 # Fabian Gundlach

Department of Mathematics
FAS, Room 233
Harvard University
Email: LASTNAME@math.harvard.edu
CV

I am a Benjamin Peirce Fellow at Harvard University. My PhD thesis, supervised by Manjul Bhargava, was on parametrizations of field and ring extensions.

## Teaching

Spring 2018
Linear Algebra with Applications
Fall 2019
Math 21a (Multivariable Calculus)
Math 223a (Algebraic Number Theory)
Spring 2020
Math 286X (Arithmetic Statistics)
Fall 2020
Math 223a (Algebraic Number Theory)

## Research Interests

Algebra, Number Theory (algebraic, analytic, and computational), Arithmetic Geometry, Geometry of Numbers, Combinatorics

## Publications

• Integral Brauer-Manin obstructions for sums of two squares and a power, J. London Math. Soc., 2(88):599-618, 2013. (arXiv)

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.

My bachelor's and master's and doctoral theses.

## Intros

In this series, I plan to upload very short introductions to beautiful mathematical ideas and subjects. The goal is to skip over as much detail as possible, while providing a range of motivational examples and outlining the underlying ideas.
• 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)