Past Events

FALL 2021 Nonlinear Analysis Seminar Series

2021-11-30
Online via Zoom

Dr. Po Lam Yung,  Australian National University

Title: Sobolev norms revisited

Abstract:

In this talk, we will describe some new ways of characterising Sobolev norms, using sizes of superlevel sets of suitable difference quotients. They provide remedy in certain cases where some critical Gagliardo-Nirenberg interpolation inequalities fail, and lead us to investigate real interpolations of certain fractional Besov spaces. Some connections will be drawn to earlier work by Bourgain, Brezis and Mironescu. Joint work with Haim Brezis, Jean Van Schaftingen, Qingsong Gu, Andreas Seeger and Brian Street.

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FALL 2021 Nonlinear Analysis Special Lecture Part 1 of 3

2021-11-24
Online via Zoom

Wednesday 10th November 2021, 10:00–11:00 JST (UTC+9), online on Zoom

Associate Professor Kabe Moen, The University of Alabama

Title: Fractional Integrals and weights Part I

Abstract:

I will introduce fractional integral operator and its related maximal operator. After developing some of the relevant background, we will discuss its boundedness on Lebesgue spaces and various related inequalities of Hedberg and Welland. We will also cover endpoint bounds and applications to Sobolev-Poincare inequalities.

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FALL 2021 Nonlinear Analysis Seminar Series

2021-11-16
Online via Zoom

Professor Galia Dafni,  Concordia University

Title: Boundedness and continuity of rearrangements in BMO and VMO

Abstract:

Joint work with Almut Burchard (Toronto) and Ryan Gibara (Cincinnati). Let \(f\) be a function of bounded mean oscillation (BMO) on cubes in \(\mathbb{R}^n\), \(n > 1\). If \(f\) is rearrangeable, we show that its symmetric decreasing rearrangement\(Sf\) belongs to \(\mathrm{BMO}(\mathbb{R}^n)\). We also improve the bounds for the decreasing rearrangement \(f^*\) by Bennett, DeVore and Sharpley, \(\|f^*\|_{ \mathrm{BMO}(\mathbb{R}_+)} \leq C_n\|f\| _{\mathrm{BMO}(\mathbb{R}^n)}\), by eliminating the exponential dependence of \(C_n\) on the dimension \(n\). The key is to switch from cubes to a comparable family of shapes. Using a family of rectangles that is preserved under bisections, one can prove a dimension-free Calder\'on-Zygmund decomposition, and the boundedness of the decreasing rearrangement with the same constant. Restricting to the subspace of functions of vanishing mean oscillation (VMO), we show that these rearrangements take VMO functions to VMO functions. Furthermore, while the map from \(f\) to \(f^*\) is not continuous in the BMO seminorm, we prove continuity when the limit is in VMO.

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FALL 2021 Nonlinear Analysis Seminar Series

2021-11-09
Online via Zoom

Tuesday 9th November 2021, 16:00–17:00 JST (UTC+9), online on Zoom

Professor Denis Serre,  The UMPA

Title:  Compensated integrability: classical and singular Divergence-BV symmetric tensors

Abstract:

Compensated Integrability is a recent tool of Functional Analysis, which extends both the Gagliardo Inequality and the Isoperimetric Inequality. It concerns the determinant of positive symmetric tensors whose row-wise Divergence is controlled in the space of bounded measures. It is somehow dual to Brenier's Theorem of Optimal Transport. Its applications cover several domains in Mathematical Physics and in Differential Geometry.

FALL 2021 Nonlinear Analysis Seminar Series

2021-11-02
Online via Zoom

Professor Itai Shafrir,  Technion-Israel Institute of Technology

Title: Minimizers of a variational problem for nematic liquid crystals with variable degree of orientation in two dimensions

FALL 2021 Nonlinear Analysis Seminar Series

2021-10-26
Online via Zoom

Professor Oscar Domingues Bonilla, The University of Lyon

Title: John–Nirenberg spaces revisited

Abstract:

We study John—Nirenberg-type spaces where oscillations of functions are controlled via covering lemmas. Our methods give new surprising results and clarify classical inequalities. Joint work with Mario Milman (Florida and Buenos Aires).

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FALL 2021 Nonlinear Analysis Seminar Series

2021-10-19
Online via Zoom

Dr. João Pedro Ramos, Eidgenössische Technische Hochschule Zürich

Title: STABILITY FOR GEOMETRIC AND FUNCTIONAL INEQUALITIES

Abstract

FALL 2021 Nonlinear Analysis Seminar Series

2021-10-12
Online via Zoom

Professor Alex Iosevich, University of Rochester

Title: Finite point configurations and the Vapnik-Chervonenkis dimension

Abstract:

The Vapnik-Chervonenkis (VC) dimension was invented in 1970 to study learning models. This notion has since become one of the cornerstones of modern data science. This beautiful idea has also found applications in other areas of mathematics. In this talk we are going to describe how the study of the VC-dimension in the context of families of indicator functions of spheres centered at points in sets of a given Hausdorff dimension (or in sets of a given size inside vector spaces over finite fields) gives rise to interesting, and in some sense extremal, point configurations. 

 

FALL 2021 Nonlinear Analysis Seminar Series

2021-10-05

FALL 2021 Nonlinear Analysis Online Seminar Seminar Series

Sagun Chanillo, Rutgers University

Title: Local Version of Courant's Nodal domain theorem.

Abstract:

Consider a smooth, compact Riemannian manifold with no boundary, endowed with a smooth metric. A famous theorem of Courant states that the k-th eigenfunction for the Laplace-Beltrami operator can have at most k nodal domains. Nodal domains are the open and connected sets where the eigenfunction does not vanish. H. Donnelly and Fefferman obtained some 30 years ago a local version of this theorem. Improvements were made by Chanillo-Muckenhoupt and others. In this talk we obtain the optimal local version of the local Courant theorem. We also relate this result to conjectures of S.-T. Yau on nodal sets, that is the zero set of eigenfunctions. The results of our talk have been obtained jointly with A. Logunov, E. Mallinikova and D. Mangoubi.

SPRING 2021 Nonlinear Analysis Seminar Series

2021-05-20

Professor Jan Kristensen,  University of Oxford

Title: Regularity and uniqueness results in some variational problems

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