Abstract: In this talk, we will discuss uniqueness properties of solutions to the linear generalized fourth-order Schrödinger equations. We show that a solution with fast enough decay in certain Sobolev spaces at two different times has to be trivial. This is a joint work with Zachary Lee. 

Abstract: This talk is about various families of limit functionals and weak type (quasi)-norms which represent the Lp norm of the gradient. This extends and unifies work by Nguyen and by Brezis, Van Schaftingen and Yung. We discuss some interesting counterexamples and open problems.

Joint work with Haïm Brezis, Jean Van Schaftingen and Po Lam Yung.

Abstract: We will present forthcoming work which proves a sharp L^7 square function estimate for the moment curve in R^3 using ideas from decoupling theory. Consider a function f with Fourier support in a small neighborhood of the moment curve. Partition the neighborhood into box-like subsets and form a square function in the Fourier projections of f onto these box-like regions. Bounding f in L^p by the square function in L^p is an important way to quantify the cancellation that f has from its specialized Fourier support. 

Abstract: It is well known that planar Besicovitch sets – sets containing a unit line segment in every direction – have Hausdorff dimension 2. In a joint work with Iqra Altaf and Marianna Csörnyei we consider Besicovitch sets of Cantor graphs in the plane– sets containing a rotated (and translated) copy of a fixed Cantor graph (its line segments of course removed) in every direction, and prove lower bounds for their Hausdorff dimension.

Abstract: Oscillatory integrals are a basic object of study in Harmonic Analysis and underpin many important problems. The goal of this talk will be to reflect on some elementary yet important observations on the role of structure in estimating oscillatory integrals, and to discuss some recent works that capture this philosophy.

Abstract:  In the 1970's Erdos asked several questions about what kind of infinite arithmetic structures can be found in every set of natural numbers with positive density. In recent joint work with Bryna Kra, Joel Moreira, and Donald Robertson we use ergodic methods to resolve some of these long-standing problems. This talk will provide an overview of our results and describe some of the dynamical structures that are used to prove them.

Slides

Abstract: Previously the L^{\infty} and Holder estimates for complex Monge-Ampere were obtained using pluri-potential theory. We consider a version of the parabolic complex Monge-Ampere on compact Kähler manifolds using PDE approach, generalizing the recent work by Guo, Phong and Tong in the elliptic case.

We prove that all bounded subsets of Q_p^n containing a line segment of unit length in every direction have Hausdorff and Minkowski dimension n. This is the analogue of the classical Kakeya conjecture with R replaced by Q_p.

Recording 

Fourier analysis has proved a fundamental tool in analytic and combinatorial number theory, usually in the guise of the Hardy-Littlewood circle method. When applicable, this method allows one to asymptotically estimate the number of solutions to a given Diophantine equation with variables constrained to a given finite set of integers. I will discuss recent work, obtained jointly with Sarah Peluse, which adapts the circle method to count the configuration x, x+y, x+y^2 in a quantitatively effective manner.

I will talk about the Brunn-Minkowski inequality, Ehrhard’s inequality, and some of their conjectured strengthenings — the Log-Brunn-Minkowski conjecture, the Dimensional Brunn-Minkowski conjecture, the “symmetric Ehrhard” conjecture, the B-conjecture, and all the various relations between them. In addition to mentioning many open problems, I will discuss the state of the art in this area, and explain some of my results in it. Finally, I will talk a bit about a new conjectured strengthening of the Brascamp-Leib inequality, its potential (significant) implications, and partial progress towards it.

We discuss the construction of sets with large Fourier dimension avoiding certain families of linear and non-linear patterns. In other words, we construct sets which do not contain a certain subset of points arranged in a particular configuration, while also supporting probability measures whose Fourier transforms exhibit polynomial decay. Our analysis involves a discussion of the concentration of measure phenomenon in probability, and some oscillatory integral estimates. As particular applications of these methods, we will construct large sets of $\mathbf{T}^d$ not containing points $x_1,\dots,x_n$ solving linear equations of the form $a_1x_1 + ... a_n x_n = b$, and large subsets of planar curves with non-vanishing curvature which do not contain three points forming an isosceles triangle.

Abstract: We report on recent progress concerning two distinct problems in sharp restriction theory to the unit sphere.

