey Tikhonov: Two weight inequalities for Fourier transforms
12:30 - 14:00 Lunch break
14:00 - 15:00 Birgit Schörkhuber: Self-similar blow up in nonlinear evolution equations
15:00 - 16:00 Thomas Duyckaerts: Profiles for bounded solutions of dispersive equations
16:00 - 16:30 Tea and cake

Abstracts

(Underlined titles can be clicked for the video recording)

One of the most fundamental facts about the Fourier transform is the Hausdorff-Young inequality, which states that for any locally compact Abelian group, the Fourier transform maps Lp boundedly to Lq where the two exponents are conjugate and p ∈ [1,2]. For Euclidean space, the optimal constant in this inequality was found Babenko for q an even integer, and by Beckner for general exponents. Lieb showed that all extremizers are Gaussian functions. This is a uniqueness theorem; these Gaussians form the orbit of a single function under the group of symmetries of the inequality.

We establish a stabler form of uniqueness for 1 < p < 2:

(i) If a function f nearly achieves the optimal constant in the inequality, then f must be close in norm to a Gaussian.

(ii) There is a quantitative bound involving the square of the distance to the nearest Gaussian.

The qualitative form (i) can be equivalently formulated as a precompactness theorem in the style of the calculus of variations. Form (ii) is a strengthening of the inequality.

Ingredients taken from additive combinatorics are at the heart of the analysis. Arithmetic progressions, of arbitrarily high rank, play an important part.

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Consider the following extremal problem: given a real-valued function f, how can one construct an entire function of prescribed exponential type in a way to minimize the L1(ℝ)-distance to f? This problem was first considered by A. Beurling in the 1930s and by A. Selberg in the 1970s (with f being the signum function and the characteristic function of an interval, respectively).

This talk will be a survey on the recent developments on this so-called Beurling-Selberg extremal problem and its applications to the theory of the Riemann zeta-function. These applications include, for instance, improvements on the (conditional) bounds for the modulus and argument of zeta on the critical line and improvements on the (conditional) bounds for the pair correlation of its zeros.

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We explain how a certain decoupling theorem from Fourier analysis finds sharp applications in PDEs, incidence geometry and analytic number theory. This is joint work with Jean Bourgain.

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In this lecture we will discuss the focusing wave equation on 1 + 5 dimensions with radial initial data as well as the one equivariant wave maps equation in 1 + 3 dimensions. In both cases this problem in H3/2 × H1/2 critical, or energy - supercritical.

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I will discuss special Hamiltonian dynamics on the Hardy space of the circle, for which singular values are conservation laws. I will explain how to define a nonlinear Fourier transform for this evolution. As an application, I will show that the evolution is almost periodic on the Sobolev space H1/2 and admits generic unbounded trajectories on Hs for every s > 1/2.

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In the prolongation of Z. Hani's talk in the previous workshop, I will discuss the equation obtained from 2D NLS set in the torus in the big box limit, that is when the size of the torus goes to infinity. In particular, I will present new results on the structure of this equation, whose nonlinearity is "diagonalized" by special Hermite functions. Adding an appropriate random force, the Kolmogorov-Zakharov, also called kinetic wave equation, can be derived via the procedure introduced by Lanford for the Boltzmann equation. This relies on an a priori estimate assumption which we hope to remove. These results were obtained in a series of works with Z. Hani, E. Faou, L. Thomann, and I. Gallagher.

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One of Paul Erdős's very favourite questions (possibly his favourite of all) was the following: Let M0 be arbitrary. Can you cover ℤ with finitely many congruence conditions a (mod m), m > M0, at most one for each m? In 2013, Bob Hough showed that the answer is no if M0 is sufficiently large. That is, there is some constant C such that in any covering of ℤ by finitely many congruences to distinct moduli m, at least one of the m's must be at most C. I will present his proof.

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A reflectionless measure for an s-dimensional singular integral operator T acting in ℝd (with s ∈ (0,d)) is, roughly speaking, a measure μ for which T(μ) is constant (in a weak sense) on the support of the measure. We shall describe the relationship between certain problems concerning the geometric properties of measures with bounded singular integral operator, and the classification of reflectionless measures. This is joint work with Fedor Nazarov.

