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Events happening on March 16, 2023

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Mar. 16, 2023 2:30pm (MATH 3…	Functional Analysis Rachel Chaiser (CU Boulder) X Matui's HK-conjecture proposes an in-principal computation of the K-theory of the reduced C-algebra of a (nice enough) groupoid in terms of the homology of the groupoid. This can be viewed as a C-algebraic analogue of a technique in algebraic topology where one can compute the K-theory of a CW-complex of dimension at most 3 explicitly from its cohomology using the Atiyah--Hirzebruch spectral sequence. However, this technique may fail for spaces of dimension 4 and greater. Given a flat manifold and an expanding self-cover, one can construct a groupoid where the invariants in the HK-conjecture are computed from the corresponding invariants of the manifold. Using this construction, a flat manifold for which the Atiyah--Hirzebruch spectral sequence fails to compute K-theory from cohomology ought to give a counterexample to the HK-conjecture. This method was introduced by Deeley to construct a counterexample given a flat manifold of dimension at least 9. This naturally leads to the question: is there a flat manifold of dimension 4 giving a counterexample? Flat manifolds and the HK-conjecture
Mar. 16, 2023 3:35pm (Online)	Probability Sung-Soo Byun (Korea Institute for Advanced Study) View Online:https://cuboulder.zoom.us/j/92389457191 X In this talk, I will discuss real eigenvalues of the real elliptic Ginibre matrix, the model which provides a natural bridge between Hermitian and non-Hermitian random matrix theories. In the maximally non-Hermitian regime, which corresponds to the matrix model with real i.i.d. Gaussian entries, it was pioneered by Edelman, Kostlan, and Shub that the number of real eigenvalues is of order $\sqrt{N}$ , where $N$ is the size of the matrix. Moreover, it can be heuristically conjectured that as a real random matrix becomes more symmetric, it gets more real eigenvalues. I will demonstrate that such a statement can be made rigorous by presenting the large- $N$ expansion of the mean and variance of the number of real eigenvalues in the almost-Hermitian regime, where one can observe a non-trivial transition between real i.i.d. and real symmetric random matrices. Furthermore, I will explain the limiting empirical distributions of the real eigenvalues which interpolate the Wigner semicircle law and the uniform distribution. Real eigenvalues of elliptic random matrices
Mar. 16, 2023 4pm (MATH 220)	Init Conditions Jon Kim Limits and Colimits