# Abstracts

**Principal Lecturer**

**Monica Visan**

University of California, Los Angeles

*Methods for Sharp Well-posedness of Completely Integrable Systems*

I will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE. Using the Korteweg-de Vries equation as our prototypical model, we first discuss the notions of macroscopic and microscopic conservation laws. We will place a particular emphasis on our new approach to these topics that remains viable for solutions with little regularity and poor spatial decay. We will then discuss important consequences of the conservation laws, including equicontinuity and local smoothing.

In the second talk, we review the notion of well-posedness, including a discussion of how continuity properties of the data-to-solution map inform our notion of non-perturbative well-posedness. The first-generation method of commuting flows will then be introduced and it will be shown how this method allowed us to complete the story of well-posedness for the Korteweg-de Vries equation. We will then elaborate on why a second-generation method was needed to complete the story for the cubic Schrödinger and the modified Korteweg-de Vries equations.

**Invited Speakers**

**Benjamin Harrop-Griffiths**

Georgetown University

*The Derivative NLS on the Line*

The derivative nonlinear Schrödinger equation arises as an effective equation for the dynamics of Alfvén waves in plasmas. While this model has been analyzed in depth since the 1970s, the last two years have seen significant progress on the behavior of solutions in Sobolev spaces. In this talk we discuss some of these recent results, culminating with a proof of well-posedness on the line for all initial data with finite mass. This is joint work with Rowan Killip, Maria Ntekoume, and Monica Vişan.

**Jason Murphy**

Missouri University of Science & Technology

*Invariance of White Noise for KdV on the Line*

We consider the problem of invariance of white noise for KdV on the real line. This problem was previously studied on the torus by Oh, Quastel, and Valkó. In the real line case, white noise fails to belong to H^{-1} (the optimal space for well-posedness of KdV). To address this difficulty, we pass through a sequence of suitable Hamiltonian approximations that are closely connected to the complete integrability of KdV. This is joint work with R. Killip and M. Visan.

**Contributed Talks**

**Animesh Biswas**

University of Nebraska-Lincoln

*Extension Problem for Fractional Power Operators*

In this talk, we will discuss the extension problem for the fractional power of operator defined on Banach Spaces. Using the semigroup property, we proved that there is a unique solution to the extension equation such that the restriction of the normal derivative of the solution on the lower dimensional space gives the fractional power of the operator. We proved this property for any fractional power s>0. This is a joint work with Pablo Raul Stinga.

**Ryan Frier**

University of Kansas

*Extremizers for the Strichartz Inequality for a Fourth-order Schrödinger Equation*We consider the Strichartz inequality for a fourth-order Schrödinger equation on R^{3}. We show that extremizers exist using a linear profile decomposition which follows from the endpoint version decomposition and the stationary phase method. Based on the existence of extremizers, we use the associated Euler-Lagrange equation to show that the extremizers have exponential decay and consequently must be analytic.

**Walton Green**

Washington University-St. Louis

*Smooth Singular Operators Restricted to Domains*

We use wavelets to analyze Sobolev mapping properties of smooth singular integral operators composed with a rough cut-off to a Lipschitz domain. The kernel singularity is dealt with by the usual smooth Calderon-Zygmund theory. The boundary singularity is handled using ideas from Carleson measures on the Dirichlet space. We give comprehensive unweighted results and partial results for weighted Sobolev spaces.

**Elizabeth Hale**

Kansas State University

*Fractional Leibniz Rules in the Setting of Quasi-Banach Function Spaces*

Fractional Leibniz rules are reminiscent of the product rule learned in calculus classes, offering estimates in the Lebesgue norm for fractional derivatives of a product of functions in terms of the Lebesgue norms of each function and its fractional derivative. We prove such estimates for Coifman-Meyer multiplier operators in the setting of Triebel-Lizorkin and Besov spaces based on quasi-Banach function spaces. As corollaries, we obtain results in weighted mixed Lebesgue spaces and Morrey spaces, where we present applications to the specific case of power weights. Other examples also include the class of rearrangement invariant quasi-Banach function spaces, of which weighted Lebesgue spaces, Lorentz spaces, and Orlicz spaces are specific examples.

**Jianhui Li**

University of Wisconsin-Madison

*Decoupling for Smooth Surfaces in $\mathbb R^3$*

We generalize the decoupling theory of Bourgain and Demeter (2015, 2017) from non-vanishing curvatures to general smooth surfaces in R^{3}. An application of the result to restriction theory will also be discussed. This is joint work with Tongou Yang.

**Zongyuan Li**

Rutgers University

*Unique Continuation for Schrödinger Equations with Variable Coefficients*

We prove that nontrivial solutions to Schrödinger equations with variable coefficients cannot have fast decay at two different times. Based on recent joint work with S. Federico (University of Bologna) and X. Yu (University of Washington).

