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Stine Marie-Berge, Leibniz Universit盲t Hannover (1/3), Antoine M茅tras, Universit茅 de Montr茅al (2/3), Theo McKenzie, UC Berkeley (3/3)

(1/3) Title: Convexity Properties for Harmonic Functions on Riemannian Manifolds

Abstract: In the 70鈥檚 Almgren noticed that for a harmonic real-valued function defined on a ball, its L 2 -norm over a sub-sphere will have an increasing logarithmic derivative with respect to the radius of mentioned sphere. We examined similar integrals over a more general class of parameterized surfaces by studying harmonic functions defined on compact subdomains of Riemannian manifolds. The integrals over spheres are also generalized to level sets of a given function satisfying certain conditions. If we consider the L 2 norms over these level sets parametrized by a generalization of the radius, we again reproduce Almgren鈥檚 convexity property. We will sketch the proof of this result and illustrate the usefulness of the convexity result by examining some explicit parameterized families of surfaces, e.g. geodesic spheres and ellipses. (2/3) Title: Steklov conformally extremal metrics in higher dimensions Abstract: Steklov extremal metrics on surfaces have been much studied due to their connection to free-boundary minimal surfaces found by Fraser and Schoen. In this talk, I will present a characterization of higher dimensional Steklov conformally extremal metrics, highlighting its similarities with the same problem for Laplace eigenvalues. To this end, I will answer the question of which normalization to use and show how the Steklov problem with boundary density appears natural in this context. This is joint work with Mikhail Karpukhin. (3/3) Title: Many nodal domains in random regular graphs Abstract: If we partition a graph according to the positive and negative components of an eigenvector of the adjacency matrix, the resulting connected subcomponents are called nodal domains. Examining the structure of nodal domains has been used for more than 150 years to deduce properties of eigenfunctions. Dekel, Lee, and Linial observed that according to simulations, most eigenvectors of the adjacency matrix of random regular graphs have many nodal domains, unlike dense Erd藵os-R麓enyi graphs. In this talk, we show that for the most negative eigenvalues of the adjacency matrix of a random regular graph, there is an almost linear number of nodal domains. Joint work with Shirshendu Ganguly, Sidhanth Mohanty, and Nikhil Srivastava.

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