Uniform boundedness in operators parametrized by polynomial curves

Multiple results in harmonic analysis involving integrals of functions over curves (such as restriction theorems, convolution estimates, maximal function estimates or decoupling estimates) depend strongly on the non-vanishing of the torsion of the associated curve.

Over the past years there has been considerable interest in extending these results to a degenerate case where the torsion vanishes at a finite number of points by using the affine arc-length as an alternative integration measure. As a model case, multiple results have been proven in which the coordinate functions of the curve are polynomials. In this case one expects the bounds of the operators to depend only on the degree of the polynomial.

In this talk I will introduce and motivate the concept of affine arclength measure, provide new decomposition theorems for polynomial curves over characteristic zero local fields, and provide some applications to uniformity results for operators in harmonic analysis.

Unique determination of ellipsoids by their dual volumes

Gusakova and Zaporozhets conjectured that ellipsoids in $\mathbb R^n$ are uniquely determined up to an isometry by their intrinsic volumes. In this talk, we will present a solution to the dual problem in all dimensions. We show that an ellipsoid is uniquely determined up to an isometry by its dual Steiner polynomial. We also discuss an alternative proof of the analogous known result of Petrov and Tarasov for classical Steiner polynomials in $\mathbb{R}^3$. This is joint work with S. Myroshnychenko and V. Yaskin.

Sharp inequalities for the mean distance of random points in convex bodies

For a convex body K the mean distance D(K) is the expected Euclidean distance between two independent and uniformly distributed in K random points. In this talk I will present an optimal lower and upper bounds for ratio between D(K) and the first intrinsic volume of K (normalized mean width). We will discuss the sharpness of obtained estimates by considering the degenerate extremal cases. This is a joint work with Gilles Bonnet, Christoph Thäle and Dmitry Zaporozhets (arXiv:2010.03351).

Refined Restricted Invertibility

In this talk, we will discuss a further refinement of the restricted invertibility principle first put forward by Bourgain and Tzafriri. Namely, we will show that any full rank matrix has a large submatrix whose smallest singular value is of the same order as the harmonic average of all singular values. We will also investigate the relation to the problem of estimating the Banach-Mazur distance to the cube. Joint work with Assaf Naor.

Strongly Convex Chains

It is a classical question to study the length of the longest monotone increasing subsequence in a random permutation on n elements, which has been studied for over half a century. From the geometric viewpoint, the question asks for the maximal number of points in a random sample of n uniform, independent points in a unit square which form an increasing chain. Based on this geometric intuition, one may study the maximal number of points (called the length) which form a convex chain, along with two fixed vertices of the unit square. In a joint work with Imre Bárány, we determined the asymptotic order of magnitude of the length of the longest convex chain, proved strong concentration estimates and a limit shape result. In a recent work, I studied the analogous question for higher order convexity, and managed to determine the expected length in this case as well (which turns out to be very aesthetic), along with concentration properties. In the talk I will give a survey of these results and present several open questions and further research directions.

FKP meets DKP

In the 80’s Dahlberg asked two questions regarding the `$L^p$ – solvability’ of elliptic equations with variable coefficients. Dahlberg’s first question was whether $L^p$ solvability was maintained under `Carleson-perturbations’ of the coefficients. This question was answered by Fefferman, Kenig and Pipher [FKP], where they also introduced new characterizations of $A_\infty$, reverse-Hölder and $A_p$ weights. These characterizations were used to create a counterexample to show their theorem was sharp.

Dahlberg’s second question was whether a Carleson gradient/oscillation condition (the `DKP condition’) was enough to imply $L^p$ solvability for some p > 1. This was answered by Kenig and Pipher [KP] and refined by Dindos, Petermichl and Pipher [DPP] (in the `small constant’ case). These $L^p$ solvability results can be interpreted in terms of a reverse Hölder condition for the elliptic kernel and therefore connected with the $A_\infty$ condition. In this talk, we discuss L^p solvability for a class of coefficients that satisfies a `weak DKP condition’. In particular, we connect the (weak) DKP condition to the characterization of $A_\infty$ in [FKP]. This allows us to treat the `large’, `small’ and ‘vanishing’ (weak) DKP conditions simultaneously and independently from the works [KP] and [DPP].

This is joint work with my co-authors Egert, Saari, Toro and Zhao. A proof of the main estimate will be sketched, but technical details will be avoided.

Time: 3 PM (NY time)
Zoom ID: 989 6529 0738
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Sharp L^p estimates for oscillatory integral operators of arbitrary signature

The restriction problem in harmonic analysis asks for L^p bounds on the Fourier transform of functions defined on curved surfaces. In this talk, we will present improved restriction estimates for hyperbolic paraboloids, that depend on the signature of the paraboloids. These estimates still hold, and are sharp, in the variable coefficient regime. This is joint work with Jonathan Hickman

Time: 3 PM (NY time)
Zoom ID: 920 5577 0985
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Strong asymptotic freeness for independent uniform variables on compact groups

Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this talk, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature, i.e. two finite sequences of non-increasing natural numbers, and for n large enough, consider the irreducible representation of Un associated to this signature. We show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. This is a joint work with Benoît Collins.

