2021-11-30
Online via Zoom

### 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.

2021-11-24
Online via Zoom

### 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.

2021-11-16
Online via Zoom

### 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.

2021-11-09
Online via Zoom

### 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.

2021-11-02
Online via Zoom

2021-10-26
Online via Zoom

### 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).

2021-10-19
Online via Zoom

2021-10-12
Online via Zoom

### 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.

2021-10-05

### 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.

2021-05-20