Spring 2025 Analysis Seminar

Organizers: Shengwen Gan, Terry Harris and Andreas Seeger

Emails:

• Shengwen Gan: sgan7 at math dot wisc dot edu
• Terry Harris: tlharris4 at math dot wisc dot edu
• Andreas Seeger: seeger at math dot wisc dot edu

Time and Room: Wed 3:30--4:30 Van Vleck B235 (new room in the spring semester!)

In some cases the seminar may be scheduled at different time to accommodate speakers.

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Links to previous seminars

Date	Speaker	Institution	Title	Host(s)	Notes (e.g. unusual room/day/time)
We, Jan.29	Thierry Laurens	UW Madison	A priori estimates for generalized KdV equations in H^{-1}	Andreas
We, Feb 5	Sergey Denisov	UW Madison	Dispersion effects for evolution equations and the case for Brascamp-Lieb bounds
We, Feb 12	Dallas Albritton	UW Madison	Progress Report on Instability and Non-uniqueness in Incompressible Fluids	Andreas
We, Feb 19	Xiaojun Huang	Rutgers University	Bounding a Levi-flat hypersurface in a Stein manifold	Xianghong
We, Feb 26	Galia Dafni	Concordia University	Some function spaces defined using means and mean oscillation	Andreas and Betsy
We, Mar 5, no seminar
We, Mar 12	Brian Simanek	Baylor University	Blaschke Products, Numerical Ranges, and the Zeros of Orthogonal Polynomials	Sergey
We, Mar 19	Shengwen Gan	UW Madison	On local smoothing estimates for wave equations
We, Mar 26, no seminar
We, Apr 2	Robert Schippa	UC Berkeley	Local smoothing for the Hermite wave equation	Andreas
We, Apr 9	Junjie Zhu	UBC	Fourier dimension of constant rank hypersurfaces	Terry
We, Apr 16	Ziming Shi	UC-Irvine	The method of homotopy formula and the Cauchy-Riemann(d-bar) equation on pseudoconvex domains of finite type in $\mathbb C^2$	Xianghong
We, Apr 23	Liding Yao	Ohio State University	Some recent works for the Cauchy-Riemann problem on distributions	Xianghong
We, Apr 30	Simon Marshall	UW Madison

Abstracts

Thierry Laurens

Title: A priori estimates for generalized KdV equations in H^{-1}

Abstract: We will discuss local-in-time a priori estimates in H^{-1} for a family of generalized KdV equations on the line. This is the first estimate for any non-integrable perturbation of KdV that matches the regularity of the sharp well-posedness theory for KdV. In particular, our analysis applies to models for long waves in a shallow channel of water with a variable bottom. This is joint work with Mihaela Ifrim.

Sergey Denisov

Title: Dispersion effects for evolution equations and the case for Brascamp-Lieb bounds

Abstract: For the Schrödinger time-dependent evolution on the torus, Bourgain proved that the Sobolev norms can grow only very slowly in time if the potential is smooth. Our motivation is to study the energy transfer phenomenon in the Euclidean case for rough potentials and generic values of a parameter. To control the main term in the Duhamel expansion, we apply the modified wave packet decomposition. If time allows, the role of Brascamp-Lieb bounds in that analysis will be explained.

Dallas Albritton

Title: Progress Report on Instability and Non-uniqueness in Incompressible Fluids

Abstract: Over the past decade, mathematical fluid dynamics has seen remarkable progress in an unexpected direction: non-uniqueness of solutions to the fundamental PDEs of incompressible fluids, namely, the Euler and Navier-Stokes equations. I will explain the state-of-the-art in this direction, with a particular focus on the relationship between instability and non-uniqueness, including our proof with E. Brue and M. Colombo that Leray-Hopf solutions of the forced Navier-Stokes equations are not unique. Time permitting, I will present forthcoming work with Colombo and Mescolini in which we investigate the non-existence of a selection principle for the forced 2D Euler equations in the vanishing viscosity limit.

Xiaojun Huang

Title: Bounding a Levi-flat hypersurface in a Stein manifold

Abstract:  Let M be a smooth real codimension two compact submanifold in a Stein manifold. We will prove the following theorem:  Suppose that M has two elliptic complex tangents and suppose CR points are non-minimal. Assume further that M is contained in a bounded strongly pseudoconvex domain. Then M bounds a unique smooth up to M Levi flat hypersurface  \widehat{M} that is foliated by Stein subspace diffeomorphic to the ball. Moreover, \widehat{M} is the hull of holomorphy of M.  This subject has a long history of investigation from E. Bishop and Harvey-Lawson. I will discuss both the history and techniques used in the proof of the above mentioned theorem.

Galia Dafni

Title: Some function spaces defined using means and mean oscillation

While the space BMO of functions of bounded mean oscillation is well-known, there are some lesser-known spaces which are defined using mean oscillation, including the John-Nirenberg (JN) spaces, and the weak-type spaces on the upper half-space recently considered by Frank. After introducing these, we will look at their relationship with other known spaces and introduce dyadic analogues and variations involving means instead of mean oscillation. This is joint work with S. Shaabani.

Manik Dhar

We are planning to reschedule the March 5 seminar on a later date.

