Most of this site is filled with topics related to my interests in mathematics. For other mathematicians, I suspect the best way to rapidly see what sort of mathematics I'm interested in is to see my research page, where I link to preprints of my work and link to related pages on talks and research discussion.

More broadly, see posts under the Math category.

Below, I give a broad introduction to my research.for a scientifically literate layperson

# Introduction to my Research

I study number theory and arithmetic geometry. This might sound like different things, but they are very strongly connected. Concretely, one problem I'm interested in the following.

If you draw a circle of radius $R$ on a sheet of graph paper, how many little boxes of the graph paper are contained inside the circle?

One way to study this problem is to *experimentally* model it: choose lots of
different radii $R$ and try to detect a pattern. Performing this experiment
suggests that there are approximately $\pi R^2$ boxes, if we suppose that
the boxes are squares of side length $1$. Then the natural question becomes
*how close to $\pi R^2$ is it?*

Further experimentation suggests that $\pi R^2$ is actually a very good
approximation,^{1}
^{1}in the sense that the size of the error is no more than
$\sqrt{R}$ or so.
but despite our efforts we don't actually *know* if
this is true or not. Interestingly, it's possible to slighly change this
problem in many ways and get very similar answers. If we use rectangular grids
instead of square grids, but the area of the rectangles is also $1$, then
essentially similar results seem to hold. Or if we scale up an ellipse (or
almost any reasonable curved shape) instead of a circle, again the results look
very similar. But if we use a shape with any straight lines instead of a
circle, then the behavior changes radically.

Problems like this have been studied for their intrinsic interest for thousands of years. The similarities in behavior in different situations suggest underlying structure, and I try to investigate that structure.

Number theory is a funny subject. Unlike many other areas of math, it's named
after the *subject matter* instead of the *tools used*. It's possible to study
the same problem from many different points of view.

For example, it turns out that there is a function closely related to the Riemann zeta function \begin{equation*} \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} \end{equation*} that encodes information about the number of graph paper squares contained inside a circle of radius $R$. This information is contained in the behavior of the variable $s$. This type of relationship was first noticed in the mid 1800s, when Riemann described how one could use the behavior of $\zeta(s)$ as a function of $s$ to count the number of prime numbers up to $X$.

The "name" of the function associated to the circle problem above is \begin{equation*} L(s) := \sum_{n = 1}^\infty \frac{r_2(n)}{n^s}, \end{equation*} where $r_2(n)$ means the number of ways of writing $n$ as a sum of $2$ squares.

A common thread in my research is to associate functions like $\zeta(s)$ or $L(s)$ to some mathematical question and then to use complex analysis to study this function, and thus learn about the question.

It turns out that "functions like $\zeta(s)$ or $L(s)$" are often called $L$-functions, and there is a very deep set of ideas linking problems in arithmetic and geometry to the analytic behaviors of $L$-functions.

Compared to many other number theorists who study similar subject matter, my
points of view are more *complex analytic* (meaning that I extract information
from the behavior of the variable $s$ using complex analysis), *real analytic*
(meaning that I also use tools from Fourier analysis and real analysis), and
*computational*. For the last several years, I've spent a lot of time computing
$L$-functions that are likely to be relevant to arithmetic applications that we
haven't considered yet.

These objects are computed and described in the $L$-function and modular form database (LMFDB).

My slides from a talk about how computation informs research give additional detail on many of the topics breached here.