May 15, 2017
Monday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


Recent developments in some multilinear problems
Anthony Carbery (University of Edinburgh)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We give an overview of some recent developments in the circle of problems which includes LoomisWhitney inequalities, BrascampLieb inequalities and multilinear Kakeya inequalities.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Jump inequalities for translationinvariant polynomial averages and singular integrals on $\mathbb Z^d$
Mariusz Mirek (Hausdorff Research Institute for Mathematics, University of Bonn)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The aim of this talk is to prove $\ell^p(\mathbb Z^d)$ inequalities with $1 < p < \infty$, for $\lambda$jumps for discrete Radon transforms. These inequalities are the $r = 2$ endpoints of the $r$variational estimates due to Mirek, Stein, and Trojan. This is a joint project with E.M. Stein and P. ZorinKranich.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


A cone restriction estimate using polynomial partitioning
Yumeng Ou (Massachusetts Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this talk I will discuss an improved restriction estimate for the cone in all dimensions (sharp in five dimensions), which uses the method of polynomial partitioning as the main tool. This is joint work with Hong Wang while in residence at MSRI.
 Supplements


03:00 PM  03:30 PM


Tea & Poster Session

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Radonlike operators of intermediate dimension
Philip Gressman (University of Pennsylvania)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We will discuss recent results establishing $L^p$improving estimates for Radonlike operators which average functions over submanifolds of intermediate dimension (e.g., neither curves nor hypersurfaces). The methods are built around an $L^p$adapted $TT^*T$ argument which is itself an instance of a Christtype method of refinements. The resulting estimates are sharp up to loss of the endpoints and provide new insights into the vector field formulation of sharp curvature conditions
 Supplements



May 16, 2017
Tuesday

09:30 AM  10:30 AM


Uniform rectifiability, bounded harmonic functions, and elliptic PDE's
Xavier Tolsa (Autonomous University of Barcelona)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this talk I will describe a recent characterization of uniform rectifiability in terms of a variant of the so called area function (or g function) for harmonic functions and an analogous result involving solutions of elliptic PDE's. As a byproduct, we obtain a new characterization of uniform rectifiability in terms of harmonic measure or elliptic measure. Some of these results are joint work with Azzam, Garnett and Mourgoglou
 Supplements


10:30 AM  11:00 AM


Break & Poster Session

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Sparse domination of singular integral operators
Francesco Di Plinio (University of Virginia)

 Location
 SLMath: Atrium
 Video

 Abstract
Singular integral operators, which are a priori signed and nonlocal, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical CalderónZygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrixvalued kernels, rough homogeneous singular integrals and critical BochnerRiesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel CondeAlonso, Yen Do, Yumeng Ou, Yannis Parissis and Gennady Uraltsev
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Rigidity, group actions and finite point configurations in thin sets
Alex Iosevich (University of Rochester)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We will use the notion of rigidity, group actions and Fourier analysis to study the problem of the existence of congruent copies of finite point configurations of a given type inside sets of a given Hausdorff dimension
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type
Lesley Ward (University of South AustraliaMawson Lakes Campus)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Much effort has been devoted to generalizing the Calder\'onZygmund theory from Euclidean spaces to metric measure spaces, or spaces of homogeneous type. Here the underlying space $\mathbb{R}^n$ with Euclidean metric and Lebesgue measure is replaced by a set $X$ with a general metric or quasimetric and a doubling measure. Further, one can replace the Laplacian operator that underpins the Calder\'onZygmund theory by more general operators~$L$ satisfying heat kernel estimates. I will present recent joint work with P.~Chen, X.T.~Duong, J.~Li and L.X.~Yan along these lines. We develop the theory of product Hardy spaces $H^p_{L_1,L_2}(X_1 \times X_2)$, for $1 \leq p < \infty$, defined on products of spaces of homogeneous type, and associated to operators $L_1$, $L_2$ satisfying DaviesGaffney estimates. This theory includes definitions of Hardy spaces via appropriate square functions, an atomic Hardy space, a Calder\'onZygmund decomposition, interpolation theorems, and the boundedness of a class of Marcinkiewicztype spectral multiplier operators.
 Supplements


04:30 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




May 17, 2017
Wednesday

09:30 AM  10:30 AM


Boundary Layers in Periodic Homogenization
Zhongwei Shen (University of Kentucky)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
This talk is concerned with a family of elliptic systems in divergence form with rapidly oscillating periodic coefficients. I will discuss recent progress on homogenization and boundary layers for Dirichlet and Neumann problems with oscillating boundary data. The problems arise naturally in the study of higherorder convergence of solutions to boundary value problems with nonoscillating data. The use of harmonic analysis techniques in obtaining sharp estimates will be highlighted
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Quantitative differentiation
Tuomas HytÃ¶nen (University of Helsinki)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
As we learn in Calculus, differentiation is about approximating a given function by an affine one in infinitesimal balls. In quantitative differentiation, we would like to do the same in "macroscopic" balls of quantified size. There are now three approaches to the problem: "geometric", "analytic", and "dynamic". I will concentrate on the latter two, which I have considered in joint work with Assaf Naor (both) and Sean Li (the analytic approach). A key to both is a quantitative elaboration of Dorronsoro's classical embedding theorem of a Sobolev space into a certain local approximation space; in the "dynamic" version, this is achieved with the help of the heat flow.
 Supplements



