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We study the number of degree $n$ number fields with discriminant bounded by $X$. In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for $6 \leq n \leq 94$., 17 pages; small updates over previous version

Modular forms are highly self-symmetric functions studied in number theory, with connections to several areas of mathematics. But they are rarely visualized. We discuss ongoing work to compute and visualize modular forms as 3D surfaces and to use these techniques to make videos flying around the peaks and canyons of these "modular terrains." Our goal is to make beautiful visualizations exposing the symmetries of these functions., 4 pages, 4 figures, for Bridges

We examine several currently used techniques for visualizing complex-valued functions applied to modular forms. We plot several examples and study the benefits and limitations of each technique. We then introduce a method of visualization that can take advantage of colormaps in Python's matplotlib library, describe an implementation, and give more examples. Much of this discussion applies to general visualizations of complex-valued functions in the plane., 20 pages, many figures, after first major set of revisions

In this article, we discuss whether a single congruent number $t$ can have two (or more) distinct triangles with the same hypotenuse. We also describe and carry out computational experimentation providing evidence that this does not occur., 15 pages; includes appendix by Brendan Hassett; to appear in Simons Symposia

The Gauss Circle Problem concerns finding asymptotics for the number of lattice point lying inside a circle in terms of the radius of the circle. The heuristic that the number of points is very nearly the area of the circle is surprisingly accurate. In this work, we describe two variants of the Gauss Circle problem that exhibit similar characteristics. The first variant concerns sums of Fourier coefficients of $\text{GL}(2)$ cusp forms. These sums behave very similarly to the error term in the Gauss Circle problem. We introduce new Dirichlet series with coefficients that are squares of partial sums of Fourier coefficients of cusp forms. We study the meromorphic properties of these Dirichlet series and use these series to give new perspectives on the mean square of the size of sums of these Fourier coefficients. The second variant concerns the number of lattice points of bounded size on one-sheeted hyperboloids. This problem is very similar to counting the number of lattice points within a spheres of various dimensions, except with the additional constraint of lying on a hyperboloid. It turns out that this problem is equivalent to estimating sums of the shape $\sum r_d(n^2 + h)$, where $r_d(m)$ is the number of representations of $m$ as a sum of $d$ squares. We prove improved smoothed and sharp estimates of the second moment of these sums, yielding improved estimates of the number of lattice points., PhD Thesis

We investigate functions with the property that for every interval, the slope at the midpoint of the interval is the same as the average slope. More generally, we find functions whose average slopes over intervals are given by the slope at a weighted average of the endpoints of those intervals. This is equivalent to finding functions satisfying a weighted mean value property. In the course of our exploration, we find connections to harmonic functions that prompt us to explore multivariable analogues and the existence of "weighted harmonic functions.", To appear in American Mathematical Monthly