We start with a simple, yet useful, concentration inequality concerning random weighted projections in high dimensional spaces. The inequality is then used to prove a general concentration inequality for random quadratic forms. In another application, we show the optimal infinity norm O(sqrt{(log n)/n}) for most unit eigenvectors of a large class of random matrices, including the adjacency matrix of random graphs. This is joint work with Van Vu.
Random weighted projections, random quadratic forms and random eigenvectors
Nov. 20, 2014 3pm (Weber 201…
Algebraic Geometry
Maria Angelica Cueto (Columbia University)
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The aim of my two talks will be to show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in the sense of Berkovich. Our proof is constructive and it relies on the combinatorial description of the tropical Grassmannian as a space of phylogenetic trees by Speyer-Sturmfels.
In the first talk, I will discuss the connection between tropical geometry and analytification of algebraic varieties, and explain the notion of faithful tropicalization. I will also describe the combinatorics of the aforementioned space of trees inside tropical projective space.
The second talk will be devoted to the proof of the main theorem and its interpretation in terms of the tropicalization map. Namely, the fiber over each point is an affinoid domain with a unique Shilov boundary point. Our homeomorphism identifies points in the tropical Grassmannian with the unique Shilov boundary point on their fibers, allowing for further generalizations of our results by Gubler-Rabinoff-Werner.