Speakers

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Talks for discussion session on March 4

Marina Iliopoulou (University of Kent), A discrete Kakeya-type inequality

Abstract: The Kakeya conjectures of harmonic analysis claim that congruent tubes that point in different directions rarely meet. In this talk we discuss the resolution of an analogous problem in a discrete setting (where the tubes are replaced by lines), and provide some structural information on quasi-extremal configurations. This is joint work with A. Carbery.

Joshua Zahl (UBC), Discretized algebraic geometry estimates for Kakeya and restriction

Abstract: I will discuss the problem of discretizing classical results from the theory of ruled surfaces, and their application to the Kakeya and restriction problem.

Jennifer Duncan (University of Birmingham), Nonlinear Brascamp-Lieb inequalities

Abstract: Inequalities that carry some sort of transversality hypothesis are ubiquitous in multilinear harmonic analysis, among which the Brascamp-Lieb inequalities are fundamental examples. In this talk we discuss their theory in both linear and nonlinear contexts, and how they relate to other subjects, in particular Fourier restriction.

Taryn Flock (Macalester College), The nonlinear Brascamp-Lieb inequality

Abstract: Brascamp-Lieb inequality generalizes many inequalities in analysis, including the Hölder, Loomis-Whitney, and Young’s convolution inequalities.The sharp constants for such inequalities have a long history and have only been determined in a few cases. The focus of the talk will be a nonlinear generalisation of the classical Brascamp–Lieb inequality. A first step in this analysis is understanding the regularity of the sharp constant in the Brascamp-Lieb constant. This work has other applications including a mutlilinear Kakeya-type inequality which is used in Bourgain, Demeter’s, Guth’s proof of the Vinogradov mean value theorem. The Brascamp-Leib constant also makes a surprise appearance in operator scaling, a generalization of the well-known matrix scaling algorithm from computer science. (Includes joint work with Jon Bennett, Neal Bez, Stefan Buschenhenke, Michael Cowling, and Sanghyuk Lee)

José Madrid (UCLA), Discrete polynomial averages and decoupling inequalities

Abstract: In this talk we will discuss some improving inequalities for discrete averaging operators along polynomials, and their connections to decoupling inequalities on the moment curve.

Hong Wang (IAS), Improved decoupling for parabola

David Beltran (UW Madison), Variation bounds for spherical averages

Abstract: Variational estimates are refinements of maximal function estimates, in which the ℓ^∞ norm is replaced by a larger V ^r norm, 1 ≤ r ≤∞. In 2008, Jones, Seeger and Wright proved that the r-variation operator associated to the spherical averages {f *σ_t}_t>0 is bounded on L^p(ℝ^d) if d∕(d- 1) < p ≤ 2d and r > 2 or p > 2d and r > p∕d, and this is sharp except for the endpoint case r = p∕d, which remains open. In this talk I will present L^p(ℝ^d) → L^q(ℝ^d) bounds for the local r-variation operator, that is, the associated with {f * σ_t}_1≤t≤2. The bounds are sharp up to endpoints (except in dimension 3), and some positive results also hold in some endpoints cases. In particular, it can be established the interesting endpoint bound L^q∕d(ℝ^d) → L^q(ℝ^d) for r = q∕d, q > (d² + 1)∕(d - 1) if d ≥ 3. Our results imply associated sparse domination and weighted inequalities. This is joint work with Richard Oberlin, Luz Roncal, Andreas Seeger and Betsy Stovall.

Kevin Hughes (University of Bristol), Discrete restriction estimates for (x,x³)

Abstract: Joint work with Trevor Wooley

Talks for discussion session on March 5

Ben Bruce (UW Madison), Restriction to hyperboloids

Abstract: This talk discusses new global Fourier restriction estimates for hyperboloids equipped with the Lorentz-invariant measure. A conditional result that converts local restriction inequalities into global ones is also presented. These results were first obtained for the one-sheeted hyperboloid in three dimensions, in joint work with Diogo Oliveira e Silva and Betsy Stovall.

Stefan Buschenhenke (Kiel University), On Fourier restriction for hyperbolic surfaces

Abstract: Restriction estimates for hyperbolic 2-surfaces were for a long time almost exclusively for the saddle z=xy, being accessible for anisotropic scalings. Unlike for for surfaces with positive curvature, much less was known for perturbations. We present a result for general perturbations adapting the polynomial partitioning method. This is the most recent one in a series of joint work with Detlef Müller and Ana Vargas.