Time: 3 PM (NY time)
Zoom ID: 872 378 5801
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The non-resonant bilinear Hilbert-Carleson operator

We introduce a new class of bilinear operators BC_a acting as a merger between two classical objects in harmonic analysis : the bilinear Hilbert transform and the linear Carleson-Stein-Wainger operator. The two opposing features (modulation invariance versus modulation of the kernel by a monomial phase with space-depending coefficients) of BC_a require a two-resolutions analysis and the use of a dilated time-frequency portrait. This is joint work with F. Bernicot, V. Lie, M. Vitturi.

Time: 3 PM (NY time)
Zoom ID: 980 3116 7550
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Variational estimates for martingale transforms

I will present Lp estimates for joint rough path lifts of martingales and deterministic paths. For motivation, I will also present some rudiments of rough integration theory, which is the deterministic version of stochastic integration.

Time: 3 PM (NY time)
Zoom ID: 989 6529 0738
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Singular Integral Operators on the Fock Space

In this talk we will discuss the recent solution of a question raised by K. Zhu about characterizing a class of singular integral operators on the Fock space. We show that for an entire function $\varphi$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator

\begin{eqnarray*}

S_{\varphi}F(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} \varphi(z- \bar{w})\,d\lambda(w), \quad z\in\mathbb{C}^n,

\end{eqnarray*}

is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that

$$

\varphi(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)^2} dx, \quadz\in\mathbb{C}^n.

$$

Here $d\lambda(w)=\pi^{-n}e^{-\left\vert w\right\vert^2}dw$ is the Gaussian measure on $\mathbb C^n$.

With this characterization we are able to obtain some fundamental results of the operator $S_\varphi$, including the normality, the $C^*$ algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of $S_{\varphi}$.

In particular, in the case $n=1$, this gives a complete solution to the question proposed by K. Zhu for the Fock space ${\mathscr F}^2(\mathbb{C})$

on the complex plane ${\mathbb C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).

This talk is based on joint work with Guangfu Cao, Ji Li, Minxing Shen, and Lixin Yan.

Time: 3 PM (NY time)
Zoom ID: 928 3597 1183
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Good and Bad Maximal Functions

In a joint work with Nazarov, Skreb and Treil, we highlight a marked difference in the presence of a matrix weight between the Doob type maximal operator in the dyadic setting and the dyadic Hardy-Littlewood type maximal operator. The former is $L^2$ bounded while the latter is not.

Time: 2 PM (NY time)
Zoom ID: 980 3116 7550
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Simultaneous dilation and translation tilings of $\R^n$

In this talk we present a solution of the wavelet set problem. That is, we characterize full-rank lattices $\Gamma\subset \R^n$ and invertible $n \times n$ matrices $A$ for which there exists a measurable set $W$ such that $\{W + \gamma: \gamma \in \Gamma\}$ and $\{A^j(W): j\in \Z\}$ are tilings of $\R^n$. The characterization is a non-obvious generalization of the one found by Ionascu and Wang, which solved the problem in the case $n = 2$. As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showing that wavelet sets exist for matrix dilations, all of whose eigenvalues $\lambda$ satisfy $|\lambda| \ge 1$. As another application, we show that the Ionascu-Wang characterization characterizes those dilations whose product of two smallest eigenvalues in absolute value is $\ge 1$. Based on a joint work with Darrin Speegle.

Time: 3 PM (NY time)
Zoom ID: 989 6529 0738
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Large deviations for triangle densities

Take a uniformly random graph with a fixed edge density e. Its triangle density will typically be about e^3, and we are interested in the large deviations behavior: what's the probability that the triangle density is about e^3 - delta? The general theory for this sort of problem was studied by Chatterjee-Varadhan and Dembo-Lubetzky, who showed that the solution can be written in terms of an optimization over certain integral kernels. This optimization is difficult to solve explicitly, but Kenyon, Radin, Ren and Sadun used numerics to come up with a fascinating and intricate set of conjectures regarding both the probabilities and the structures of the conditioned random graphs. We prove these conjectures in a small region of the parameter space. Joint work with Charles Radin and Lorenzo Sadun.

Time: 3 PM (NY time)
Zoom ID: 970 5223 9016
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Spectral gap in random regular graphs and hypergraphs

Random graphs and hypergraphs are models of choice for a variety of machine learning problems, from algorithms testing and benchmarking to understanding the kind of conditions one must impose in order to obtain guarantees for such algorithms. In particular, regular graphs and hypergraphs have desirable properties like expansion, which is connected to rapid mixing, and which is controlled by the spectral gap (the distance between the Perron-Frobenius eigenvalue and the rest of the others). I will explain how we know that these regular (hyper)graphs are expanders, and mention some of the proof techniques and tools that we use to determine that.