Professor Oscar Domingues Bonilla, The University of Lyon
Title: John–Nirenberg spaces revisited
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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Professor Alex Iosevich, University of Rochester
Title: Finite point configurations and the Vapnik-Chervonenkis dimension
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 Online Seminar Seminar Series
Sagun Chanillo, Rutgers University
Title: Local Version of Courant's Nodal domain theorem.
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.