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Time and venue: Wednesday 3pm, via Zoom; watch out for Zoom meeting links in announcements emails. Programme: 27 January 2021 Eugenia Malinnikova (Stanford) On Yau's conjecture for the Dirichlet Laplacian on C^1 domains Abstract. Let D be a bounded domain in R^n with C^1 boundary and let u be a Dirichlet Laplace eigenfunction in D with eigenvalue \lambda. We show that the (n-1)-dimensional Hausdorff measure of the zero set of u does not exceed C\sqrt{\lambda}. The opposite estimate follows from the work of Donnelly and Fefferman. The talk is based on a joint work with A. Logunov, N. Nadirashvili, and F. Nazarov. 10 February 2021 Dan Mangoubi (Einstein Institute of Mathematics) A Local version of Courant's Nodal domain Theorem Abstract. Let u_k be an eigenfunction of a vibrating string (with fixed ends) corresponding to the k-th eigenvalue. It is not difficult to show that the number of zeros of u_k is exactly k+1. Equivalently, the number of connected components of the complement of $u_k=0$ is $k$. In 1923 Courant found that in higher dimensions (considering eigenfunctions of the Laplacian on a closed Riemannian manifold M) the number of connected components of the open set $M\setminus {u_k=0}$ is at most $k$. In 1988 Donnelly and Fefferman gave a bound on the number of connected components of $B\setminus {u_k=0}$, where $B$ is a ball in $M$. However, their estimate was not sharp (even for spherical harmonics). We describe the ideas which give the sharp bound on the number of connected components in a ball. The talk is based on a joint work with S. Chanillo, A. Logunov and E. Malinnikova, with a contribution due to F. Nazarov. 3 March February 2021 Derek Harland (Leeds) Monopoles: construction, dynamics, transforms Abstract. This talk is adapted from one given at a recent BIRS workshop and will be part pedagogical introduction and part survey. I will review some of the foundational tools used to construct and study monopoles, including the Nahm transform, spectral curves, and rational maps. I will go on to survey recent progress on the problems of explicitly constructing monopoles and classifying their dynamics, both on Euclidean R^3 and on other geometries. 5 May 2021 Tobias Colding (MIT) Higher codimension mean curvature flow and the search for stable structures Abstract. We will discuss a new circle of ideas that gives a new way of attacking mean curvature flow in higher codimnesion. Higher codimension mean curvature flow is a complicated parabolic system where much less is known than for hypersurfaces. |