Zoom link. https://umn.zoom.us/j/95724260234

Title. Lojasiewicz Inequalities and the Zero Sets of Harmonic Functions

Abstract. Whereas $C^\infty$ functions can vanish (almost) arbitrarily often to arbitrarily high order (e,g, $f(x) = e^{-1/x}$ vanishes to infinite order at zero), the zero sets of analytic functions have a lot more structure. For example, you learn in intro to complex analysis that the zeroes of a Holomorphic function are isolated.

The Lojasiewicz inequalities (partially) quantify this extra structure possessed by analytic functions. Developed originally by algebraic geometers, Lojasiewicz inequalities have been used with great success to study geometric flows. In this talk, I will give a brief introduction to these inequalities and then discuss some joint work (and maybe some work in progress) with Matthew Badger (UConn) and Tatiana Toro (U Washington), in which we use Lojasiewicz inequalities to study the zero sets of harmonic functions and, more interestingly, sets which are infinitesimally approximated by the zero sets of harmonic functions.

Zoom link. https://umn.zoom.us/j/97330666898

Title. Sparse bounds for the Discrete Spherical Maximal Function

Abstract. We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.

Zoom link. https://umn.zoom.us/j/97809481449

Title. New Developments in Parabolic Uniform Rectifiability

Abstract. In the 1980’s the $L^2$ boundedness of the Cauchy integral was established (by Coifman, McIntosh and Meyer) and this $L^2$ boundedness was quickly generalized to `nice’ singular integral operators (Coifman, David and Meyer / David) on Lipschitz graphs. David and Semmes then asked the natural question: For what sets are all `nice’ singular integral operators $L^2$ bounded? They were remarkably successful in this endeavor, providing more than 15 equivalent notions and called these sets “uniformly rectifiable” or UR. These sets are still studied extensively today, most recently in their connection to elliptic partial differential equations.

Among the characterizations of UR sets provided by David and Semmes is a quadratic estimate on the so-called $\beta$-numbers, which measure the flatness of the set at a particular location and scale. In a paper of Hofmann, Lewis and Nyström a notion of “parabolic uniform rectictifiable sets” was introduced by taking the definition as a “quadratic estimate on the parabolic $\beta$-numbers”. There are no correct proofs of ANY of the analogues of the David Semmes theory; for instance, it is not known if $L^2$ boundedness of parabolic singular integral operators characterizes parabolic uniformly rectifiable sets.

In this talk I will discuss some recent progress in the direction of establishing the parabolic David-Semmes theory and some open problems that remain. On one hand, we have made significant progress and provided some useful characterizations of parabolic uniform rectifiability. On the other hand, we have also discovered that many of the `elliptic’ characterizations do not hold in this parabolic setting. This is joint work with J. Hoffman, S. Hofmann, J.L. Luna and K. Nyström.

Zoom link. https://umn.zoom.us/j/91341120335

Title. Weak endpoint estimates for Calderón-Zygmund operators in von Neumann algebras

Abstract. The classical Calderón-Zygmund decomposition is a fundamental tool that helps one study endpoint estimates for numerous operators near L1. In this talk, we will discuss an extension of the decomposition to a particular operator valued setting where noncommutativity makes its appearance. Noncommutativity will allow us to get rid of the -usually necessary- UMD property of the Banach space where functions take values. Based on joint work with L. Cadilhac and J. Parcet.

Zoom link. https://umn.zoom.us/j/95248478720

Title. The uncentered spherical maximal function and Nikodym sets

Abstract. Stein's spherical maximal function is an analogue of the Hardy-Littlewood maximal function, where the averages are taken over spheres instead of balls. While the uncentered Hardy-Littlewood maximal function is bounded on Lp for all p>1 and pointwise equivalent to its centered counterpart, the corresponding uncentered spherical maximal function is not as well-behaved.

We provide multidimensional versions of the Kakeya, Nikodym, and Besicovitch constructions associated with spheres. These yield counterexamples indicating that maximal operators given by translations of spherical averages are unbounded on Lp for all finite p.

However, for lower-dimensional sets of translations, we obtain Lp boundedness for the associated maximally translated spherical averages for a certain range of p that depends on the Minkowski dimension of the set of translations. This is joint work with A. Chang and J. Kim.

Zoom link. https://umn.zoom.us/j/92095338432

Title. A quantification of the Besicovitch projection theorem and its generalizations

Abstract. The Besicovitch projection theorem asserts that if a subset E of the plane has finite length in the sense of Hausdorff and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every linear projection of E to a line will have zero measure. As a consequence, the probability that a randomly dropped line intersects such a set E is equal to zero. This shows us that the Besicovitch projection theorem is connected to the classical Buffon needle problem. Motivated by the so-called Buffon circle problem, we explore what happens when lines are replaced by more general curves. This leads us to discuss generalized Besicovitch theorems and the ways in which we can quantify such results by building upon the work of Tao, Volberg, and others. This talk covers joint work with Laura Cladek and Krystal Taylor.