Firstly, the classical estimate of Agmon-Hörmander for the adjoint restriction operator to the sphere is in general not saturated by constants. We describe the surprising intermittent behaviour exhibited by the optimal constant and the space of maximizers, both for the inequality itself and for a stable form thereof.

Secondly, the Stein-Tomas inequality on the sphere is rigid in the following rather strong sense: constants continue to maximize the weighted inequality as long as the perturbation is sufficiently small and regular, in a precise sense to be discussed. We present several examples highlighting why such assumptions are natural, and describe some consequences to the (mostly unexplored) higher dimensional setting.

This talk is based on joint work with E. Carneiro and G. Negro.

Abstract: In this talk, I will discuss my recent work on top-degree global solvability for $\partial_b$ operator defined on a generic sub-manifold of the complex space as well as for the differential complex associated with a locally integrable structure over a smooth manifold. The main assumptions are that the locally integrable structure is hypocomplex and that the differential complex is locally solvable in degree one. One of the main tools is an adaptation of a sheaf theoretical argument due to Ramis-Ruget-Verdier. This is a joint work with Prof. Paulo Domingos Cordaro.

In recent work we construct a measure that is p-adic and q-adic doubling for any coprime p and q, yet not doubling overall.  The proof involves an intricate interplay of number theory, geometry and analysis, and here we give an overview of some of the key features. 

Let F be a finite subset of an additive group G, and let E be a subset of G.  A (translational) tiling of E by F is a partition of E into disjoint translates a+F, a∈ A of F.  The periodic tiling conjecture asserts that if a periodic subset E of G can be tiled by F, then it can in fact be tiled periodically; among other things, this implies that the question of whether E is tileable by F at all is logically (or algorithmically) decidable.  This conjecture was established in the two-dimensional case G = Z^2 by Bhattacharya by ergodic theory methods; we present a new and more quantitative proof of this fact, based on a new structural theorem for translational tilings.  On the other hand, we show that for higher dimensional groups the periodic tiling conjecture can fail if one uses two tiles F_1,F_2 instead of one; indeed, the tiling problem can now become undecidable.  This is established by developing a "tiling language" that can encode arbitrary Turing machines.

This is joint work with Rachel Greenfeld. 

In this talk we revisit a result of Muckenhoupt and Wheeden, which gives a weighted version of the classical John-Nirenberg Theorem (specifically for Ap weights). We will discuss a modern proof of this result, using the recent machinery of sparse operators.

We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is joint work with Marianna Csörnyei.

Given a fractal set E on the plane and a set F of directions, can we find one direction \theta\in F such that the orthogonal projection \Pi_{\theta} E is large?

We will survey some classical and modern projection theorems and discuss their applications. 

A natural 3-dimensional analogue of Bourgain’s circular maximal function theorem in the plane is the study of the sharp L^p bounds in R^3 for the maximal function associated with averages over dilates of the helix (or, more generally, of any curve with non-vanishing curvature and torsion). In this talk, we present a sharp result, which establishes that L^p bounds hold if and only if p>3. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.

Abstract 

We will discuss some geometric similarities between the sequence {\log n}_{n=N+1}^{N+N^{1/2}} (and sequences with similar convexity properties) and the parabola from a decoupling point of view. Based on those observations we present decoupling inequalities for those sequences. The sequence {\log n}_{n=N+1}^{2N} is closely connected to a conjecture of Montgomery on Dirichlet polynomials but we see some difficulties in studying the sequence {\log n\}_{n=N+1}^{N+N^{\alpha}} for \alpha > 1/2. This is joint work with Larry Guth and Dominique Maldague.

Given a curve C with nonvanishing geodesic curvature in the unit sphere of R^3, it is an open problem whether the Hausdorff dimension of an arbitrary set A is almost surely preserved under projection onto the orthogonal complements of vectors in C. In this talk I will outline some recent progress on this problem, which makes use of some Fourier restriction tools such as decoupling and wave packet decompositions.  Toward the end of the talk I will mention a couple of open problems suggested by the approach.  

In this survey talk we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic

theory and probability.  We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point

of view of oscillation estimates in both the single and several parameter  settings. We establish a number of new oscillation inequalities

and give new proofs for known results with elementary arguments.

