Dr Dan Fretwell

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, 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.

I like to work on explicit problems concerning the above (e.g. How do I compute this thing? How do I count these things? How do I tell whether two of these things are different?), but I also like much deeper problems (e.g. Does what I just did apply for more general families of automorphic form/Galois representations? How does what I just did fit into the general framework of the Langlands program? Have I just proved a special case of the Bloch-Kato conjecture?). Some of my work is motivated by deep conjectures in Mathematics, and the practical need to compute explicit evidence for these. I have very wide interests, extending beyond Number Theory. For example, I have recently been working on projects relating to Combinatorial Probability. In particular, I have studied a specific family of Markov chains called Interacting Particle Systems. These are useful in Statistical Mechanics and Physics (e.g. they can be used to model traffic flow). By comparing dynamics/probability measures on 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!