May 18, 2017
Thursday

09:30 AM  10:30 AM


The Analyst's Traveling Salesman Theorem for large dimensional objects
Jonas Azzam (University of Edinburgh)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The classical Analyst's Traveling Salesman Theorem of Peter Jones gives a condition for when a subset of Euclidean space can be contained in a curve of finite length (or in other words, when a "traveling salesman" can visit potentially infinitely many cities in space in a finite time). The length of this curve is given by a square sum of quantities called betanumbers that measure how nonflat the set is at each scale and location. Conversely, given such a curve, the square sum of its betanumbers is controlled by the total length of the curve, giving us quantitative information about how nonflat the curve is. This result and its subsequent variants have had applications to various subjects like harmonic analysis, complex analysis, and harmonic measure. In this talk, we will introduce a version of this theorem that holds for higher dimensional surfaces. This is joint work with Raanan Schul.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


The hidden landscape of localization of eigenfunctions
Svitlana Mayboroda (University of Minnesota, Twin Cities)

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic methods could predict the exact spatial location and frequencies of the localized waves.
In this talk I will present recent results revealing a new criterion of localization, tuned to the aforementioned questions, and will illustrate our findings in the context of the boundary problems for the Laplacian and bilaplacian, $div A\nabla$, and (continuous) Anderson and AndersonBernoulli models on a bounded domain. Via a new notion of ``landscape" we connect localization to a certain multiphase free boundary problem and identify location, shapes, and energies of localized eigenmodes. The landscape further provides estimates on the rate of decay of eigenfunctions and delivers accurate bounds for the corresponding eigenvalues, in the range where both classical Agmon estimates and Weyl law may fail.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Approximation of $\dot{W}^{s,n/s}$ functions by bounded functions on $\mathbb{R}^n$
Po Lam Yung (Chinese University of Hong Kong)

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
The homogeneous Sobolev spaces $\dot{W}^{s,n/s}(\mathbb{R}^n)$ all fail to embed into $L^{\infty}$ when $0<s<n$, but only barely so. In dimension $n \geq 2$, Bourgain and Brezis found a rather useful remedy for this failure when $s = 1$, by considering how well a general $\dot{W}^{1,n}$ function can be approximated by an $L^{\infty}$ function on $\mathbb{R}^n$. In this talk we give an extension of their results to $\dot{W}^{s,n/s}$ for all $0 < s < n$. This is joint work with Pierre Bousquet, Emmanuel Russ and Yi Wang.
 Supplements


03:00 PM  03:30 PM


Tea & Poster Session

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Applications of decouplingtype estimates to the cubic NLSE
Bobby Wilson (University of Washington)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this talk, we will discuss how decoupling inequalities allow for the establishment of wellposedness results for the nonlinear Schrodinger equation in the case when data is defined on a twodimensional irrational torus.
 Supplements



May 19, 2017
Friday

09:30 AM  10:30 AM


Two weight norm inequalities for singular and fractional integral operators in $R^n$.
Ignacio UriarteTuero (Michigan State University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will report on recent advances on the topic, related to proofs of T1 type theorems in the two weight setting for Calder\'{o}nZygmund singular and fractional integral operators, with side conditions, and related counterexamples. Joint work with Eric Sawyer and ChunYen Shen.
 Supplements



10:30 AM  11:00 AM


Break & Poster Session

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


An improved bound on the Hausdorff dimension of Besicovitch sets in R^3
Joshua Zahl (University of British Columbia)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
A Besicovitch set is a compact set in R^d that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that every Besicovitch set in R^d must have Hausdorff dimension d. I will discuss a recent improvement on the Kakeya conjecture in three dimensions, which says that every Kakeya set in R^3 must have Hausdorff dimension at least 5/2 + \eps, where \eps is a small positive constant. This is joint work with Nets Katz
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


On the failure of lower square function estimates in the nonhomogenous weighted setting.
Stefanie Petermichl (UniversitÃ© de Toulouse III (Paul Sabatier))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We show that the classical A_infinity condition is not sufficient for a lower square function estimate in the nonhomogeneous weighted L^2 space. We also show that under the martingale A_2 condition, an estimate holds true, but the optimal power of the characteristic jumps from 1 / 2 to 1. This is in a sharp contrast to recent positive results in this direction on the discrete time nonhomogeneous martingale transforms. Joint work with Domelevo, Ivanisvili, Treil, Volberg while in residence at MSRI.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Recent developments in decoupling theory
Ciprian Demeter (Indiana University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will review some of the recent progress on decouplings.
 Supplements