**Jeffrey Oregero**

University of Central Florida

*Spectral Theory of a Non-self-Adjoint Dirac Operator with a Jacobi Elliptic Potential*

One of the prototypical integrable nonlinear evolution equations is the focusing nonlinear Schrödinger (NLS) equation, which is a universal model for weakly nonlinear dispersive wave packets, and as such it arises in a variety of physical settings, including deep water, optics, acoustics, plasma, condensed matter, etc. When solving the focusing NLS equation with periodic initial data a special role is played by the so-called finite-band solutions, which can be described in terms of theta functions. The finite-band solutions arise from the spectral theory of a non-self-adjoint Dirac operator with periodic or quasi-periodic potential. In this talk, I will discuss (i) the spectral theory of non-self-adjoint Dirac operators on the circle, and (ii) a two-parameter family of Jacobi elliptic finite-band potentials.

**Bryanna Petentler**

University of Iowa

*Compactness Method for Boundary Regularity of the Navier-Stokes Equation*

In this talk, we look at suitable weak solutions of the Navier-Stokes equation in three dimensions and investigate the Hölder regularity of these solutions up to the boundary. The Hölder regularity up to flat boundaries was already proven by Seregin in 2002. We prove boundary regularity result with a compactness argument and monotonicity properties of harmonic functions.

**Majed Sofiani**

University of Kansas

*Existence and Regularity of Solution to Parabolic PDE with Non-constant Hölder Diffusion Coefficient*

The study of the regularity of solutions to parabolic PDEs is quite extensive. We consider a particular 1-dimensional parabolic differential equation with non-constant Hölder diffusion coefficient that appears in studying the 1-dimensional full Ericksen-Leslie model for nematic liquid crystals. In this talk, we will discuss the existence and regularity of weak solutions for which we make use of explicitly the Hölder continuity of the coefficient.

**Trung Truong**

Kansas State University

*A Sampling-type Method Combined with Deep Learning for Inverse Scattering with One Incident Wave*

We are interested in the inverse scattering problem that aims to reconstruct the geometry of a bounded object from measured data of the scattered wave. When the scattered wave data was obtained for only one incident wave, existing reconstruction methods cannot provide satisfactory results for relatively complex geometries. This lack of data is common in practice due to technical difficulties. Therefore, we study a sampling-type method that is fast, simple, regularization-free, stable against high levels of noise, and combine it with a deep neural network to solve the inverse scattering problem in which the scattering data is only provided for one incident wave. This combined method can be understood as a network using the image computed by the sampling method for the first layer and followed by the U-net architecture for the remaining layers. The network is trained on simulated data sets of simple scattering objects and is validated by objects with more complex geometries and real data without any additional transfer training. This is joint work with Thu Le, Dinh-Liem Nguyen, and Vu Nguyen.

**Liding Yao**

Ohio State University

*Sharp Hölder Regularity for Nirenberg's Complex Frobenius Theorem*

Nirenberg’s famous complex Frobenius theorem gives necessary and sufficient conditions on a locally integrable structure for when it is equal to the span of some real and complex coordinate vector fields. For a $C^{k,s}$ complex Frobenius structure, we show that there is a $C^{k,s}$ coordinate chart such that the structure is spanned by coordinate vector fields which are $C^{k,s-\varepsilon}$ for all $\varepsilon>0$. Here the $\varepsilon>0$ loss in the result is optimal.

**Poster Presentations**

**Chad Berner**

Iowa State University

*Fourier Series from Singular Measures*

Any square integrable function on the torus is a norm limit of its Fourier series, but what if you change the measure from Lebesgue measure to a singular measure? It turns out you will lose orthogonality of the exponentials, but by the Kaczmarz algorithm, any function that is square integrable in this new measure space has a Fourier series converging in norm. We discuss further results in higher dimensions as well as analytic operators and their relation to the Hardy space and Fourier series.

**Michael Jesurum**

University of Wisconsin-Madison

*Linear Operators and Curves*

Harmonic analysis is full of linear operators associated with manifolds—Fourier restriction, convolution operators, and more. Those manifolds usually need some sort of curvature, but there are useful results to be had for manifolds which are not well-curved. We discuss methods for bounding linear operators associated with ill-behaved curves in Euclidean space.

**Jeongsu Kyeong**

Temple University

*Multilayer Potentials Associated with the Poly-Cauchy Operator in Non-smooth Domains*

The subject of this talk is the analysis of multilayer potentials associated with integer powers of the classical $\overline{\partial}$ operator in non-smooth domains in the complex plane. This analysis includes integral representation formulas, jump relations, higher-order Fatou theorems, and higher-order Hardy spaces. This is joint work with Irina Mitrea (Temple University), Dorina Mitrea and Marius Mitrea (Baylor University).

