Sato--Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height

We obtain asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of the elliptic curve \E_{a,b} : Y^2 = X^3 + aX + b satisfies certain ``natural'' properties, on average over integers $a$ and $b$ with $|a|\le A$ and $|b| \le B$, where $A$ and $B$ are small relative to $x$. Specifically, we investigate behavior with respect to the Sato--Tate conjecture, cyclicity, and divisibility of the number of points by a fixed integer $m$

Fractional parts of Dedekind sums

Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec~(1997) on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums $s(m,n)$ with $m$ and $n$ running over rather general sets. Our result extends earlier work of Myerson (1988) and Vardi (1987). Using different techniques, we also study the least denominator of the collection of Dedekind sums $\bigl\{s(m,n):m\in(\mathbb Z/n \mathbb Z)^*\bigr\}$ on average for $n\in[1,N]$.Comment: Using recent results of S. Bettin and V. Chandee, arXiv 1502.00769, we have improved some of our result

Integers with a large smooth divisor

We study the function $\Theta(x,y,z)$ that counts the number of positive integers $n\le x$ which have a divisor $d>z$ with the property that $p\le y$ for every prime $p$ dividing $d$. We also indicate some cryptographic applications of our results

Supersymmetry, the Cosmological Constant and a Theory of Quantum Gravity in Our Universe

There are many theories of quantum gravity, depending on asymptotic boundary conditions, and the amount of supersymmetry. The cosmological constant is one of the fundamental parameters that characterize different theories. If it is positive, supersymmetry must be broken. A heuristic calculation shows that a cosmological constant of the observed size predicts superpartners in the TeV range. This mechanism for SUSY breaking also puts important constraints on low energy particle physics models. This essay was submitted to the Gravity Research Foundation Competition and is based on a longer article, which will be submitted in the near future