 # Fabian Gundlach

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

I am a Lecturer at Harvard University. Before that, I was a Benjamin Peirce Fellow. 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)
Spring 2021
Math 137 (Algebraic Geometry)
Math 223b (Algebraic Number Theory)
Fall 2021
Math 228Y (Algorithms in Algebra and Number Theory)
Spring 2022
Math 137 (Algebraic Geometry)
Math 229 (Introduction to Analytic Number Theory)
Fall 2022
Math 21a (Multivariable Calculus)
Spring 2023
Math 280Y (Arithmetic Statistics)

## Research Interests

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

## Publications and Preprints

• Malle's conjecture with multiple invariants. (arXiv)

Abstract: We define invariants $$\operatorname{inv}_1,\dots,\operatorname{inv}_m$$ of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle's conjecture which asymptotically predicts the number of extensions that satisfy $$\operatorname{inv}_i\leq X_i$$ for all $$X_i$$. The resulting conjecture is proved for abelian Galois groups. We also describe refined Artin conductors that carry essentially the same information as the invariants $$\operatorname{inv}_1,\dots,\operatorname{inv}_m$$.

• Polynomials vanishing at lattice points in a convex set. (arXiv)

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.

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

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)

## Software

• Sauklaue, an application for hand-written online lecturing using an external graphics tablet.
• Contributed to CMS, a contest management system for informatics olympiads, as well as to the fork used at German informatics olympiad training camps.