The 1d cubic nonlinear Schrödinger equation (NLS) and the modified Korteweg-de Vries equation (mKdV) are two of the most intensively studied nonlinear dispersive equations. Not only are they important physical models, arising, for example, from the study of fluid dynamics and nonlinear optics, but they also have a rich mathematical structure: they are both members of the ZS-AKNS hierarchy of integrable equations. In this talk, we discuss an optimal well-posedness result for the cubic NLS and mKdV on the line. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, which in turn rests on the discovery of a one-parameter family of microscopic conservation laws. This is joint work with Rowan Killip and Monica Vișan. 

The restriction conjecture is a major open problem in harmonic analysis concerning interactions between the Fourier transform and curved surfaces.  While the case of elliptic, or positively curved, surfaces has been studied most, this talk will describe some recent results from non-elliptic settings.  In particular, global restriction estimates for hyperboloids will be presented.

Slides  Recording 

The Bergman kernel function of a domain D in the complex plane is the reproducing integral kernel for the Hilbert space of square integrable holomorphic functions on D.  It is easily shown that the (on-diagonal) Bergman kernel is bounded below by the reciprocal of the volume of the domain D.  In this talk, I geometrically characterize the domains whose Bergman kernels achieve the lower bound.

Recording 

Abstract  Recording 

It is known that the oscillation of the Green function for the Laplacian in a domain is related to the flatness of the boundary of the domain. In a joint work with Guy David and Svitlana Mayboroda, we consider the Green function for a second-order elliptic operator in the half-space. We show that if the coefficients satisfy a quadratic Carleson condition, then the Green function is almost affine, in the sense that the normalized difference between the Green function with a sufficiently far away pole and a suitable affine function at every scale satisfies a Carleson measure estimate. Our results are optimal, in the sense that the class of the operators considered cannot be improved.

This work is motivated mainly by finding PDE characterizations of uniformly rectifiable sets with higher co-dimension, yet our result is new of this kind in the co-dimension one setting as well.

Recording 

In this talk, I'll survey recent progress on problems in additive combinatorics, harmonic analysis, and ergodic theory related to Bergelson and Leibman's polynomial generalization of Szemerédi's theorem.

Recording 

A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.

Recording 

We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtain some new cases of the fractal uncertainty principle in odd dimensions. This is joint work with Terence Tao. 

Recording

The investigation of polynomial extensions of the Roth’s theorem was started by Bourgain and Chang, and has seen a lot of recent advancements. The most striking of these are a series of results of Peluse and Prendiville which prove quantitative versions of the polynomial Roth and Szemerédi theorems in the integer setting. There is yet no corresponding result for corners, the two dimensional setting for polynomial Roth's Theorem, where one considers progressions of the form (x, y), (x+t, y), (x, y+t^2) in  [1 ,..., N]^2, for example. 

We will talk about a recent result on the corners version of the result of Bourgain and Chang, showing an effective bound for a three term polynomial Roth’s theorem in the finite field setting. This is based on joint work with Michael Lacey and Fan Yang.

Recording

The classical Strichartz estimates show that a solution to the linear Schrodinger equation on Euclidean space is in certain Lebesgue spaces globally in time provided the initial data is in L^2. On compact manifolds one can no longer have global control, and some loss of derivatives is necessary in interesting cases (meaning the initial data needs to be in a Sobolev space rather than L^2). On non-compact manifolds it is a challenging problem to understand when one can have good space-time estimates with no loss of derivatives. 

In this talk we discuss an endpoint Strichartz-type estimate for the linear Schrodinger equation on the infinite cylinder (or, equivalently, with one periodic component and one Euclidean component). Our estimate is sharp, scale-invariant, and requires only L^2 data. This contrasts the purely periodic case where some loss of derivatives is necessary at the endpoint, as originally observed by Bourgain.

Joint work with M. Christ and B. Pausader.  

Recording available at: https://drive.google.com/file/d/1L2FMsxCnbybtnR6vmCiLUi9Ss1y5h81_/view?usp=sharing

We discuss possible ways to define a notion of 'uniform completeness' as a dual notion for uniform minimality. We contrast these definitions with a well known density theorem of Landau, and a quantified version of this theorem due to Olevskii and Ulanovskii. We show that analogs of these results can be obtained for an appropriate notion of 'uniform completeness'.

Abstract: Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming the number $d$ of variables is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a ``stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

Recording 

Abstract: Given a first order partial differential equation system, Nirenberg's complex Frobenius theorem gives necessary and sufficient conditions on whether we can rewrite this system as the derivatives along some real and complex coordinate vector fields. In the talk I will introduce the complex Frobenius structure and talk about what is the optimal regularity estimate of the coordinate chart and coordinate vector fields in the non-smooth setting.

