Abstracts

Principal Lecturer

Christopher Sogge
Johns Hopkins University

The Role of Curvature in Harmonic Analysis on Manifolds
Curvature has played a key role in Euclidean harmonic analysis since the 1970s. The first results included breakthroughs on Fourier restriction problems by C. Fefferman, E. M. Stein and P. Tomas in the 1970s. This continued through the work of Stein and Bourgain on spherical and circular maximal functions, the work on establishing and applying Strichartz estimates in PDEs, and more recently, proving and applying decoupling estimates through the work of Wolff and Bourgain and Demeter and many others. In all of these results, the analysis centered on kernels or Fourier transforms which were highly focused on curved hypersurfaces in R^n.

We shall focus on related problems arising in harmonic analysis on Riemannian manifolds. In this case the curvature assumptions involve sectional curvatures. Classical results, going back to Hörmander and Bérard from the 1960s and 1970s concern sup-norm estimates for eigenfunctions, and, more generally quasimodes. These were extended to L^p estimates for p > p_c = 2(n + 1)/(n − 1) on (Mⁿ, g), an n-dimensional compact Riemannian manifold, by Hassell and Tacy in the previous decade. Note that p_c is the critical exponent for the Stein-Tomas extension theorem. The work of Bérard and Hassell-Tacy show that if one assumes that all of the sectional curvatures of Mⁿ are non-positive then there is improved L^p estimates compared to the universal bounds obtained by Sogge in the 1980s which are saturated on the sphere (positive curvature). Subsequently, Blair and Sogge were able to extend these results to include the critical exponent p = p_c, which, by interpolation leads to improved eigenfunction estimates for all exponents p > 2 under the assumption of non-positive sectional curvatures.

In addition to going over these now classical results, we shall present new sharp estimates of X. Huang and Sogge for p ∈ (2,p_c] that allow a new classification of compact space forms (manifolds of constant sectional curvature). One can determine the sign of the curvature of a compact space form by observing the growth rate of L^p-norms of L²-normalized quasimodes for p ∈ (2, p_c], even though no such classification is possible for any p > p_c. We also will show that such a classification is possible by measuring the growth rate of L²-norms of L¹-normalized quasimodes, as well as through L²-concentration near geodesic tubes. The proof of the new estimates require variable coefficient bilinear estimates which generalize Euclidean ones of Tao, Vargas and Vega, and, to classify space forms, we show that our estimates are sharp using constructions on compact space forms of negative, zero or positive curvature which are natural variants of the Knapp construction for Euclidean space. These problems are also related to new Strichartz estimates for solutions of time-dependent Schrödinger equations, generalizing the classical universal estimates of Burq, Gérard and Tzvetkov.

Invited Speakers

Hong Wang
Courant Institute of Mathematics
New York University

Some Structure of Kakeya Sets in R³
A Kakeya set in Rⁿ is a set of points that contains a unit line segment in every direction. We study the structure of Kakeya sets in R³ and show that for any Kakeya set K, there exists well-separated scales 0 < δ < ρ ≤ 1 so that the δ-neighborhood of K is almost as large as the ρ-neighborhood of K. As a consequence, every Kakeya set in R³ has Assouad dimension 3. This is joint work with Josh Zahl.

Ruixiang Zhang
University of California, Berkeley

A New Conjecture to Unify Fourier Restriction and Bochner-Riesz
The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions ≥ 3, and notably have the same conjectured exponents. In the 1970s, Hörmander asked if a more general class of operators (known as Hörmander type operators) all satisfy the same L^p- boundedness as in the above two conjectures. A positive answer to Hörmander’s question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately Hörmander’s question is known to fail in all dimensions ≥ 3 by the work of Bourgain and many others. It continues to fail in all dimensions ≥ 3 even if one adds a “positive curvature” assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension 3 one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it “Bourgain’s condition”. We unify Fourier restriction and Bochner-Riesz by conjecturing that any Hörmander type operator satisfying Bourgain’s condition should have the same L^p-boundedness as in those two conjectures. As evidence, we prove that the failure of Bourgain’s condition immediately implies the failure of such an L^p-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

Contributed Talks

Blake Allan
Baylor University

Approximating Eigenvalues via Truncations
Suppose we approximate a complicated operator by simpler ones (e.g., finite truncations of an infinite matrix) - do their spectra approximate the spectrum of the original? We study this phenomenon (and its failure) in general, formulate sufficient conditions to guarantee convergence of the spectrum, and apply these results to computing the critical coupling constant for a quantum-mechanical point dipole. Based on joint work with Fritz Gesztesy.

