We prove that, when genus two curves $C/\mathbb{Q}$ with a marked Weierstass point are ordered by height, the average number of rational points $\#|C(\mathbb{Q})|$ is bounded. The argument follows… Expand

The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic L-functions agree with the scaling limits of eigenvalue distributions of… Expand

We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact is less than $66$ (and at most… Expand

We prove the Alon-Tarsi conjecture that the number of even and odd Latin squares, while conjecturally not equal in even dimensions, are equal to leading order asymptotically.Expand

A recent paper by Hanusa and Nath states many conjectures in the study of self-conjugate core partitions. We prove all but two of these conjectures asymptotically by number-theoretic means. We also… Expand

We prove that the number of partitions of an integer into at most b distinct parts of size at most n forms a unimodal sequence for n sufficiently large with respect to b.Expand

In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219… Expand