We establish a central limit theorem of (1/\sqrt{h_p})\sum_{X< n \leq X+h_p}\big(\tfrac{n}{p}\big) for almost all the primes $p$, with $X$ uniformly random in $[g(p)]$, $g(p)$ an arbitrary divergent function growing slower than any power of $p$, provided $(\log h_p)/(\log g(p))\rightarrow 0, \, h_p \rightarrow \infty$ as $p \rightarrow \infty$. This improves the recent results of Basak, Nath and Zaharescu, who established this for g(p) = (\log p)^A, A>1. We also use the best currently available tools to expand the original central limit theorem of Davenport and Erdős for all the primes to a shorter interval of starting points. In this paper we exploit a Selberg's sieve argument, recently used by Harper, an intersection result due to Evertse and Silverman and some consequences of the Weil bound on general character sums.

In this paper we give an algorithm to find the 3-torsion subgroup of the Jacobian of a smooth plane quartic curve. We describe $3-$torsion points in terms of cubics which triply intersect the curve, and use this to define a system of equations whose solution set corresponds to the coefficients of these cubics. We compute the points of this zero-dimensional, degree $728$ scheme first by approximation, using homotopy continuation and Newton-Raphson, and then using lattice reduction or continued fractions to obtain accurate expressions for these points. We describe how the Galois structure of the field of definition of the $3$-torsion subgroup can be used to compute local conductor exponents, including at $p=2$.

We consider smooth solutions to the Monge-Amp`ere equation subject to mixed boundary conditions on annular domains. We establish global $C^2$ estimates when the boundary of the domain consists of two smooth strictly convex closed hypersurfaces.

We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space $H$. We prove a version of Azzam and Schul's $d$-dimensional Analyst's Travelling Salesman Theorem in this setting by showing for any lower $d$-regular set $E \subseteq H$ that $$\text{diam}(E)^d + \beta^d(E) \sim \mathscr{H}^d(E) + \text{Error},$$ where $\beta^d(E)$ give a measure of the curvature of $E$ and the error term is related to the theory of uniform rectifiability (a quantitative version of rectifiability introduced by David and Semmes). To do this, we show how to modify the Reifenberg Parametrization Theorem of David and Toro so that it holds in Hilbert space. As a corollary, we show that a set $E \subseteq H$ is uniformly rectifiable if and only if it satisfies the so-called Bilateral Weak Geometric Lemma, meaning that $E$ is bi-laterally well approximated by planes at most scales and locations.

We consider the computational complexity of computing Bayes-Nash equilibria in first-price auctions, where the bidders' values for the item are drawn from a general (possibly correlated) joint distribution. We show that when the values and the bidding space are discrete, determining the existence of a pure Bayes-Nash equilibrium is NP-hard. This is the first hardness result in the literature of the problem that does not rely on assumptions of subjectivity of the priors, or convoluted tie-breaking rules. We then present two main approaches for achieving positive results, via bid sparsification and via bid densification. The former is more combinatorial and is based on enumeration techniques, whereas the latter makes use of the continuous theory of the problem developed in the economics literature. Using these approaches, we develop polynomial-time approximation algorithms for computing equilibria in symmetric settings or settings with a fixed number of bidders, for different (discrete or continuous) variants of the auction.