Zoom link. https://umn.zoom.us/j/91471008662

Title. The Analysts' Traveling Salesman Problem in Banach spaces

Abstract. This talk discusses recent work (joint with Matthew Badger, UCONN) on generalizations of the Analysts' Traveling Salesman Theorem to uniformly smooth and uniformly convex Banach spaces (e.g., l_p spaces). In 1990, motivated by problems in Singular Integral Operators, Peter Jones posed and solved his celebrated Analysts' Traveling Salesman Problem: namely, to characterize all subsets of rectifiable curves in the plane. Since then, many authors have contributed, proving similar results in Euclidean spaces, Hilbert Spaces, Carnot groups, for 1-rectifiable measures, etc. This talk will give a broad overview of some of these results and their core ideas. In the end, we will discuss the challenges in Banach spaces and what generalizations hold there. This talk will include lots of pictures and examples.

Zoom link. https://umn.zoom.us/j/97214138395

Title. Mathematical foundations of slender body theory

Abstract. Slender body theory (SBT) facilitates computational simulations of thin filaments in a 3D viscous fluid by approximating the hydrodynamic effect of each fiber as the flow due to a line force density along a 1D curve. Despite the popularity of SBT in computational models, there had been no rigorous analysis of the error in using SBT to approximate the interaction of a thin fiber with fluid. In this talk, we develop a PDE framework for analyzing the error introduced by this approximation. In particular, given a 1D force along the fiber centerline, we define a notion of `true' solution to the full 3D slender body problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing. In addition, we perform a complete spectral analysis of the slender body PDE in a simple geometric setting, which sheds light on the use of SBT in approximating the `slender body inverse problem,' where we instead specify the fiber velocity and solve for the 1D force density. Finally, we make some comparisons to the method of regularized Stokeslets and offer thoughts on improvements to SBT.

Zoom link. https://umn.zoom.us/j/97459532983

Title. A Minkowski-type problem for measure associated to A-harmonic PDEs

Abstract. The classical Minkowski problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness, and regularity.

In this talk, we study a Minkowski problem for certain measure associated with a compact convex set E with nonempty interior and its A-harmonic capacitary function in the complement of E. Here A-harmonic PDE is a non-linear elliptic PDE whose structure is modelled on the p-Laplace equation. If \mu_E denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure \mu on S^(n-1), find necessary and sufficient conditions for which there exists E as above with \mu_E =\mu. We will discuss the existence, uniqueness, and regularity of this problem in this setting.

Zoom link. https://umn.zoom.us/j/96861877118

Title. On the compactness threshold in the critical Kirchhoff equation

Abstract. We study a class of critical Kirchhoff problems with a general nonlocal term. The main difficulty here is the absence of a closed-form formula for the compactness threshold. First we obtain a variational characterization of this threshold level. Then we prove a series of existence and multiplicity results based on this variational characterization.

Zoom link. https://umn.zoom.us/j/91459188075

Title. Regularized Distances and Geometry of Measures

Abstract. I will present joint work with Max Engelstein and Svitlana Mayboroda where we generalize the notion of the regularized distance function $D_{\mu, \alpha}(x) = ( \int | x - y |^{-d-\alpha} d\mu(y))^{-1/ \alpha}$ to functions with more general integrands. We provide a large class of integrands for which the corresponding distance functions contain geometric information about $\mu$. In particular, we produce examples that are in some sense far from the original kernel $| x - y|^{-d - \alpha}$ but still characterize the geometry of $\mu$ since they have nice symmetries with respect to flat sets. In co-dimension 1, these examples are explicit, but in higher co-dimensions, our proof of existence of such examples is non-constructive, and thus we have no additional information about their structure.

Zoom link. https://umn.zoom.us/j/96811202870

Title. Carleson measure estimates for the Green function

Abstract. We are interested in the relations between an elliptic operator on a domain, the geometry of the domain, and the boundary behavior of the Green function. In joint work with Guy David and Svitlana Mayboroda, we show that if the coefficients of the operator satisfy a quadratic Carleson condition, then the Green function on the half-space is almost affine, in the sense that the normalized difference between the Green function with a sufficiently far away pole and a suitable affine function at every scale satisfies a Carleson measure estimate. We demonstrate with counterexamples that our results are optimal, in the sense that the class of the operators considered are essentially the best possible.

This work is motivated mainly by finding PDE characterizations of uniform rectifiable sets with higher co-dimension. I’ll talk about this motivation and backgrounds, our recent results, as well as possible directions in the future.

Zoom link. https://umn.zoom.us/j/95260802468

Title. Mean-Value Inequalities for Harmonic Functions

Abstract. The mean-value theorem for harmonic functions says that we can bound the integral of a harmonic function in a ball by the average value on the boundary (and, in fact, there is equality). What happens if we replace the ball by a general convex or even non-convex set? As it turns out, this simple question has connections to classical potential theory, probability theory, PDEs and even mechanics: one of the arising questions dates back to Saint Venant (1856). There are some fascinating new isoperimetric problems: for example, the worst case convex domain in the plane seems to look a lot like the letter "D" but we cannot prove it. I will discuss some recent results and many open problems.