Jonathan Hickman (The University of Edinburgh), Restriction estimates for hyperbolic paraboloids

Abstract: In this talk I describe some recent joint work with Marina Iliopoulou which provides new restriction/estimates for hyperbolic paraboloids. I give a brief indication of some features of the problem which distinguish it from the elliptic case.

Alex Barron (UI Urbana-Champaign), On weighted Fourier restriction estimates for the hyperbolic paraboloid

Abstract: We discuss certain weighted L² estimates for the Fourier extension operator associated to the hyperbolic paraboloid. These bounds extend estimates of Du and Zhang to hyperbolic setting where the surface is allowed to have principal curvatures of mixed signs. When the signature of the surface is as small as possible our estimates are sharp in all dimensions. Joint with M.B. Erdogan and T.L.J. Harris.

Jeremy Schwend (University of Georgia), Surfaces with Vanishing Curvature–The Convolution Case

Abstract: Analyzing the effects of manifolds having vanishing curvature has been an active area of harmonic analysis, with significance to many common operators, like the convolution operator and Fourier restriction operator. In this talk, we will focus on the convolution operator, presenting a method for finding the precise range of (p,q) for which local averages along graphs of a class of two-variable polynomials in ℝ³ are of restricted weak type (p,q), given the hypersurfaces have Euclidean surface measure. We derive these results using non-oscillatory, geometric methods, for a model class of polynomials bearing a strong connection to the general real-analytic case. Many of the ideas shown here can also be applied to other operators, with some having already been applied to the Fourier restriction operator.

Luz Roncal (BCAM), Directional square functions

Abstract: We present a general approach, based on a directional embedding theorem for Carleson sequences, to study time-frequency model square functions associated to conical or directional Fourier multipliers. The estimates obtained for these square functions are applied to obtain sharp or quantified bounds for directional Rubio de Francia type square functions. In particular, a precise logarithmic bound for the polygon multiplier is shown, improving previous results. This is joint work with Natalia Accomazzo, Francesco Di Plinio, Paul Hagelstein, and Ioannis Parissis.

Constantin Bilz (University of Birmingham), Large sets without Fourier restriction theorems

Abstract: We construct a function that lies in L^p(ℝ^d) for every p > 1 and whose Fourier transform has no Lebesgue points in a compact set of full Hausdorff dimension. We discuss some recent results in maximal restriction theory and fractal restriction theory and combine them with our construction to show that Hausdorff dimension and Fourier restriction theorems are mostly independent properties of sets.

Talks for discussion session on March 12

Terence Harris (Cornell University), Hausdorff dimension of projections and Fourier restriction

Abstract: Let γ : [0,1] → S² be a curve in the unit sphere of ℝ³ satisfying the non-degeneracy condition det(γ,γ′,γ′′). Given a measurable set A ⊆ R³ of Hausdorff dimension at most 2, it is conjectured that the orthogonal projection of A onto the 2-dimensional plane orthogonal to γ(t) has the same Hausdorff dimension as A, for almost every t ∈ [0,1]. In this talk, I will present some progress made towards this problem via a Fourier analytic approach. Toward the end of the talk I will discuss some possibilities for generalisation and further improvement.

Zane Li (Indiana University), Decoupling for fractal subsets of the parabola

Abstract: We study the problem of decoupling a fractal set on the parabola. The analogous number theory problem was studied by Kirsti Biggs using efficient congruencing in her PhD thesis. We reduce the problem of decoupling a fractal set on the parabola to a fractal set on the line. Our result generalizes and improves upon the Bourgain-Demeter decoupling theorem for the parabola due to sparsity of the Fourier supports we consider. This is joint work with Alan Chang, Jaume de Dios Pont, Rachel Greenfeld, Asgar Jamneshan, and José Madrid.

Jaume de Dios (UCLA), Decoupling for Cantor sets

Abstract: In this talk we discuss sharp ℓ²L²ⁿ estimates for Cantor sets. These estimates are related to the work of Biggs bounding the number of solutions to a certain type of Diophantine equations for integers contained in Ellipsephic sets, sets of numbers missing certain digits in base p. We discuss the connection between both problems, and exploit it to find computational methods to find sharp decoupling estimates. Joint work with A. Chang, R. Greenfeld, A. Jamneshan, Z.K. Li and J. Madrid.