Chian-Yeong Chuah
Ohio State University

Compactness of Sobolev Embedding into the Variable Lorentz 
The study of Sobolev embedding into the Lebesgue spaces and Lorentz spaces plays a fundamental role in the field of PDE and approximation theory. In the classical setting, the Rellich–Kondrachov theorem provides condition on which the Sobolev embedding is compact. In the case where the embedding is non-compact, there are various levels to which one can measure the quality of non-compactness. In this talk, we discuss the compactness of Sobolev embedding into the variable Lorentz space where the behavior of non-compactness is concentrated around a single point. The quality of non-compactness is also discussed in this case. This is a joint work with Jan Lang.

Sam Craig
University of Wisconsin

Failure of Weak-type Endpoint Restriction Estimates for Quadratic Manifolds
Simple examples demonstrate that the Fourier extension operator associated with a quadratic manifold cannot map L^p to L^q when q ≤ 2d/(d-1). However, those examples do not rule out the possibility of the extension operator mapping L^p to weak L^q. A paper of Beckner, Carbery, Semmes, and Soria rules out such a bound for quadratic hypersurfaces by proving a bound would contradict the existence of measure zero Kakeya sets. We will generalize their proof to prove that quadratic manifolds of dimension n cannot map L^p to weak L^q when q = 2d/n by constructing a measure zero set with a translated copy of every normal plane to the manifold and proving that the existence of such sets contradicts a weak-type bound at q = 2d/n.

Jacob Fiedler
University of Wisconsin

Universal Sets for Pinned Distances
An important problem in geometric measure theory is bounding the size of pinned distance sets. We discuss recent work on this problem in the plane which shows that, as long as the pin point x satisfies certain properties, the pinned distance set of Y at x will be as large as possible. In particular we show that any AD-regular set X of dimension more than 1 has the property we call universality: for any Borel Y, there is an x in X such that the pinned distance set of Y at x has maximal Hausdorff dimension. This can be thought of as in a certain sense giving a weak version of Falconer's conjecture for dimension in the plane. This is based on joint work with Don Stull.

Jiaqi Hou
University of Wisconsin

Restrictions of Eigenfunctions on Arithmetic Hyperbolic 3-manifolds
In this talk, I will talk about the L² restriction norm problems for Laplace eigenfunctions with large eigenvalues on Riemannian manifolds. Let X be a compact arithmetic hyperbolic 3-manifold. Let f be a Hecke-Maass form on X, which is a joint eigenfunction of the Laplacian and Hecke operators. Using methods from number theory, we prove a power saving over the local bound by Burq, Gérard, and Tzvetkov for the L² norm of f restricted to a totally geodesic surface in X. I will also present a bound for f restricted to geodesic tubes. This bound can be applied to improve the L^p bounds of f by Sogge for  2<p<4.

Kaiyi Huang
University of Wisconsin

Brascamp-Lieb Inequalities on the Heisenberg Groups
I will talk about recent developments in Brascamp-Lieb inequalities on the Heisenberg groups. This is a nonlinear analogue of linear Brascamp-Lieb inequalities, and a version of multilinear Radon-like transforms with various coranks.

Joshua Jordan
University of Iowa

On Canonical Metrics of Complex Surfaces with Split Tangent and Related Geometric PDEs
In this paper, we study bi-Hermitian metrics on complex surfaces with split holomorphic tangent bundle and construct 2 types of metric cones. We introduce a new type of fully non-linear geometric PDE on such surfaces and establish smooth solutions. As a geometric application, we solve the prescribed Bismut Ricci problem. In various settings, we obtain canonical metrics on 2 important classes of complex surfaces: primary Hopf surfaces and certain Inoue surfaces.

Hanna Kim
University of North Carolina, Chapel Hill

Upper Bounds of Second Eigenvalues on the Sphere and the Real Projective Space
In this talk, we discuss sharp upper bounds for the second nonzero eigenvalue of the Laplacian on higher dimensional spheres and real projective spaces. These isoperimetric inequalities of eigenvalues on the closed manifold arise from the classical isoperimetric problem and extending this problem to differential geometry. Our method consists of constructing trial functions based on recent developments in the hyperbolic center of mass and on topological degree theory used to verify that the trial functions are valid. We lastly talk about open problems related to these topics.

Zachary Lee
University of Texas, Austin

On Uniqueness Properties of Solutions of the Generalized Fourth-order Schrödinger Equations
We study uniqueness properties of solutions to certain fourth-order Schrödinger equations in any dimension, both of linear type with a time-dependent and complex potential and of non-linear type with a rather general nonlinearity. We show that a linear solution with fast enough decay in certain Sobolev spaces at two different times has to be trivial. Consequently, we show that if the difference between two nonlinear solutions decays sufficiently fast at two different times, they must be identical.

