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 https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars   https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars 

 
 =Abstracts=
 
 ===Alexei Poltoratski===
 

 
 Title: Dirac inner functions
 

 
 Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.
 
 We will discuss connections between problems in complex function theory, spectral and scattering problems for differential
 
 operators and the nonlinear Fourier transform.
 

 
 ===Polona Durcik and Joris Roos===
 

 
 Title: A triangular Hilbert transform with curvature, I & II.
 

 
 Abstract: The triangular Hilbert is a twodimensional bilinear singular
 
 originating in timefrequency analysis. No Lp bounds are currently
 
 known for this operator.
 
 In these two talks we discuss a recent joint work with Michael Christ
 
 on a variant of the triangular Hilbert transform involving curvature.
 
 This object is closely related to the bilinear Hilbert transform with
 
 curvature and a maximally modulated singular integral of SteinWainger
 
 type. As an application we also discuss a quantitative nonlinear Roth
 
 type theorem on patterns in the Euclidean plane.
 
 The second talk will focus on the proof of a key ingredient, a certain
 
 regularity estimate for a local operator.
 

 
 ===Andrew Zimmer===
 

 
 Title: Complex analytic problems on domains with good intrinsic geometry
 

 
 Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many wellstudied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is nonsmooth. In particular, it is possible to characterize the domains in this family where the dbarNeumann operator on (0, q)forms is compact (which generalizes an old result of FuStraube for convex domains).
 

 
 ===Hong Wang===
 

 
 Title: Improved decoupling for the parabola
 

 
 Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$.
 
 We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.
 

 
 ===Kevin Luli===
 

 
 Title: Smooth Nonnegative Interpolation
 

 
 Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Nonnegativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets.
 

 
 ===Niclas Technau===
 

 
 Title: Number theoretic applications of oscillatory integrals
 

 
 Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding finescale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.
 

 
 ===Terence Harris===
 

 
 Title: Low dimensional pinned distance sets via spherical averages
 

 
 Abstract: An inequality is derived for the average tenergy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.
 

 
 ===Yuval Wigderson===
 

 
 Title: New perspectives on the uncertainty principle
 

 
 Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and wellstudied notions include the variance (and higher moments), the entropy, the supportsize, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.
 

 
 ===Oscar Dominguez===
 

 
 Title: New BrezisVan SchaftingenYung inequalities via maximal operators, Garsia inequalities and CaffarelliSilvestre extensions
 

 
 Abstract: The celebrated BourgainBrezisMironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weaktype Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higherorder Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsiatype inequalities and an application of the CaffarelliSilvestre extension. This is joint work with Mario Milman.
 

 
 ===Tamas Titkos===
 

 
 Title: Isometries of Wasserstein spaces
 

 
 Abstract: Due to its nice theoretical properties and an astonishing number of
 
 applications via optimal transport problems, probably the most
 
 intensively studied metric nowadays is the pWasserstein metric. Given
 
 a complete and separable metric space $X$ and a real number $p\geq1$,
 
 one defines the pWasserstein space $\mathcal{W}_p(X)$ as the collection
 
 of Borel probability measures with finite $p$th moment, endowed with a
 
 distance which is calculated by means of transport plans \cite{5}.
 

 
 The main aim of our research project is to reveal the structure of the
 
 isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although
 
 $\mathrm{Isom}(X)$ embeds naturally into
 
 $\mathrm{Isom}(\mathcal{W}_p(X))$ by pushforward, and this embedding
 
 turned out to be surjective in many cases (see e.g. [1]), these two
 
 groups are not isomorphic in general. Kloeckner in [2] described
 
 the isometry group of the quadratic Wasserstein space
 
 $\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$
 
 is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$
 
 is extremely rich. Namely, it contains a large subgroup of wild behaving
 
 isometries that distort the shape of measures. Following this line of
 
 investigation, in \cite{3} we described
 
 $\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and
 
 $\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.
 

 
 In this talk I will survey first some of the earlier results in the
 
 subject, and then I will present the key results of [3]. If time
 
 permits, I will also report on our most recent manuscript [4] in
 
 which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)
 
 and D\'aniel Virosztek (IST Austria).
 

 
 [1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein
 
 spaces: isometric rigidity in negative curvature}, International
 
 Mathematics Research Notices, 2016 (5), 13681386.
 

 
 [2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean
 
 spaces}, Annali della Scuola Normale Superiore di Pisa  Classe di
 
 Scienze, Serie 5, Tome 9 (2010) no. 2, 297323.
 

 
 [3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of
 
 Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373
 
 (2020), 58555883.
 

 
 [4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of
 
 Wasserstein spaces: The Hilbertian case}, submitted manuscript.
 

