Academic stuff
BSc & ARCS, Imperial
College, University of London,
August 1999.
PhD & DIC, Imperial College, University of London, November 2002.
Bateman Research Instructor
at the California Institute of
Technology, September 2002 -- July 2004.
Visitor at the Max-Planck-Institut fuer Mathematik, August
2004 -- September 2004.
Fellow at Merton College,
Oxford (and member of the Mathematical Institute and the
Number Theory Group),
September 2004 -- September 2007.
Research Fellow at the University of Bristol, September 2007 --.
Research Interests
I am interested in Number
Theory; more
specifically in
such topics as (the arithmetic of) Modular
Forms, Elliptic
Curves and Arithmetic Algebraic
Geometry -- especially the computational aspects of these
subjects. I often use computer algebra packages like MAGMA or PARI/GP or SAGE in my work.
I am listed as one of the many contributors to SAGE 4.0.
At the moment I am working on Hecke operators acting on quaternionic
modular forms, and considering certain non-Gorenstein Hecke algebras.
I am also considering how best one should approximate certain Hecke operators
acting on classical modular forms of level p, where X_0(p) has genus 1.
Here are some programs for computing the Hecke
operator U_p
on quaternionic modular forms.
Here is some code to calculate U_p acting on
overconvergent p-adic modular forms, for p such that
X_0(p) has genus 1 (in other words, p = 11, 17 or 19).
Teaching
I taught the course Introduction to
Modular Forms in Oxford, as a trial for an undergraduate
course to run in 2005--2006, and as a graduate course again in 2006--2007.
I taught the Part C course Introduction to Modular
Forms; the lectures were at noon on Tuesdays and Fridays in the
Mathematics Institute in Oxford.
In my previous post at Caltech, I taught courses in graduate algebra,
Galois theory, elliptic curves for undergraduates and number theory.
Book
I have written a book, Modular Forms: A classical and computational introduction, available on Amazon. It is a graduate-level introduction to the theory of modular forms, starting with the construction of Eisenstein series and working up to the action of Hecke operators. It also covers the computational side of the theory, and shows how SAGE and MAGMA can be used to compute modular forms.
The cover was created using SAGE by Tomas Boothby; I am very grateful to him for his work.
I am maintaining an errata list; please contact me if you find other errors.
Publications/Preprints
- Counting Mod $\ell$ Solutions via Modular Forms, with Edray Goins, in preparation.
- On mod $p$ modular
representations which are defined over $\F_p$: II, with Gabor Wiese, to appear in the Glasgow Mathematical Journal.
- Experimental finding of modular forms for noncongruence subgroups.
- Computing the action of the $U_p$ operator on $p$-adic
overconvergent
modular forms when $X_0(p)$ has genus~1, to appear in the Journal
of Number Theory.
- On a $p$-adic extension of the Jacquet-Langlands correspondence to
weight 1, submitted.
- Generating
spaces of modular forms with $\eta$-quotients, JP journal of Algebra, Number Theory and Applications, Volume 8 No. 2 , 2007, pages
213--216.
- On the failure of
the Gorenstein property for Hecke algebras
of prime weight, with Gabor
Wiese, math.NT/0612317, Experimental
Mathematics, Volume 17, number 1, pages 37--52.
- On the slopes of
the $U_5$ operator acting on overconvergent modular forms,
math.NT/0606363, Journal de Theorie des Nombres de Bordeaux,
Volume 20, number 1, 2008, pages 165--182.
- On mod $p$ modular
representations which are defined over $\F_p$, math.NT/0606052,
Glasnik Matematicki,
Volume 43, number 1, 2008, pages 1--6.
- The 2-adic eigencurve
at
the boundary of weight space, with Kevin Buzzard, Compositio Mathematica, Volume
141(3), 2005, pages 605--619.
- A
Mathematical Tourist in Oxford [about the Merton College
astrolabes],
Mathematics
Today, Volume 41, Number 4, page 116.
- A
Mathematical Tourist in Germany [about Gallo-Roman dodecahedra],
Mathematics
Today, Volume 40, Number 6, December 2004, page 204.
- Explicit computations of the slopes of the
U_5 operator acting on overconvergent modular
forms,
Preprint number MPIM2004-91 of the Max-Planck-Institut preprint series.
- A generalization of a necessary
and sufficient condition for primality due to Vantieghem, International Journal of Mathematics and
Mathematical Sciences, Volume 70, pages 3889--3892, 2004.
- Computing
with overconvergent modular forms, Communications in Computer Algebra (the SIGSAM Bulletin),
Volume 38, Number 3, September 2004.
- Slopes of 2-adic overconvergent modular forms
with
small level. Mathematics Research
Letters, Volume 11, Issue 6, November 2004, pages 723--739.
- Slopes of overconvergent modular
forms, PhD thesis, Imperial College of Science, Technology and
Medicine, October 2002.
Available as DVI or PDF or PS.
- Some non-Gorenstein Hecke algebras attached to
spaces of modular forms, Journal of
Number Theory, Volume 97, No. 1, November
2002, pages 157--164.
Professional information
Mathematical Links
- ASCIIMathML
- provides a nice way to write mathematics on web pages. See an
example here.
- The Art Of
Problem-Solving website has a very useful online
application called the
TeXer, which creates .gif images from LaTeX input. Here is an example:
We can only see a
short distance ahead, but we can see plenty there that needs to be
done. - Alan Turing.
This page last modified
by L. J. P. Kilford
Tuesday, 09-June-2009 10:39:00 BST