Title: Spread Furstenberg Sets over F_p, Z_p, and R.

Abstract: A Kakeya set in R^n is a set that contains a line segment in every direction. The Kakeya conjecture states that these sets have dimension n (after recent breakthroughs for n=3 by Wang and Zahl, open for n>=4). Over F_q^n, a Kakeya set is similarly defined as containing a line in every direction and is known to have a positive fraction of points for all n. In this talk, we consider some works on a generalization of this problem for higher dimensional flats. Spread Furstenberg sets in F_q^n, R^n, Z_p^n are sets that have large intersections with k flats in every direction. For k>= 2 in F_q^n, Z_p^n, and k >= log_2 n for R^n we will show that these sets are large and give a very simple description of all tight examples. These results over finite fields have recently had applications in the study of lattice coverings and linear hash functions. Based on works with Zeev Dvir, Ben Lund, and Paige Bright.

Brian Simanek

Title: Blaschke Products, Numerical Ranges, and the Zeros of Orthogonal Polynomials

Abstract: Our main object of interest will be the location of zeros of orthogonal polynomials on the unit circle. After recalling some fundamental results, we will discuss more recent developments that relate these zeros to Blaschke products, numerical ranges of matrices, algebraic curves, and quadrature measures. Special attention will be paid to the relationship between paraorthogonal polynomials on the unit circle and Poncelet's Theorem. Some open problems will be mentioned along the way. This is based on joint work with Markus Hunziker, Andrei Martinez-Finkelshtein, Taylor Poe, and Barry Simon.

Shengwen Gan

Title: On local smoothing estimates for wave equations

Abstract: The previously known exponent in the local smoothing estimate is the decoupling exponent p=2(n+1)/(n-1)​. In this talk, I will first introduce local smoothing estimates and compare them with restriction estimates. Then, I will discuss a new method that incorporates wave packet density and refined decoupling, and briefly outline how this approach leads to a local smoothing estimate with an exponent that improves upon the decoupling exponent.

Robert Schippa

Title: Local smoothing for the Hermite wave equation Abstract: We consider L^p-smoothing estimates for the wave equation with harmonic potential. For the proof, we linearize an FIO parametrix, which yields Klein-Gordon propagation with variable mass parameter. We obtain decoupling and square function estimates depending on the mass parameter, which yields local smoothing estimates with sharp loss of derivatives. The obtained range is sharp in 1D, and partial results are obtained in higher dimensions.

Junjie Zhu

Title: Fourier dimension of constant rank hypersurfaces

Abstract: Any hypersurface in $\mathbb{R}^{d+1}$ has a Hausdorff dimension of $d$. However, the Fourier dimension depends on the finer geometric properties of the hypersurface. For example, the Fourier dimension of a hyperplane is 0, and the Fourier dimension of a hypersurface with non-vanishing Gaussian curvature is $d$. Recently, Fraser, Harris, and Kroon showed that the Euclidean light cone in $\RR^{d+1}$ has a Fourier dimension of $d-1$, which leads one to conjecture that the Fourier dimension of a hypersurface equals the number of non-vanishing principal curvatures. We prove this conjecture for all constant rank hypersurfaces. Our method involves substantial generalizations of their strategy.

Ziming Shi

Title: The method of homotopy formula and the Cauchy-Riemann(d-bar) equation on pseudoconvex domains of finite type in $\mathbb C^2$ Abstract: The method of homotopy formula is one of the most powerful theories in several complex variables. In the first part of the talk, we will introduce the basics of the method, as well as some recent advances that allow us to solve a wide range of problems from the d-bar equation to the deformation theory of complex structures on manifolds with boundary. In the second part, we will present a new homotopy formula which yields almost sharp estimates in Sobolev and H\"older-Zygmund spaces for the d-bar equation on finite type domains in $\mathbb C^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main ingredient of our proof is the construction of holomorphic support functions that admit precise boundary estimates when the parameter variable lies in a thin shell outside the domain. If time allows, we will also briefly talk about the promise of our method in solving the d-equation with good estimates on a large class of pseudoconvex domains in $\mathbb C^n$, called the "h-extendable" domains.

Liding Yao

Title: Some recent works for the Cauchy-Riemann problem on distributions Abstract: The Cauchy-Riemann problem, also known as the $\overline\partial$-problem, is a central problem in several complex variables. It concerns the regularity estimates to the equation $\overline\partial u=f$ on forms in a certain bounded domain $\Omega\subset\mathbb C^n$. We will talk about some background of the $\overline\partial$-regularity theory, and some recent results for solution operators $[f\mapsto u]$ where f is a generic distribution on $\Omega$. Specifically, for every $\overline\partial$ closed form $f$ on $\Omega$, we know there is a $u$ on $\Omega$ such that $\overline\partial u=f$, when $\Omega$ is one of the following domains:

- A planar domain $\Omega\subset\mathbb C^1$ with Lipschitz boundary (this is a well-known result from the past);

- A strongly pseudoconvex domain with $C^2$ boundary;

- A smooth convex domain of finite type;

- The product of the above domains.

Moreover, the regularity estimates for the above cases are optimal.

The talk is based on several different results, including some joint works with Ziming Shi and with Yuan Zhang.