# Kenneth Maples

Postdoc
Institut für Mathematik, Universität Zürich
kenneth.maples@math.uzh.ch
Office: 27 K 42

# Final Exam for Mat628: Harmonic Analysis

## Research interests

I am a postdoc at the institute for mathematics of the university of Zürich. My research interests include random matrices, additive combinatorics, and number theory.

I graduated from UCLA in June 2011 with a PhD in mathematics under Terry Tao. I also graduated from Harvey Mudd College in 2006 with a bachelor's degree in mathematics and engineering.

## Preprints and papers

with J. Najnudel and A. Nikeghbali. Limit operators for circular ensembles. pdf

We consider a coupling between unitary matrices of different dimensions where each matrix is marginally Haar distributed. With this coupling, we show that the powers of the matrices converge in a suitable sense to a flow of operators on an explicit random sequence space. The eigenvalues of this flow are shown to be distributed like a sine kernel point process. This appears to be the first example of such a flow of operators constructed from finite dimensional random matrix theory.

Quantum unique ergodicity for random bases of spectral projections. pdf

Zelditch has introduced a model of random waves on compact Riemannian manifolds to study the expected behavior of typical'' eigenfunctions of the laplacian in the high energy limit. This is done by considering the sequence of functions generated by taking short linear combinations (quasi-modes) of eigenfunctions using a family of random unitary matrices. We show that these functions obey the quantum unique ergodicity property, so that with probability one there do not exist exceptional sequences of random waves that do not converge to the uniform measure.

Announcement: Symmetric random matrices over finite fields. pdf

In the 60s, Komlos proved that a random $n \times n$ iid matrix with discrete, non-degenerate entries is invertible with probability $1 - o(1)$. Costello, Tao, and Vu extended this result to show that symmetric matrices with iid coefficients on and above the diagonal are also invertible with probability $1 - o(n^{-1/8})$. We consider the analogous problem over finite fields of prime order, and show that the ranks of $n \times n$ symmetric matrices $Q_n$ with iid entires above the diagonal converges in total variation to a universal distribution. We also prove an analogous result for antisymmetric matrices, where the universal distribution depends on the parity of $n \bmod 2$.

with A. Nikeghbali and D. Zeindler. On the number of cycles in a random permutation. Electron. Commun. Probab., 2012 (17). no. 20, 1--13. arXiv pdf

We compute the limiting distribution on the number of cycles of random permutations generated according to a generalization of Ewens measure. The method also produces large deviation estimates. This answers a question of Betz, Ueltschi, and Velenik.

A Brun-Titchmarsh inequality for discrete random matrices. pdf

We prove the following analogue of the Brun-Titchmarsh inequality. For all $n$ sufficiently large, let $A$ be a random $n \times n$ matrix with iid entries in the integers taken from a probability distribution that is not concentrated on an arithmetic progression. Then we have the bound $\mathbb{P}(\det A \text{ is prime and } \equiv a \bmod q) \leq C \frac{1}{\varphi(q)} \frac{1}{n}$ for all positive integers $q \leq O(e^{cn})$ and invertible residues $a \in (\mathbb{Z}/q\mathbb{Z})^\times$, where the constants are absolute.

Cokernels of random matrices satisfy the Cohen-Lenstra heuristics. arXiv pdf

Let $A$ be an $n \times n$ random matrix with iid entries taken from the $p$-adic integers $\mathbb{Z}_p$ or $\mathbb{Z}/N\mathbb{Z}$. Then under mild non-degeneracy conditions the cokernel of $A$ has a universal probability distribution. In particular, the $p$-part of a random matrix over $\mathbb{Z}$ has cokernel distributed according the Cohen-Lenstra measure, $\mathbb{P}(\text{coker } A \cong G) = \frac{1}{|\text{Aut } G|} \prod_{k = 1}^\infty (1 - p^{-k}) + O(e^{-cn})$ where the constants are absolute.

Singularity of random matrices over finite fields. arXiv pdf

Let $A$ be an $n \times n$ random matrix with iid entries over a finite field of order $q$. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - \alpha$ for some fixed $0 < \alpha < 1$. We show that the singularity probability converges to the uniform limit with error bounded by $O(e^{-c\alpha n})$, where the implied constant and $c > 0$ are absolute. We also show that the determinant of $A$ assumes each non-zero value with probability $q^{-1} \prod_{k=2}^\infty (1 - q^{-k}) + O(e^{-c\alpha n})$, where the constants are absolute.