- Theresa (Tess) Anderson (UW Madison)
- Title: A spherical maximal function along the primes
- Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior. The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example. In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to. We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory. This is joint work with Cook, Hughes, and Kumchev.

- Xianghong Chen (UW Milwaukee)
- Title: On almost everywhere divergence of spherical harmonics expansion
- Abstract: We discuss summability of spherical harmonics expansion for integrable functions on the n-sphere. It is well known that the critical summability index is (n-1)/2 for L^1 convergence. In the case of super-critical index, Bonami and Clerc showed that almost everywhere convergence holds. In the case of sub-critical index, Meaney showed that almost everywhere divergence can happen. At the critical index, a general result of Christ and Sogge implies that weak L^1 convergence holds. In this talk we present a recent result showing that almost everywhere divergence can happen at the critical index. This is joint work with Dashan Fan and Juan Zhang.

- Laura Cladek (UCLA)
- Title: Quantitative additive energy estimates for regular sets and connections to discretized sum-product theorems
- Abstract: We prove new quantitative additive energy estimates for a large class of porous measures which include, for example, all Hausdorff measures of Ahlfors-David subsets of the real line of dimension strictly between 0 and 1. We are able to obtain improved quantitative results over existing additive energy bounds for Ahlfors-David sets by avoiding the use of inverse theorems in additive combinatorics and instead opting for a more direct approach which involves the use of concentration of measure inequalities. We discuss some connections with Bourgain's sum-product theorem.

- Xiumin Du (IAS)
- Title: A sharp Schrödinger maximal estimate in R^2
- Abstract: Joint with Guth and Li, we showed that the solution to the free Schrödinger equation converges to its initial data almost everywhere, provided that the initial data is in the Sobolev space H^s(R^2) with s>1/3. This is sharp up to the endpoint, due to a counterexample by Bourgain. This pointwise convergence problem can be approached by estimates of Schrödinger maximal functions, which have some similar flavor as the Fourier restriction estimates. The original estimate in three-dimensional space-time physical space can be reduced to an essentially two-dimensional one, via polynomial partitioning method which helps us identify some algebraic structures where the solutions are most concentrated. The reduced problem asks how to control the size of the solution on a sparse and spread-out set, and it can be solved by refined Strichartz estimates derived from l^2 decoupling theorem and induction on scales. In this talk, I'll focus on the reduced two-dimensional problem in the planar case, which is the most interesting scenario.

- Polona Durcik (CalTech)
- Title: Singular Brascamp Lieb inequalities: a special case
- Abstract: Brascamp Lieb inequalities have been studied extensively in recent years. One can consider singular variants of the Brascamp Lieb inequalities, by which we mean Lp bounds for the Brascamp Lieb integrals where one of the functions is replaced by a singular integral kernel. We discuss some special cases of such integrals which are associated with the unit cube in two and three dimensions. This is joint work in progress with Vjeko Kovac, Kristina Skreb and Christoph Thiele.

- Taryn Flock (U Mass Amherst)
- Title: The nonlinear Brascamp-Lieb inequality for simple data
- 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. Time permitting, we will discuss these connections as well. (Joint work with Jon Bennett, Neal Bez, Stefan Buschenhenke, Michael Cowling, and Sanghyuk Lee)

- Maxim Gilula (Michigan State)
- Title: Higher decay inequalities for multilinear oscillatory integrals
- Abstract: It has been known for decades that the Newton polyhedron predicts optimal decay rates for some large class of oscillatory integrals in higher dimensions, very few nontrivial concrete examples of the conditions we need to assume on phases to obtain such decay rates are known even today. One of the most well-known conditions was used by Varchenko in 1976 to prove sharp bounds for a class of oscillatory integrals. His condition is a first order nondegeneracy assumption on real-analytic phases. Another well-studied result is a decay rate for multilinear oscillatory integral operators, due to Phong-Stein-Sturm from 2001; it is amazing because it is basically uniform over polynomials of the same degree. However, the L_p spaces that form the domain of the oscillatory integral operator appear at first sight to be somewhat limited. I will discuss a result obtained with two coauthors, Philip Gressman and Lechao Xiao, that combines Varchenko's work with that of Phong-Stein-Sturm: the condition we assume on the phase is a second order nondegeneracy condition analogous to Varchenko's, and the result obtained is a sharp decay rate for the oscillatory integral operator for a range of L_p spaces on the "opposite end of the spectrum" compared to Phong-Stein-Sturm. The techniques used are quite different from the ones used in the above two results: they are completely real-analytic and don't rely on any algebraic geometry.

- Shaoming Guo (Indiana University)
- Title: Parsell-Vinogradov systems in higher dimensions
- Abstract: I will report some recent progress on counting the number of integer solutions of Parsell-Vinogradov systems in higher dimensions. Applications to Waring's problem and to the problem of counting rational linear subspaces lying on certain hyper-surfaces will be discussed. Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.

