I am a Benjamin Peirce Fellow at Harvard University. My PhD thesis, supervised by Manjul Bhargava, was on parametrizations of field and ring extensions.
Abstract: We use Brauer-Manin obstructions to explain failures of the integral Hasse principle and strong approximation away from \(\infty\) for the equation \(x^2+y^2+z^k=m\) with fixed integers \(k\geq3\) and \(m\). Under Schinzel's hypothesis (H), we prove that Brauer-Manin obstructions corresponding to specific Azumaya algebras explain all failures of strong approximation away from \(\infty\) at the variable \(z\). Finally, we present an algorithm that, again under Schinzel's hypothesis (H), finds out whether the equation has any integral solutions.
Dirichlet series: They can be used to answer questions such as the following: given a (number theoretic) sequence \(a_1,a_2,\dots\) of nonnegative integers, what is the asymptotic behavior of \(\sum_{n\leq N}a_n\) as \(N\) goes to infinity? We in particular cover the following sequences: $$ \begin{aligned} \sigma_n &= \text{number of divisors of $n$},\\ \Delta_n &= \#\{(k,f_1,\dots,f_k)\mid k\geq0,\quad f_1,\dots,f_k\geq2,\quad n=f_1\cdots f_k\}\\ &= \text{number of ordered factorizations of $n$ into any number of factors bigger than 1},\\ \mathbb P_n &= \text{1 if $n$ is prime, 0 otherwise}.\qquad\text{(We sketch how to prove the Prime Number Theorem.)} \end{aligned} $$ Prerequisites: Basic Number Theory, Complex Analysis, (familiarity with Ordinary Generating Functions a plus)