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We will discuss an ongoing program with Jean Bourgain to study local-global phenomena in orbits that are "thin". Consequences include applications to numerical integration, pseudorandom sequences, and Diophantine geometry.

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It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer m there should be infinitely many sets of m primes, with each set contained in an interval of size roughly m\log{m}. Although proving these conjectures seems to be beyond our current techniques, recent progress has enabled us to obtain some partial results. We will introduce a refinement of the 'GPY sieve method' for studying these problems. This refinement will allow us to show (amongst other things) that

for any integer m, and so there are infinitely many bounded length intervals containing m primes. We also discuss some extensions of this result.

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Tadahiro Oh: Invariant Gibbs measures for the defocusing NLS on the real line

In 1994, Bourgain constructed invariant Gibbs measures for NLS on the circle. Then, in 2000, he considered the limit of these invariant statistics, by taking larger and lager periods, and constructed unique solutions for the defocusing (sub-)cubic NLS on the real line. His result, however, focuses on the construction of solutions and does not discuss the limiting Gibbs measures on the real line or their invariance.

In this talk, we construct Gibbs measures for the defocusing NLS on the real line as a stationary diffusion process in x. Then, we show that these Gibbs measures are invariant for the defocusing (sub-)quintic NLS on the real line. We also discuss the limit Gibbs measures for the Dirichlet boundary value problem on the real line as well as the half line, allowing us to construct new rough solutions in these settings.

This is a joint work with Jeremy Quastel (University of Toronto) and Philippe Sosoe (Harvard University).

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Oana Pocovnicu: A modulated two-soliton with transient turbulent regime for a focusing cubic nonlinear half-wave equation on the real line

In this talk we discuss work in progress regarding a nonlocal focusing cubic half-wave equation on the real line. Evolution problems with nonlocal dispersion naturally arise in physical settings which include models for weak turbulence, continuum limits of lattice systems, and gravitational collapse. The goal of the present work is to construct an asymptotic global-in-time modulated two-soliton solution of small mass, which exhibits the following two regimes: (i) a turbulent regime characterized by an explicit growth of high Sobolev norms on a finite time interval, followed by (ii) a stabilized regime in which the high Sobolev norms remain stationary large forever in time. This talk is based on joint work with P. Gerard (Orsay, France), E. Lenzmann (Basel, Switzerland), and P. Raphael (Nice, France).

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We describe a notion of lacunarity in all dimensions, and use it to characterize sets of directions that give rise to bounded directional maximal operators on Lebesgue spaces. This is joint work with Edward Kroc.

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Birgit Schörkhuber: Self-similar blow up in nonlinear evolution equations

For many different types of PDEs singularity formation via self-similar solutions is known to occur. In this lecture I discuss the stability analysis of self-similar blow up profiles focusing on energy supercritical wave equations. This is joint work with Roland Donninger (EPF Lausanne).

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Vedran Sohinger: The Gross-Pitaevskii hierarchy on the three-dimensional torus

In this talk, we will study the Gross-Pitaevskii hierarchy. This is an infinite system of linear partial differential equations which occurs in the derivation of the nonlinear Schrodinger equation from the dynamics of N-body Bose systems. We will study this hierarchy on the three-dimensional torus.

The first part of the talk will be denoted to the analysis of the uniqueness of solutions. Our work builds on the previous study of this problem on R3 by Erdos, Schlein, and Yau, as well as by Klainerman and Machedon, and the study of this problem on T2 by Kirkpatrick, Schlein, and Staffilani. The first uniqueness result that we prove is a conditional uniqueness resolt for a class of density matrices of regularity strictly greater than 1. The proof of this fact is based on a spacetime bound, which we prove for a sharp range of regularity exponents. This result is obtained in a joint work with Philip Gressman and Gigliola Staffilani. The second uniqueness result that we prove is in a different class, in which the level of regularity is 1. The latter approach is based on the application of the Quantum de Finetti theorem, which was recently applied in the non-periodic setting by T. Chen, Hainzl, Pavlovic, and Seiringer. The latter uniqueness result allows us to obtain the last step in the derivation of the defocusing cubic NLS on T3 from the dynamics of N-body Bose systems, following the program of Elgart, Erdos, Schlein, and Yau.

