Elizabeth Mansfield

Contact information page

I am now a Fellow of the IMA, that is, Institute of Mathematics and its Applications.

In January 2015, I was elected to a Vice Presidency of the IMA, with responsibility for Learned Societies. This means I chair the IMA Research Committee and attend Executive Board and Council meetings, amongst other activities. I care deeply about access of girls to advanced mathematics teaching in schools, about socially aware mathematics, about funding mathematics research across the whole country, and about dealing with implicit bias.

Book on Moving Frames

My book for Cambridge University Press, A Practical Guide to the Invariant Calculus. Thank you everyone for contributing the typos and other errors you found, in particular Tania Goncalves, Francis Valiquette, Evelyne Hubert, Peter Hydon and Peter Olver! Here are the Table of Contents and the first ``what's in this book" chapter.

The book is now available from CUP (above link), also Amazon

An errata page will appear soon :(

My two former graduate students, Tania Goncalves and Jun Zhao pursue their careers, congratulations to them both! Here is a great photo of us at the FoCM '08 banquet in Hong Kong, where Jun, Tania and Andy Wheeler presented posters.

Recent research funding :

The EPSRC grant, "Group actions on function approximation spaces". A description of the project is is here. I worked with Tristan Pryer on this grant.


I masterminded our MSc in Mathematics and its Applications . Designed to be accessible, relevant, interesting and to foster creativity and communication skills, students choose from 11 options and take 2 compulsory project modules.

This coming year, 2015-16 I am teaching on MA593/MA561 Lie groups and algebras, and MA588, Mathematical Techniques and Differential Equations. I alternate teaching Lie groups and algebras with teaching Mathematics and Music, which involves Chladni patterns on drums, digital signal processing ie recording music, and the geometry of music and music composition.

Professional Activities

I have been on the Council of the London Mathematical Society, for two terms, from November, 2011 to November, 2015.

I was Vice Chair of ACM's SIGSAM, Special Interest Group in Symbolic and Algebraic Manipulation, for two years.

I served on the EPSRC Strategic Advisory Team for the mathematics program, for five years (at least!!).

I served as a board member for the Society for the Foundations of Computational Mathematics for maybe ten years!! I co-organised the Symbolic Analysis Workshop of three (!) FoCM conferences.

I was part of the official delegation of the London Maths Society to the International Congress of Mathematics in Hyderabad, India.

Editorial Advisory Boards

I am on the editorial boards of Journal of Computation and Mathematics and Foundations of Computational Mathematics.


Research Interests:
My research is the development of algorithms for Analysis, in the context of symbolic computation and increasingly numerical computation; recent papers are on various forms of the discrete variational calculus and moving frames. The mathematics that I use comes from commutative algebra, differential geometry, variational calculus, integrable systems and geometric integration.

Earlier Research Funding
My previous EPSRC grant, "Symmetric variational problems", which funded Tania and Jun, is now just finshed. A brief description is here


diffgrob2 is a MAPLE package to simplify overdetermined systems of nonlinear differential equations of polynomial type. The algorithms are based on those by Buchberger for a Gröbner basis of a polynomial ideal. This package is no longer being maintained and is not at present available for public use. Packages which perform related functions are Maple's diffalg package maintained by Evelyne Hubert, and the Maple package rif which is available from Allan Wittkopf's home page.

The MAPLE package Indiff is now available. This is a set of functions designed to calculate reductions and compatibility conditions of systems of equations referred to a moving frame. The theory is discussed in " Algorithms for symmetric differential systems", J. Foundations of Comp. Math.,1 (2001) 335-383. The Short Manual contains installation instructions for UNIX, a guide to the procedures and three worked examples. There are two versions of the code, a readlib version and a version suitable for generic platforms (non readlib version, ie, maple just reads the file in whole, and you don't use the with command). I currently have this working for Maple 9. The Maple worksheets for the examples in the manual are for invariant differentation, an over determined system, and a classification problem.

Recent talks and currently available titles.

On the mathematics of handwriting recognition.

Noether's Second Theorem for smooth and discrete systems.

Digital Atlases and Difference Forms Plenary talk at ISSAC '08, Linz.

Discrete Variational Methods Plenary talk at FoCM, Santander.

Moving Frames and Noether's Theorem, and/or invariant ODE and/or curvature flows.

Discrete gradients

Moment maps for discrete symplectic mappings. Available from the Isaac Newton Institute website, Discrete Integrable Systems and Special Functions workshop.

On a variational complex for difference systems

Towards a variational complex for finite element systems

A simple criterion for involutivity

Selected interesting papers

Difference Forms with Peter Hydon.

Extensions of Noether's Second Theorem: from continuous to discrete systems with Peter Hydon. In the Proceedings of the Royal Society.

On Moving Frames and Noether's Theorem with Tania Goncalves. In Studies in Applied Mathematics.

Discrete variational calculus for B-spline Approximated Curves with Jun Zhao.

Noether's Theorem for Smooth, Difference and Finite Element Systems , in FoCM Santander 2005. Eds: Pardo, Pinkus, Suli and Todd. CUP 2006.

A variational complex for difference equations with Peter Hydon (Surrey). Now available from Journal of Foundations of Computational Mathematics.

Towards a variational complex for the Finite Element Method with Reinout Quispel (LaTrobe). In a CRM Proceedings for the workshop on Group Theory and Numerical Analysis

Evolution of Curvature Invariants and Lifting Integrability, with Peter van der Kamp, Journal of Geometry and Physics

Symmetry Group Analysis of the Shallow Water and Semi-Geostrophic Equations with Nicoleta Bila and Peter Clarkson, in Quarterly Journal of Mechanics and Applied Mathematics


Since I am still receiving requests for my PhD thesis, Differential Groebner Bases here it is! Many thanks to Katya Krupchyk for making this TeX version possible. Please note this is an historical document (submitted 1991) and appears here without amendment, with the exception that Chapter 6 is not included, since it was published in its entirety (A Simple Criterion for Involutivity, Journal of the London Mathematical Society, 54, pages 323-345, 1996). Superceded computer code is also not included.

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