It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered "standard" tiling complement. It is not known whether this behavior persists for all tilings with no restrictions on the number of prime factors of M.

In an ongoing collaboration with Izabella Laba, we proved that this is true when M=(pqr)^2. In my talk I will discuss this problem and introduce the main ingredients in the proof.

In this talk, we discuss the construction and invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree-nonlinearity.

In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the mutual singularity of the Gibbs measure and the so-called Gaussian free field. In contrast, the Gibbs measure for one or two-dimensional wave equations is absolutely continuous with respect to the Gaussian free field.

In the second part of the talk, we discuss the probabilistic well-posedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. At the moment, this is the only theorem proving the invariance of any singular Gibbs measure under a dispersive equation.

Recording 

Abstract: There is a class of geometric problems in harmonic analysis associated with some curved manifolds such as the sphere or the paraboloid. In the study of these problems, relevant geometric maximal functions play a central role. In this talk, we consider maximal averaging operators along line segments oriented in a set of directions and their singular integral counterparts. How do operator norms of these maximal functions depend on the number and the distribution of directions? I will discuss some results in this direction and a divide-and-conquer approach for L² estimates. 

Abstract: We exhibit the necessary range for which functions in the Sobolev spaces Lˢₚ can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle-Lemarié wavelets. We also consider the natural extensions to Triebel-Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.

Abstract available here 

Abstract available here

Abstract: Restriction problems, which are introduced by Stein in 1970s, play key model problems in harmonic analysis. In the first half of the talk, we will discuss restriction estimates for hypersurfaces. In the second half of the talk, we will talk about restriction estimates for surfaces with codimension larger than one. 

The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we use are polynomial partitioning, and the Bourgain Demeter l^2 decoupling theorem. 

Abstract: Whitney's Extension Problem asks the following: Given a compact set E⊂ ℝⁿ and a function f:E→ ℝ, how can we tell if there exists F∈ Cᵐ(ℝⁿ) such that f is the restriction of F to E? The classical Whitney Extension theorem tells us that, given potential Taylor polynomials Pˣ at each x∈E, there is such an extension F if and only if the Pˣ's are compatible under Taylor's theorem. However, this leaves open the question of how to tell solely from f. A 2006 paper of Charles Fefferman answers this question. We explain some of the concepts of that paper, as well as recent work of the speaker, joint with Fushuai Jiang and Garving K. Luli, which establishes the analogous result when f≥0 and we require F≥0. 

The celebrated decoupling theorem of Bourgain and Demeter allows for a decomposition in the Lᵖ norm of functions Fourier supported near curved hypersurfaces M ⊂ ℝⁿ. In this project, we find that the condition of non-vanishing principal curvatures may be weakened. When M ⊂ ℝ³, we may allow one principal curvature at a time to vanish, and it is assumed additionally that M is foliated by a canonical family of orthogonal curves having nonzero curvature at every point. We find that ℓᵖ decoupling over nearly flat subsets of M holds within this context.

Abstract:

Multiple results in harmonic analysis involving integrals of functions over curves (such as restriction theorems, convolution estimates, maximal function estimates or decoupling estimates) depend strongly on the non-vanishing of the torsion of the associated curve. Over the past years there has been considerable interest in extending these results to a degenerate case where the torsion vanishes at a finite number of points by using the affine arc-length as an alternative integration measure. As a model case, multiple results have been proven in which the coordinate functions of the curve are polynomials. In this case one expects the bounds of the operators to depend only on the degree of the polynomial. 

 In this talk I will introduce and motivate the concept of affine arclength measure, provide a new decomposition theorem for polynomial curves over characteristic zero local fields, and show some applications to uniformity results in harmonic analysis.

Slides from the talk

In this talk I will present the new notion of computable harmonic approximation, and show that for an arbitrary domain, computability of the harmonic measure for a single point implies its computability for any point. Nevertheless, different points may require different algorithms, which gives rise to surprisingly natural examples of continuous functions whose values can be computed at any point but cannot be computed using same algorithm on their entire domain. I will present counter examples supporting this and study the conditions under which the harmonic measure is computable uniformly, that is by a single algorithm, and characterize them for regular domains with a computable boundary.

This talk is based on a joint work with I. Binder, C. Rojas, and M. Yampolsky.

Slides