In the second part of the talk, we will apply randomization techniques in order to study randomized forms of the Gross-Pitaevskii hierarchy at low regularities. This was motivated by the use of randomization techniques in the study of nonlinear dispersive equations starting with the work of Bourgain, which was motivated by the work of Lebowitz, Rose, and Speer and Zhidkov. In the randomized context, we can extend the range of regularity exponents in the main spacetime estimate. Using these ideas, we study randomized Duhamel expansions and we show that they converge to zero. This result is obtained in a joint work with Gigliola Staffilani. Finally, we will construct local-in-time solutions to the randomized Gross-Pitaevskii hierarchies. This is achieved by using the truncation argument first developed by T. Chen and Pavlovic in the deterministic setting. As a result, we are able to obtain local-in-time solutions evolving from low regularity initial data. These solutions belong to a space which contains a random component.

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One of the basic questions in analytic number theory is to understand how the prime numbers are distributed in arithmetic progressions; this information can be combined with sieve-theoretic tools to obtain results such as the recent establishment of an infinite sequence of bounded gaps between the prime numbers. For progressions of small modulus, one can obtain satisfactory results using the theory of Dirichlet L-functions, but for progressions of large modulus, even the generalised Riemann hypothesis is insufficient to obtain useful distributional results for all progressions. However, a celebrated and very useful theorem of Bombieri and Vinogradov unconditionally gives equidistribution of arithmetic progressions _on the average_, as long as the spacing of the progression is less than the square root of the magnitude of the entries.

It has been a major challenge to break this "square root barrier" and obtain stronger equidistribution estimates on the primes.  Limited results in this direction were initially obtained by Bombieri, Fouvry, Friedlander, and Iwaniec, but last year there was a significant advance by Yitang Zhang, who obtained a robust family of such estimates, by combining the dispersion method of Linnik with known estimates on exponential sums. These estimates have since been strengthened, with somewhat simplified proofs, by the online collaborative Polymath project, and have been used to improve the bounds on gaps between primes.  In this talk, we will survey these equidistribution estimates, and give some indication of their proofs.

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Sergey Tikhonov: Two weight inequalities for Fourier transforms

We discuss recent results on weighted Fourier inequalities, including Pitt type and two sided Boas type inequalities.

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Ignacio Uriarte-Tuero: Two weight norm inequalities for singular and fractional integral operators in R^N

I will report on recent advances on the topic, related to proofs of T1 type theorems in the two weight setting for Calderón-Zygmund singular and fractional integral operators, with side conditions, and related counterexamples. Joint work with Eric Sawyer and Chun-Yen Shen.

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Armen Vagharshakyan: Lower bounds for L_1 discrepancy

We find the best constant of the leading term of the asymptotical lower bound for the L_1 norm of two-dimensional discrepancy that could be obtained by K.Roth's "test function" method among a large class of test functions.

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As a consequence of recent work concerning the proof of the \ell^2 decoupling conjecture, Bourgain and Demeter show that for each fixed

k\geq 2 and C>0, the Diophantine system

\begin{align*}
 |n_1^k+n_2^k+n_3^k-n_4^k-n_5^k-n_6^k| & \le CN^{k-2} \\
 n_1+n_2+n_3-n_4-n_5-n_6 & =0
 \end{align*}

has O(N^{3+\epsilon}) integral solutions with n_i\le N. We explore the consequences of the efficient congruencing method (from Vinogradov's mean value theorem) for this problem and its generalisations.

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We survey results (some old, some new) for maximal functions associated to lines and curves.

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Must the Fourier series of an L^2 function converge pointwise almost everywhere? In the 1960's, Carleson answered this question in the affirmative, by studying a particular maximal oscillatory singular integral operator, which has since become known as a Carleson operator. The analysis of this operator has led to some spectacular development in the past 40 years, culminating in the development of what is now known as time-frequency analysis. In this talk, we will introduce the Carleson operator and survey several of its generalizations. We will then describe some recent joint work with Lillian Pierce that introduces curved structure to the setting of Carleson operators

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