Bingyuan Liu
University of Texas Rio Grande Valley

The Diederich-Fornæss Index and the dbar-Neumann Problem
A domain W in Cⁿ is said to be pseudoconvex if -log(-d(z)) is plurisubharmonic in W where d is a signed distance function of W. The study of global regularity of dbar-Neumann problem on bounded pseudoconvex domains is dated back to the 1960s. However, a complete understanding of the regularity is still absent. On the other hand, the Diederich-Fornæss index was introduced in 1977 originally for seeking bounded plurisubharmonic functions. Through decades, enormous evidence has indicated a relationship between global regularity of the dbar-Neumann problem and the Diederich-Fornæss index. Indeed, it has been a long-lasting open question whether the trivial Diederich-Fornæss index implies global regularity. In this talk, we will introduce the backgrounds and motivations. The main theorem of the talk proved recently by Emil Straube and me answers this open question for (0, n-1) forms.

Alex McDonald
Kennesaw State University

Prescribed Projections and Efficient Coverings of Sets by Curves
A remarkable result of Davies shows that an arbitrary measurable set in the plane can be covered by lines in such a way that the union of the lines minus the original set has measure zero. This theorem has an equivalent dual formulation which says that one can find a single set in the plane with given "prescribed" projections in almost every direction, up to measure zero errors. We extend these results to a non-linear setting and prove that a set in the plane can be covered efficiently by translates of a single curve satisfying a mild curvature assumption.

Blanca Radillo-Murguia
Baylor University

Kakeya-type Sets and Maximal Operators
We will discuss a recent result that provides a sufficient condition on a set of directions to show the admissibility of Kakeya-type sets, using a probabilistic argument inspired by the ideas of Michael Bateman and Nets Katz. This condition guarantees that the associated directional maximal operator is unbounded on L^p(R²⁾ for every 1 < p < infinity. This work is joint with Paul Hagelstein and Alex Stokolos.

Tristan Reynoso
Louisiana State University

Strichartz Estimates for the N-body Wave Equation with Small Interacting Potentials
Dispersive equations are useful for describing a variety of physical systems. The Schrödinger equation and the Wave equation being two cornerstone models. The single particle variant of these equations has been studied extensively in recent history. Over the past few years progress has been made in extending single particle dispersive equations to cover their many body counterparts. Space-time Strichartz estimates for the N-body Schrödinger Equation with small interacting potentials was recently established for the Euclidean and Tori settings. We look to similarly prove Strichartz estimates for a proposed N-body extension of the wave equation with small interacting potentials. In order to prove these estimates the standard mixed Lebesgue spaces are insufficient and we must utilize the space of bounded p-Variation and their close cousin the atomic space.

Seung-Yeon Ryoo
California Institute of Technology

Lipschitz Padded Nested Random Partitions
We introduce the concept of a "Lipschitz Padded Nested Random Partition," which is a multiscale version of padded decompositions. Often in geometry, when one wishes to construct an object globally on a geometric object such as a manifold or a discrete metric space, one first constructs it locally and pastes it via some partition of unity; control on the partition of unity leads to control on the resulting global object. We show that the existence of a Lipschitz Padded Nested Random Partition on a doubling metric space implies a sharp version of the Assouad embedding theorem, and that the Heisenberg group admits a Lipschitz Padded Nested Random Partition. This gives an alternative construction of the sharp Assouad embedding for the Heisenberg group. This is a forthcoming joint work with Alan Chang.

Pedro Takemura
Baylor University

Composite Herz Spaces and Singular Integrals
In this talk, we introduce a new class of Herz-type spaces, which we refer to as Composite Herz spaces. We provide various examples to illustrate how our theory generalizes the existing body of work on Herz spaces. As an application, we present some boundedness results for a generic class of sublinear operators, as well as for singular integral operators of Calderón-Zygmund type. This is joint work with Marius Mitrea.

Sasanka Widuranga Wijesiri Gunawardana
University of Cincinnati

Monotone Rearrangement for Flattening Weight
We consider the class of flattening weights F_2 on an interval, which are those Muckenhoupt A_2 weights whose logarithms are in VMO. We show that the F_2 characteristic (the natural scale-dependent analog of the usual A_2 characteristic) does not increase under monotone rearrangement. This allows one to work with monotone functions when proving integral estimates for the class F_2.

Yunfeng Zhang
University of Cincinnati

Semiclassical Fun with SU(3)
This talk is intended to give an overview of several "semiclassical analytic" questions posted on compact Lie groups, SU(3) being a typical example, such as Strichartz estimates for the Schrödinger flow, bounds of Laplacian eigenfunctions, etc.

Poster Presentations

Katheryn Beck
University of Kansas

Stability and Dynamics of the Ericksen-Leslie Model
In this talk we consider nematic liquid crystal placed between two parallel plates. We investigate this set-up by use of the Ericksen-Leslie model with certain boundary conditions to describe this physical system. We will look into steady states in the system and then study the stability of these steady states using analytical and numerical methods. These steady states fall into a variety of cases which the study of shows a variety of dynamics including the presence of a saddle node bifurcation and further dynamics of the unstable states shows the evidence of the existence of heteroclinic orbit.

