# Dr Anthony Ashton

## Career

**2015****- :**College Lecturer and DoS, Homerton College.**2013-15:**Stokes Fellow, Pembroke College.**2010-****13:**Junior Research Fellow, Emmanuel College.**2009-10:**EPSRC Prize Fellow, DAMTP.**2006-09:**PhD Candidate, DAMTP.**2005-06:**Part III Mathematics, Peterhouse.**2002-05:**BA Mathematics, Peterhouse.

## Research

Anthony is a member of the Department of Applied Mathematics and Theoretical Physics and works in the Applied and Computational Analysis group. His research interests include: novel approaches to elliptic boundary value problems, Lie groups in PDE, new approaches to rigorous problems in Linear PDE theory and certain aspects of mathematical physics.

## Teaching

I am lecturing the Part III course *Distribution Theory & Applications*, which focuses on classical distribution theory and application to the analysis of linear PDEs. Example sheets are here and the corresponding solutions will appear below

There are also some handouts

I've also lectured Part II *Integrable Systems *which includes topics such as: the Arnold-Liouville theorem, inverse scattering, infinite dimensional Hamiltonian systems and Lie group methods in PDE. Handouts were

- Generating functions
- Arnold-Liouville theorem and worked example
- Evolution of scattering data
- From Lax pairs to zero curvature
- Painlevé equations

In Lent term 2019 I am lecturing Part IA *Vector Calculus, *where we take a look extending our ideas about calculus to two and three dimensions. We go on to discuss some aspects of PDE and right towards the end we start to look at Cartesian tensors. Handouts are

I will update this set of notes as we progress through the course. They currently cover lectures 1-14.

## Selected Publications

- Ashton, 2015.
*A new weak formulation of the Dirichlet-Neumann map on convex polyhedra with explicit coercivity constants.*(in preparation). - Ashton & Crooks, 2014.
*Numerical Analysis of Fokas' Unified Method for Linear Elliptic PDEs*, Appl. Num. Math. (in press). - Ashton, 2014.
*Laplace's Equation on Convex Polyhedra via the Unified Method*, Proc. Roy. Soc. A**471**(2176). - Ashton, 2014.
*Elliptic PDEs*J. Math. Anal. & Appl.**425**(1). - Ashton & Fokas, 2014.
*Elliptic Equations with Low Regularity Boundary Data via the Unified Method,*Complex Var. & Elliptic Eq.**60**(5)*.* - Ashton, 2013.
*The spectral Dirichlet-Neumann map for Laplace's equation in a convex polygon,*SIAM J. Math. Anal.**45**(6). - Ashton, 2012.
*On the rigorous foundations of the Fokas method for linear elliptic PDEs*, Proc. Roy. Soc. A**468**(2142). - Ashton & Fokas, 2011.
*A Nonlocal Formulation of Rotational Water Waves,*J. Fluid. Mech.**689**(1). - Ashton, 2011.
*Regularity of Elliptic and Hypoelliptic Operators via the Global Relation*, J. Part. Diff. Eq.**24**(1). - Ashton, 2011.
*On The Non-Existence of Three Dimensional Water Waves with Finite Energy*, Nonlin. Anal. B**12**(4). - Ashton, 2010.
*Stability of Parallel Fluid Loaded Plates: A Nonlocal Approach,*Stud. App. Math.**125**(3). - Ashton & Fokas, 2009.
*A Novel Approach to the Fluid Loaded Plate,*Proc. Roy. Soc. A**465**(2112). - Ashton, 2008.
*Conservation Laws and Non-Lie Symmetries*, J. Nonlin. Math. Phys.**15**(3). - Ashton, 2008.
*The Fundamential k-form and Global Relations,*SIGMA**4**(33).