- Jonathan Hickman (University of Chicago)
- Title: Anisotropic Fourier decay and L^2 Fourier restriction for surfaces of intermediate dimension.
- Abstract: I will discuss some work with Jim Wright concerning variants of the classical Stein--Tomas restriction theorem for surfaces of intermediate dimension.

- Irina Holmes (Michigan State)
- Title: On an inequality for the dyadic square function
- Abstract: In this talk we discuss the weak (1, 1) sharp inequality for the dyadic square function. Specifically, we outline a new approach via Bellman functions, inspired by an older paper of Bollobas. The new approach involves two Bellman functions to tackle the same inequality, instead of the usual one. Many of the properties of the two Bellman functions are mirror images of one another, and their intertwined behavior yields the sharp inequality. Joint with Paata Ivanisvili and Alexander Volberg.

- Marina Iliopoulou (UC Berkeley)
- Title: Sharp estimates for Hörmander-type operators with positive-definite phase
- Abstract: In the heart of harmonic analysis lies the restriction problem: the study of Fourier transforms of functions that are defined on curved surfaces. The problem came to life in the late 60s, when Stein observed that such Fourier transforms have better behaviour than if the surfaces were flat. Soon after, Hörmander conjectured that oscillatory integral operators with more general phase functions should also demonstrate similar agreeable behaviour. Surprisingly, 20 years later Bourgain disproved Hörmander's conjecture. However, under additional assumptions on the phase function one can expect better estimates than the sharp ones by Bourgain. In this talk, we present such better estimates in the sharp range, under the assumption that the phase function is positive definite. This is joint work with Larry Guth and Jonathan Hickman.

- Jongchon Kim (IAS)
- Title: Derivative estimates on the averaged Green's function for an elliptic equation with random coefficients.
- Abstract: We consider a divergence form elliptic difference operator on the lattice Z^d, where the coefficient matrix is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques to obtain a simple representation of the averaged operator whose Green's function coincides with the averaged Green's function for this model. In this talk, I will present an improved decay estimate regarding the averaged operator, which is conjectured to be nearly optimal. As an application, we obtain (discrete) derivative estimates for the averaged Green's function which go beyond the second derivatives due to Delmotte-Deuschel. This is a joint work with Marius Lemm.

- Benjamin Krause (CalTech)
- Title: Discrete analogues in Harmonic Analysis: Radon transforms and beyond
- Abstract: Motivated by Bourgain's work on pointwise ergodic theorems, and the work of Stein and Stein-Wainger on maximally modulated singular integrals without linear terms, we develop a complete high $\ell^p, p \geq 2$ theory, and a partial low $\ell^p$ theory, for maximally monomially modulated discrete Hilbert transforms.

- Joris Roos (UW Madison)
- Title: Variation-norm estimates for a Stein-Wainger type oscillatory integral
- Abstract: In this talk I will present variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Corresponding maximal operators were first studied by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. The proof relies on square function estimates for a family of Schrödinger-like equations due to Lee, Rogers and Seeger. An additional ingredient in one dimension are local smoothing estimates for these equations. This is a recent joint work with Shaoming Guo and Po-Lam Yung.

- Hong Wang (MIT)
- Title: A restriction estimate in R^3 using brooms
- Abstract: If f is a function supported on a truncated paraboloid, what can we say about Ef, the Fourier transform of f? Stein conjectured in the 1960s that for any p>3, \|Ef\|_{L^p(R^3)} \lesssim \|f\|_{L^{\infty}}. We observe that if Ef is large, then Ef is concentrated on the thin neighborhood of many low degree algebraic surfaces and the wavepackets of Ef are organized into large brooms. We analyse this broom structure and make a little progress on Stein's conjecture for p> 3+3/13\approx 3.23. In the proof, we combined polynomial partitioning techniques introduced by Guth and two ends argument introduced by Wolff and Tao.

- Bobby Wilson (MIT)
- Title: Arbitrarily slowly decaying Favard length.
- Abstract: In this talk, we will discuss the concept of quantifying the visibility of a planar set and its relation to planar geometry. We will present the classical theorem of Besicovitch characterizing regularity of planar sets by the properties of their orthogonal projections which can be extrapolated to a characterization based on what is known as Favard length. This will be followed by an exploration of a similar characterization using approximations of sets in place of the original sets we want to study.

- Ruixiang Zhang (IAS)
- Title: Refined Strichartz estimate and its variants
- Abstract: The refined Strichartz estimate by Du-Guth-Li is done by an induction of the Bourgain-Demeter decoupling theorem over different scales. It gives one a good understanding about functions with Fourier support on the unit paraboloid, especially when we only want to understand their behavior on a subset of the ball of radius $R$. Recently several interesting variants have been obtained. We will review the basic toolbox in restriction type problems and talk about three results: The multilinear refined Strichartz (joint with Xiumin Du, Larry Guth and Xiaochun Li); a "refined decoupling inequality" by Guth-Iosevich-Ou-Wang that has application on the 2D Falconer distance problem; and a "fractal L^2 restriction estimate" which solves Carleson's pointwise convergence problem of free Schrödinger Solutions in general dimension, up to the endpoint (joint with Xiumin Du).