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Dr. Tony Lyons

Title:

Lecturer in Mathematics

School/ Department:

School of Science & Computing - Dept Computing & Mathematics

Room Name/ Num:

311 1969 - Main Building

Campus:

Main Campus

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My Biography

I received a BA in Theoretical Physics from Trinity College Dublin (2004) and a MA in Theoretical Physics from Vrije Universiteit Amsterdam (2007). In 2013 I was awarded a PhD in Mathematical Sciences from Dublin Institute of Technology.

Following this I was an Irish Research Council Postdoctoral Fellow in University College Cork. Since September 2015, I have been a lecturer in mathematics in the Department of Computing and Mathematics here in Waterford Institute of Technology.

Research

My main area of research is in the nonlinear analysis of fluid models, especially in relation to hydrodynamics. Some of my research interests include

Inverse scattering - A method of solving weakly nonlinear hydrodynamic models from an associated spectral problem. This may be thought of as the nonlinear version of the Fourier transform method used to solve linear Partial Differential Equations (PDE).

Nonlinear analysis - This applies methods from the analysis of PDE, such as maximum principles and conformal mapping theory, which allows us to make rigorous statements about fully nonlinear hydrodynamic models.

Geometric analysis of fluid models - The Euler equation of hydrodynamics along with several of its weakly nonlinear approximations may be represented as geodesic flows in appropriately chosen infinite dimensional spaces. Using these methods, my collaborators and I have recently formalised this geometric structure for a family of fluid models with vorticity (fluid rotation).

Geophysical fluid dynamics - The famous Gerstner solution discovered in 1802 (and re-discovered several times since) has recently been extended to incorporate numerous hydrodynamic and geophysical effects. These waves may also posses cusps at the wave troughs which lead to several mathematical complications due to lack of regularity. In recent work I have shown that these extreme waves are possible as internal waves in stratified water waves near the equator.

Teaching

Lecturing

Some of the courses I lecture here in WIT include

  • Engineering Maths
  • Maths Calculus
  • Maths Advanced Calculus
  • Topics in Engineering Maths
  • Maths Ordinary Differential Equations
  • Data Visualisation
  • Network Theory Fundamentals

 

Project Supervision

Recently, as a research collaboration with a final year Physics for Modern Technology student, we succesfully constructed soliton solutions for a nonlinear shallow-water model known as the Kaup-Boussinesq equation. This work has recently been accepted for publication in a peer review journal (Haberlin, J. & Lyons, T. Eur. Phys. J. Plus (2018) 133: 16), and a preprint is available at

https://www.researchgate.net/publication/316955511_Solitons_of_shallow-water_models_from_energy-dependent_spectral_problems