I am a fourth-year PhD student under the supervision of Dr. Simon Rydin Myerson and Dr. Sam Chow. In Fall 2024 I will join Leibniz Universität Hannover as a postdoc in the research group of Prof. Dr. Ulrich Derenthal.

E-mail address: Nuno.Arala-Santos(at)warwick.ac.uk.

Interests

Number Theory, especially interactions between Analytic Number Theory and Diophantine Geometry.

Background

I obtained my Bachelor's degree from the University of Porto in 2018. I then did my Master's at the University of Bonn, which I finished in 2020. I wrote my Master's Thesis under the supervision of Valentin Blomer, who introduced me to the delta method.

For more information on my background see my CV.

Papers and Preprints

1. (with Sam Chow) Incidence geometry and polynomial expansion over finite fields (2024) (arXiv, .pdf), submitted
2. (with Jayce R. Getz, Jiaqi Hou, Chun-Hsien Hsu, Huajie Li and Victor Y. Wang) A nonabelian circle method (2024) (arXiv, .pdf), submitted
3. (with Sam Chow) Expansion properties of polynomials over finite fields (2024) (arXiv, .pdf), submitted
4. The analytic Hasse Principle for certain singular intersections of quadrics in $\mathbb{P}^9$ (2023) (arXiv, .pdf), submitted
5. A maximal extension of the Bloom-Maynard bound for sets with no square differences (2023) (arXiv, .pdf), Functiones et Approximatio Commentarii Mathematici (to appear)
6. Lines of polynomials with alternating Galois group (2022) (arXiv, .pdf), Annali della Scuola Normale Superiore di Pisa - Classe de Scienze (to appear)
7. On the density of a set of primes associated to an Elliptic Curve (2021) (arXiv, .pdf), Journal of Number Theory 243 (2023) 241-255.

Miscellaneous writing

Below are some math-related texts I wrote over the years, for various purposes. Unfortunately, some of them are written in Portuguese.

• (2014, in Portuguese) Three handouts on Euclidean Geometry topics, intended for Olympiad students: Geometry for Beginners, Random configurations and Brute Force. These are joint work with Miguel Moreira.
• (2015, in Portuguese) A short handout on the "Lifting the Exponent" Lemma, intended for Olympiad students: Lifting the Exponent.
• (2019, in Portuguese) Affine and Projective Geometry. This text was loosely inspired in a course I took in the final year of my Bachelor's.
• (2020) A short text on the classification of Quaternion Algebras over Local Fields.
• (2020) My Master's Thesis on Representation Numbers of Quadratic Forms.
• (2021, in Portuguese) Lecture notes on p-adic numbers and general local fields.