Tongou Yang (UBC), Uniform ℓ² decoupling in ℝ² for polynomials

Changkeun Oh (UW Madison), Decoupling inequalities for quadratic forms

Abstract: In this talk, we will discuss recent work on decoupling inequalities for quadratic forms. A new ingredient of the proof is scale-independent Brascamp-Lieb inequalities by Dominique Maldague. Joint work with Shaoming Guo, Ruixiang Zhang, and Pavel Zorin-Kranich.

Laura Cladek (UCLA), Additive Energy of Regular Measures and the Fractal Uncertainty Principle in High Dimensions

Abstract: 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.

Yu Deng (USC), Strichartz estimates on generic irrational tori

Abstract: We prove Strichartz estimates on irrational tori with generic parameters, which goes beyond the theorem of Bourgain-Demeter. As an application, we also prove improved polynomial growth bounds for higher Sobolev norms of solutions to nonlinear Schrodinger equations, including the first such result in the energy critical setting. This is joint work with P. Germain, L. Guth and S. Myerson.

Talks for discussion session on March 13

Shohei Nakamura (Tokyo Metropolitan University), Maximal estimates for the Schrödinger equation with orthonormal initial data

Abstract: This talk is based on the joint work with Professors Neal Bez (Saitama University) and Sanghyuk Lee (Seoul National University). We are interested in the pointwise convergence problem for infinitely many particles (Fermion). We formulate the problem in terms of the density function and then provided a non-trivial result on ℝ. For this purpose, we also generalise the maximal in time estimate for the Schrödinger propagator and the one-dimensional endpoint Strichartz estimate with orthonormal system input. The later one provides almost sharp answer to an endpoint problem raised by Frank-Sabin.

Yakun Xi (Zhejiang University), Square function estimates and local smoothing for Fourier Integral Operators

Abstract: We discuss some recent progress on the local smoothing conjecture for FIOs. In particular, we prove a variable coefficient version of the square function estimate of Guth–Wang–Zhang, which implies the full range of sharp local smoothing estimates for 2+1 dimensional Fourier integral operators satisfying the cinematic curvature condition. As a consequence, the local smoothing conjecture for wave equations on compact Riemannian surfaces is settled. This is a joint work with Chuanwei Gao, Bochen Liu and Changxing Miao.

Danqing He (Fudan University), Maximal operators related to bilinear multipliers

Abstract: Bilinear spherical maximal functions and bilinear maximal Bochner–Riesz operators are typical bilinear maximal operators related to bilinear multipliers and also oscillatory integrals. In this talk we plan to discuss some recent progress on them.

Eyvindur Ari Palsson (Virginia Tech), Geometric averaging operators motivated by point configurations

Abstract: Two classic questions—the Erdős distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem—both focus on the distance, which is a simple two point configuration. When studying the Falconer distance problem, a geometric averaging operator, namely the spherical averaging operator, arises naturally. Questions similar to the Erdős distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will give a brief introduction to the spherical averaging operator and its direct multilinear analogues, as well as the motivating point configuration questions and then report on some novel geometric averaging operators and their mapping properties.

Shukun Wu (UI Urbana-Champaign), Maximal Bochner-Riesz operator in ℝ²

Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. In this talk, I will present an improvement to the maximal Bochner-Riesz conjecture based on a joint work with Xiaochun Li. The main method we use is the Bourgain-Demeter ℓ² decoupling theorem.

Shaozhen Xu (University of the Chinese Academy of Sciences), Decay estimates for degenerate oscillatory integral operators

Abstract: This talk serves an introduction to the subject “Decay estimates for degenerate oscillatory integral operators”. A joint work with Zuoshunhua Shi and Dunyan Yan is also included in the last section. In fact, by establishing endpoint estimates for suitable damped oscillatory integral operators, we are able to give a new proof of the sharp L^p estimates for oscillatory integral operators with real-analytic phases which have been given previously by Xiao.

Cheng Zhang (University of Rochester), Pointwise Weyl Laws for Schrödinger operators with singular potentials

Abstract: We consider the Schrödinger operators H_V = -Δ_g + V with singular potentials V on general n-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this pertubation. We prove that the pointwise Weyl law holds for potentials in the Kato class, which is the minimal assumption to ensure that H_V is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. Moreover, we show that the pointwise Weyl law with the standard sharp remainder term O(λ^n-1) holds for potentials in Lⁿ(M). This is a joint work with Xiaoqi Huang (JHU).