**Thu Thi Anh Le**

Kansas State University

*Orthogonality Sampling Method for Inverse Elastic Scattering from Anisotropic Media*

In this talk, we investigate the inverse scattering problem of time-harmonic elastic waves for anisotropic inclusions in an isotropic homogeneous background. The elasticity tensor and mass density are allowed to be heterogeneous inside the inclusion and may be discontinuous across the background-inclusion interface. We derive the far-field pattern of the scattered wave using the Lippmann-Schwinger integral equation of the scattering problem. Using the far-field pattern as the data for the inverse problem we construct indicator functions of orthogonality sampling types for the scattering objects. These functions are very robust to noise, computationally cheap, do not involve any regularization process. We provide some theoretical analysis as well as numerical simulations for the proposed indicator functions. This is joint work with Dinh-Liem Nguyen and Trung Truong.

**Gary Moon**

University of Oklahoma

*Local Well-posedness of Damped Water Waves *

We will discuss the local well-posedness of the water waves system (i.e., the free-surface Euler Euler equations) subject to damping and consider the lifespan of solutions, primarily in the small data regime.

**Michael Penrod**

University of Alabama

*The **e**-Maximal Operator and Haar Multipliers on Variable Lebesgue Spaces*

We prove the *e*-maximal operator is bounded on variable Lebesgue spaces under assumptions that are weaker than log-Holder continuity. With an additional assumption, we prove that the Haar multiplier is locally compact on variable Lebesgue spaces.

**Mohammad Mahabubur Rahman**

Texas Tech University

*Global Regularity Issue of the Two-and-a-half Dimensional Hall-magnetohydrodynamics System*

Whether or not the solution to the two-and-a-half dimensional Hall-magnetohydrodynamics system starting from any smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on component reduction of regularity criteria for Navier-Stokes equations and magnetohydrodynamics system has caught much attention recently, the Hall term has presented many difficulties. In this manuscript, we discover a certain cancellation within the Hall term and obtain various new regularity criteria: first, in terms of a gradient of only the third component of the magnetic field; second, in terms of only the third component of the current density; third, in terms of only the third component of the velocity field; fourth, in terms of only the first and second components of the velocity field. As another consequence of the cancellation that we discovered, we are able to prove the global well-posedness of the two-and-a-half dimensional Hall-magnetohydrodynamics system with hyper-diffusion only for the magnetic field in the horizontal direction; we also obtained an analogous result in the 3-dimensional case via the discovery of additional cancellations. These results extend and improve various previous works. This is the joint work with Prof. Kazuo Yamazaki.

**James Tautges**

University of Wisconsin-Madison

*Extremizers for Radon-like Inequalities for Compact, Curved Hypersurfaces*

It is well-known that convolution with surface measure on compact, curved hypersurfaces extends to a bounded L* ^{p}*-L

*operator for certain*

^{q}*p, q*. Using a concentration-compactness argument and derivative estimates for the associated Euler-Lagrange equation, we prove that extremizing sequences are pre-compact modulo translation and extremizers are smooth in a certain range of exponents. This work builds on the techniques used by Christ and Xue to answer a similar question for the paraboloid.

**Anh Vo**

University of Nebraska-Lincoln

In recent years nonlocal conservation laws have been studied extensively in theory and widely applied in several fields such as physics, biology, computer science, etc. Nonlocal behavior is modeled through integral operators which collect information from neighborhood of a point. Furthermore, nonlocal theories can be used in conservation laws to overcome some challenges encountered by classical theories such as shock development. However, there are still open questions about bridging the gap between nonlocal theories and local theories. In particular, we are interested in studying convergence of nonlocal solutions to classical counterparts in the limit of the vanishing horizon of interaction. Under appropriate conditions, one expects the nonlocal model to provide an approximation to its classical counterpart. While these results are nontrivial even in the linear setting due to the lack of compactness for nonlocal operators, the nonlinear framework brings additional challenges. We will show conditions under which the nonlocal nonlinear divergence operator converges to the local counterpart (with rates of convergence depending on different degrees of smoothness for the data); these results will then be employed to show convergence of nonlocal solutions to corresponding local ones.

**Lingxiao Zhang**

University of Wisconsin-Madison

*Spectral Multipliers for Maximally Subelliptic Operators*

Given a non-negative, self-adjoint operator L, for every bounded Borel function m on [0,\infty), m(L) is defined by spectral theorem. Maximal subellipticity is a far-reaching generalization of ellipticity. Suppose L is maximally subelliptic acting on a compact manifold. In this talk, I will discuss the Mihlin-Hormander type conditions on m such that m(L) is bounded on L* ^{p}* Sobolev spaces. In particular, m(L) will be an “infinitely smooth” singular integral operator if m satisfies the Mihlin-Hormander type condition of infinite order. I will also discuss the equivalence between L

*Sobolev spaces adapted to L and those adapted to a Carnot-Caratheodory geometry.*

^{p}