 
 [5] C. Villani, \emph{Optimal Transport: Old and New,}
 
 (Grundlehren der mathematischen Wissenschaften)
 
 Springer, 2009.
 

 
 ===Shukun Wu===
 

 
 Title: On the BochnerRiesz operator and the maximal BochnerRiesz operator
 

 
 Abstract: The BochnerRiesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the BochnerRiesz conjecture and the maximal BochnerRiesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem.
 

 

 
 ===Jonathan Hickman===
 

 
 Title: Sobolev improving for averages over space curves
 

 
 Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^pSobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some nontrivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.
 

 
 ===Hanlong Fang===
 

 
 Title: Canonical blowups of Grassmann manifolds
 

 
 Abstract: We introduce certain canonical blowups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi)positivity of the anticanonical bundles, the existence of KählerEinstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification.
 

 
 ===Bingyang Hu===
 

 
 Title: Some structure theorems on general doubling measures.
 

 
 Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are padic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly nontrivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.
 

 
 ===Krystal Taylor===
 

 
 Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting
 

 
 Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections.
 
 It turns out that many nonlinear projectiontype operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multiscale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projectiontype operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the fourcorner Cantor set.
 

 
 ===Dominique Maldague===
 

 
 Title: A new proof of decoupling for the parabola
 

 
 Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and inductiononscales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.
 

 
 ===Diogo Oliveira e Silva===
 

 
 Title: Global maximizers for spherical restriction
 

 
 Abstract: We prove that constant functions are the unique realvalued maximizers for all $L^2L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying EulerLagrange equation being smooth, a fact of independent interest which we discuss. We further show that complexvalued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.
 

 
 ===Oleg Safronov===
 

 
 Title: Relations between discrete and continuous spectra of differential operators
 

 
 Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other
 
 by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.
 

 
 ===Ziming Shi===
 

 
 Title: Sharp Sobolev $1/2$estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary
 

 
 Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$boundary, for all $1< p < \infty$ and $s>1/p$.
 
 We also show that the same solution operator gains a $1/2$derivative in the H\"olderZygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work of X. Gong.
 
 The main ingredients used in our proof are a HardyLittlewood lemma of Sobolev type and a new commutator estimate.
 
 Joint work with Liding Yao.
 

 
 ===Xiumin Du===
 

 
 Title: Falconer's distance set problem
 

 
 Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.
 

 
 ===Etienne Le Masson===
 

 
 Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces
 

 
 Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces BenjaminiSchramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the WeilPetersson model of random surfaces introduced by Mirzakhani. They also apply to congruence covers of the modular surface, where we recover a result of Nelson on the equidistribution of Maass forms (with weaker convergence rate). The proof is based on ergodic theory methods.
 
 Joint work with Tuomas Sahlsten.
 

 
 ===Theresa Anderson===
 

 
 Title: Dyadic analysis (virtually) meets number theory
 

 
 Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first, which we will spend the most time motivating and discussing, involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. If time remains, secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "padic" doubling for any prime p (in a finite set of primes), yet not doubling overall.
 

 
 ===Nathan Wagner===
 

 
 Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness
 

 
 Abstract: The Bergman and Szegő projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the BékolléBonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operatortheoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weighted weaktype (1,1) estimates. Here we use a modified version of singularintegral theory and a generalization of the RieszKolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale.
 

 
 ===David Beltran===
 

 
 Title: Sobolev improving for averages over curves in $\mathbb{R}^4$
 

 
 Abstract: Given a smooth nondegenerate space curve (that is, a smooth curve whose n1 curvature functions are nonvanishing), it is a classical question to study the smoothing properties of the averaging operators along a compact piece of such a curve. This question can be quantified, for example, by studying the $L^p$Sobolev mapping properties of those operators. These are well understood in 2 and 3 dimensions, and in this talk, we present a new sharp result in 4 dimensions. We focus on the positive results; the nontrivial examples which show that our results are best possible were presented by Jonathan Hickman in December 1st. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.
 

 
 ===Yumeng Ou===
 

 
 Title: On the multiparameter distance problem
 

 
 Abstract: In this talk, we will describe some recent progress on the Falconer distance problem in the multiparameter setting. The original Falconer conjecture (open in all dimensions) says that a compact set $E$ in $\mathbb{R}^d$ must have a distance set $\{xy: x,y\in E\}$ with positive Lebesgue measure provided that the Hausdorff dimension of $E$ is greater than $d/2$. What if the distance set is replaced by a multiparameter distance set? We will discuss some recent work on this problem, which also includes some new results on the multiparameter radial projection theory of fractal measures. This is joint work with Xiumin Du and Ruixiang Zhang.
 

 

 =Extras=   =Extras= 