Strong asymptotic freeness for independent uniform variables on compact groups

Speaker: 

Charles Bordenave

Institution: 

Institute of Mathematics of Marseille

Time: 

Monday, October 11, 2021 - 12:00pm

Location: 

Zoom

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.

 

To see zoom ID and psscode click on the name of the speaker on the webstie of the seminar:
https://sites.google.com/view/paw-seminar/

Sharp L^p estimates for oscillatory integral operators of arbitrary signature

Speaker: 

Marina Iliopoulou

Institution: 

University of Kent, UK

Time: 

Monday, October 4, 2021 - 12:00pm

Location: 

Zoom

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

 

Zoom ID is available here:
https://sites.google.com/view/paw-seminar

FKP meets DKP

Speaker: 

Simon Bortz

Institution: 

University of Alabama

Time: 

Monday, September 27, 2021 - 12:00pm

Location: 

Zoom

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.

 

https://sites.google.com/view/paw-seminar

Refined Restricted Invertibility

Speaker: 

Pierre Youssef

Institution: 

NYU Abu Dhabi

Time: 

Monday, September 13, 2021 - 12:00pm

Location: 

Zoom

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.

https://sites.google.com/view/paw-seminar

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