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Current Postgraduate Research Students

Dan Fretwell supervises 1 postgraduate research students. If these students have produced research profiles, these are listed below:

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Dr Dan Fretwell FHEA

Lecturer - Security and Protection Science

Dan Fretwell

Fylde College

LA1 4YF

Lancaster

ORCID: 0000-0001-6473-0430

PhD supervision

Projects in Algebraic Number Theory and related areas (in particular projects concerning the arithmetic of Modular Forms and Galois Representations). Interested in projects concerning applications to Cryptography and Cyber Security.

Research Interests

My main area of research is (Algebraic) Number Theory. In particular, I work on problems that involve a wide range of objects, from extremely explicit to mind bogglingly abstract:

  • Quadratic forms, lattices, error correcting codes, integer valued polynomials, diophantine equations.
  • Modular forms, elliptic curves, number fields, clifford algebras, quaternion algebras.
  • Automorphic forms, Galois representations, L-functions, Bloch-Kato Selmer groups.
Some of my work is motivated by the Langlands program, which aims to explain deep links between these objects. One of my main focusses is the computation of explicit evidence towards these conjectures, and concrete applications of the general theory.
 
I have very wide interests, extending beyond number theory. For example, I have recently been working with interacting particle systems. These are continuous-time Markov chains on discrete state spaces, having many applications in Statistical Mechanics and Physics (e.g. modelling traffic flow, data flow in networks and tumour growth). By comparing dynamics/probability measures coming from such systems one can often prove q-series identities of combinatorial significance (e.g. the famous Jacobi triple product identity).
 
I am currently thinking about how my research can be useful in the real world (e.g. Cybersecurity, Cryptography and Quantum Computation). Have any ideas about this? Send me an email!

Career Details

I received my PhD in Mathematics from the University of Sheffield in 2015, under the supervision of Prof. Neil Dummigan. After, I was a Heilbronn Research Fellow at the University of Bristol until 2022, working on research of both theoretical and practical importance in Mathematics. I was a Lecturer of Mathematics at the University of South Wales 2022-2023.

Web Links

https://www.danfretwell.com

Current Teaching

MATH320: Mathematical Cryptography

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Selected Publications and Research Outputs

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Activities

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