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Open question: What is a quantum honeycomb?
This problem lies in the highly interconnected interface between algebraic combinatorics (esp. the combinatorics of Young tableaux and related objects, including honeycombs and puzzles), algebraic...
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Random matrices: the circular law
Van Vu and I have recently uploaded our joint paper, “Random matrices: the circular law“, submitted to Contributions to Discrete Mathematics. In this paper we come close to fully resolving the circular...
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Milliman Lecture II: Additive combinatorics and random matrices
This is my second Milliman lecture, in which I talk about recent applications of ideas from additive combinatorics (and in particular, from the inverse Littlewood-Offord problem) to the theory of...
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When are eigenvalues stable?
I was asked recently (in relation to my recent work with Van Vu on the spectral theory of random matrices) to explain some standard heuristics regarding how the eigenvalues of an matrix A behave under...
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Random matrices: universality of local eigenvalue statistics
Van Vu and I have just uploaded to the arXiv our paper “Random matrices: universality of local eigenvalue statistics“, submitted to Acta Math.. This paper concerns the eigenvalues of a Wigner matrix ,...
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Random matrices: Localization of the eigenvalues and the necessity of four...
Van Vu and I have just uploaded to the arXiv our paper “Random matrices: Localization of the eigenvalues and the necessity of four moments“, submitted to Probability Theory and Related Fields. This...
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Random matrices: Sharp concentration of eigenvalues
Van Vu and I have just uploaded to the arXiv our paper Random matrices: Sharp concentration of eigenvalues, submitted to the Electronic Journal of Probability. As with many of our previous papers,...
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The spectral proof of the Szemeredi regularity lemma
Perhaps the most important structural result about general large dense graphs is the Szemerédi regularity lemma. Here is a standard formulation of that lemma: Lemma 1 (Szemerédi regularity lemma) Let...
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Random matrices have simple spectrum
Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple eigenvalues“. Recall that an Hermitian matrix is said to have simple eigenvalues if all of its eigenvalues are...
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Random matrices: tail bounds for gaps between eigenvalues
Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we...
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Eigenvectors from eigenvalues
Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues“. This note gives two proofs of a general eigenvector...
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Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra
Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv a completely rewritten version of our previous paper, now titled “Eigenvectors from Eigenvalues: a survey of a basic...
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Sums of GUE matrices and concentration of hives from correlation decay of...
Hariharan Narayanan, Scott Sheffield, and I have just uploaded to the arXiv our paper “Sums of GUE matrices and concentration of hives from correlation decay of eigengaps“. This is a personally...
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