Brett Ehrman
University of Kansas

Orbital Stability of Smooth Solitary Waves for the Novikov Equations
We study the orbital stability of smooth solitary wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. These solitary waves are shown to exist as a one-parameter family (up to spatial translations) parameterized by their asymptotic endstate, and are encoded as critical points of a particular action functional. As an important step in our analysis we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operator acting on the space of square-integrable functions on the real line. We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses always holds for the Novikov equation. Together with a detailed study of the associated Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary wave solutions of the Novikov equation are nonlinearly orbitally stable.

Ryan Frier
Arizona State University

Strichartz Inequality for the Schrödinger Equation on R²
We study the extremal problem for the Strichartz inequality for the Schrödinger equation on R². We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi in 2007 and Hundertmark-Zharnitsky in 2006.

Le Gong
Vanderbilt University

Dynamical Sampling for Source Recovery
In this talk, I will give a brief introduction of dynamical sampling and present a recent work in source term recovery. Inspired by environmental monitoring applications for identifying the magnitude of pollution sources, we consider the problem of recovering constant source terms in a discrete dynamical system represented by x_n+1 = Ax_n + w. Here, x_n is the n-th state in a Hilbert space H, A is a bounded linear operator in B(H), and wrepresents a source term within a closed subspace W of H. I will present the necessary and sufficient conditions for the reconstruction of w from time-space sample measurements, independent of the unknown initial state x₀ and applicable to any wÎW.

Walton Green
Washington University, St. Louis

Observability of Wave and Heat Equations from Very Small Sets
We discuss recent developments due to Logunov-Malinnikova and Burq-Moyano concerning the observability of parabolic equations from sets of positive codimension. We extend these results in the special case of the heat equation to sets of any codimension strictly less than 1. Furthermore, we exhibit special sets of even larger codimension for which wave and heat equations are observable on the 2-torus.

Chris Mayo
University of Kansas

Well-Posedness of a Higher-Order Nonlinear Schrödinger Equation on a Finite Interval
The higher-order nonlinear Schrödinger (HNLS) equation is a more accurate alternative to the standard NLS equation when studying wave pulses in the femtosecond regime. It arises in a variety of applications ranging from optics to water waves to plasmas to Bose-Einstein condensates. In this talk, we consider the initial-boundary value problem for HNLS on a finite interval in the case of a power nonlinearity. We establish the local well-posedness of this problem in the sense of Hadamard (existence and uniqueness of the solution as well as its continuous dependence on the data) for initial data in the Sobolev space H^s on a finite interval and boundary data in suitable Sobolev spaces determined by the regularity of the initial data and the HNLS equation. The proof relies on a combination of estimates for the linear problem and nonlinear estimates, which vary depending on whether s > 1/ or 0 ≤ s < 1/2. The linear estimates are established by using the explicit solution formula obtained via the unified transform method of Fokas. This is a joint work with Dionyssis Mantzavinos and Turker Ozsari.

Nhung Nguyen
Kansas State University

Fast and Stable Imaging of Objects in 2D Acoustic Waveguides Using Scattering Data
This talk addresses the inverse problem of reconstructing extended objects from scattering data at a fixed frequency within a 2D acoustic waveguide. This problem has significant applications in physics and engineering, including radar, sonar, and nondestructive testing. We propose an imaging function to efficiently determine the shape of the unknown object. The method is both fast and stable, with a simple implementation that avoids the need of solving an ill-posed problem. Our numerical studies demonstrate that the proposed imaging function can provide more accurate reconstructions compared to the direct sampling method and Kirchhoff migration.

Daniel Pezzi
Johns Hopkins University

Sharp Spectral Projection Estimates on the Torus
We prove sharp spectral projection estimates on general tori in all dimensions for all windows except the endpoint at the critical exponent for decoupling. These estimates with a sub-polynomial loss are a fairly direct consequence of decoupling, but such a loss is mandatory in applications of decoupling. We use microlocal estimates, a bilinear decomposition, and results for Fourier series to remove this loss in the context of this problem.

Jonathan Schillinger
Florida State University

Measures and Asymptotics for the Gaussian Kernel
For the continuous energy of gaussian kernels we will go over some new results concerning their equilibrium measures, and the order of their asymptotics for the interval, cube, disc, and ellipse. Some preliminary results over self-similar fractal sets will also be discussed.

Liding Yao
Ohio State University

On Seeley‐type Universal Extension Operators for the Upper Half Space
The Seeley's extension operator is an operator that extends function defined on the upper half space to the total space Rⁿ using reflection principle. We give a construction of such operator which is bounded in all C^k-spaces, Sobolev spaces, Besov and Triebel-Lizorkin spaces. This is joint work with Haowen Lu.