# Dr Anthony Ashton

## Career

**2013-2016:**Stokes Fellow, Pembroke College, University of Cambridge.**2010-****2013:**Junior Research Fellow, Emmanuel College, University of Cambridge.**2009-2010:**EPSRC Prize Fellow, DAMTP.**2006-2009:**PhD Candidate, DAMTP.**2005-2006:**Part III Mathematics, Peterhouse, University of Cambridge.**2002-2005:**BA Mathematics, Peterhouse, University of Cambridge.

## Research

Anthony is a member of the Department of Applied Mathematics and Theoretical Physics and works in the Applied and Computational Analysis group. He did his PhD under the supervision of Prof. A.S. Fokas. 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

He is 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 are below.

There are also some handouts:

- Hand Out 1 (notation)
- Hand Out 2 (distributions supported at a point)
- Hand Out 3 (fundamental solution to heat operator)

He also lectures 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.

- Hand Out 1 (generating functions)
- Hand Out 2 (Arnold-Liouville worked example)
- Hand Out 3 (evolution of scattering data)
- Hand Out 4 (from Lax pairs to zero curvature)
- Hand Out 5 (Painleve transcendents)

## 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 study of the spectral Dirichlet-Neumann Map*, submitted. - Ashton, 2014.
*Laplace's Equation on Convex Polyhedra via the Unified Method*, Proc. Roy. Soc. A (in press). - 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. (in press)*.* - 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).