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?
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 , where 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- 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.