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VUA

PhD Position in Number Theory and Formalization

Vrije Universiteit Amsterdam Department of Mathematics
✓ Fully Funded ⏰ Closing Soon 🎓 Mathematics number theory formalization proof assistant lean roccq diophantine problems algorithmic number theory arithmetic geometry

Explore the formalization of number theory with proof assistants to tackle Diophantine problems. Join a dynamic team at Vrije Universiteit Amsterdam working at the intersection of pure and algorithmic mathematics.

AI-generated overview

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Why This Research Matters

This research addresses the need for formal verification in pure mathematics, enhancing the reliability and reproducibility of results in number theory. The formalization of Diophantine problems paves the way for new computational tools and algorithms that can benefit cryptography and theoretical computer science.

Mathematics

Project Description

Project Overview

The Department of Mathematics at Vrije Universiteit Amsterdam offers a 4-year PhD position focused on Number Theory and Formalization. Research involves developing and formalizing mathematical theories necessary for Diophantine problem-solving, primarily in effective and algorithmic number theory, with pure mathematical results also contributing. The project is supervised by Dr. Sander Dahmen and Assia Mahboubi, supported by an NWO-funded Vici project aiming to bridge the gap in formalized Diophantine research. The candidate will work within the Center for Topology, Algebra, and their Applications, joining an inclusive and interdisciplinary group committed to diversity and internationalism.

What You Will Do

Your duties include conducting original research aimed at a PhD thesis, using proof assistants such as Lean or Rocq to formalize number theory. You will also participate in teaching activities, including supervising exercise classes, which comprise roughly 15% of your working time.

Expected Outcomes

The research aims to advance the formalization and algorithmic understanding of number theory relevant to Diophantine problems, contributing novel mathematical insights and proof formalizations. The position combines theoretical and computational approaches within an international research environment.

Why This Matters

Formalizing number theory using modern proof assistants enhances mathematical rigor and accessibility, fosters reproducibility, and facilitates complex problem-solving in mathematics. The research supports developments in both pure and effective number theory, with potential applications in cryptography, algorithm design, and mathematical logic.

Entry Requirements

A (prospective) MSc degree in Mathematics or an equivalent. Good communication skills in English. Priority to candidates with background in number theory/arithmetic geometry and affinity or experience with proof assistants (or motivation to learn).

Eligibility

UK/Home
EU
International

Supervisor Profile

DS
Dr. Sander Dahmen
Vrije Universiteit Amsterdam, Department of Mathematics

Dr. Sander Dahmen is a researcher in number theory and formalization at Vrije Universiteit Amsterdam. He focuses on bridging formalized mathematics and classical mathematical problems such as those in Diophantine equations, using tools like Lean. He leads the NWO-funded Vici project aimed at advancing formalized Diophantine research and is deeply involved in computational approaches to pure mathematics.

Key Publications

2014 80 citations
Generalized Fermat equations: a miscellany
2008 49 citations
Classical and modular methods applied to Diophantine equations
2013 33 citations
Klein forms and the generalized superelliptic equation
2007 29 citations
Counting integral Lamé equations by means of dessins d’enfants
2022 24 citations
A formalization of Dedekind domains and class groups of global fields

Research Contributions

Advanced the study of Generalized Fermat equations using a miscellany of methods.
Enhanced understanding of complex Diophantine problems with implications in number theory.
Applied classical and modular methods to solve Diophantine equations.
Provided frameworks used in the resolution of long-standing number theory problems.
Formalized aspects of Dedekind domains and class groups of global fields.
Contributed to mathematical logic and automated reasoning in algebraic number theory.
Counted integral Lamé equations using dessins d’enfants and modular forms.
Offered new combinatorial and modular perspectives in the study of differential equations.

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