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: Let \(P\) be a bounded convex subset of \(\mathbb R^n\) of positive volume. Denote the smallest degree of a polynomial \(p(X_1,\dots,X_n)\) vanishing on \(P\cap\mathbb Z^n\) by \(r_P\) and denote the smallest number \(u\geq0\) such that every function on \(P\cap\mathbb Z^n\) can be interpolated by a polynomial of degree at most \(u\) by \(s_P\). We show that the values \((r_{d\cdot P}-1)/d\) and \(s_{d\cdot P}/d\) for dilates \(d\cdot P\) converge from below to some numbers \(v_P,w_P>0\) as \(d\) goes to infinity. The limits satisfy \(v_P^{n-1}w_P \leq n!\cdot\operatorname{vol}(P)\). When \(P\) is a triangle in the plane, we show equality: \(v_Pw_P = 2\operatorname{vol}(P)\). These results are obtained by looking at the set of standard monomials of the vanishing ideal of \(d\cdot P\cap\mathbb Z^n\) and by applying the Bernstein-Kushnirenko theorem. Finally, we study irreducible Laurent polynomials that vanish with large multiplicity at a point. This work is inspired by questions about Seshadri constants.
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)