We will discuss several ensembles of non-Hermitian random matrices with block structures, motivated from math and physics. We give a characterization of the limiting spectral measure and compute their spectral radius.
Eigenvalue densities of Structured Random Matrices
Apr. 23, 2015 3pm (CSU Webbe…
Algebraic Geometry
Richard Hain (Duke University)
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Suppose that k is a field and that 2g-2+n > 0. Then one has the moduli space M_{g,n}/k of n pointed smooth projective curves of genus g over k. The generic curve of type (g,n) over k is defined to be the restriction of the universal curve over M_{g,n}/k to its generic point Spec k(M_{g,n}). This curve has n tautological rational points. Results in Teichmuller theory imply that when g > 2 and char k = 0, these are the only rational points. Recent results of Watanabe imply that this also holds when char k is positive. When n = 0, results of Hain in char 0 and Watanable in positive characteristic imply that Grothendieck's Section Conjecture holds for the generic curve of genus g > 2. This talk will survey these results and introduce the tool of weighted completion that was used to prove them.