====== Exponential asymptotics for the physical sciences ======

This will be a course page for the Autumn 2024 [[https://www.maths.ox.ac.uk/groups/tcc|Taught Centre Course]] 

==== Outline ====

There are many interesting situations where exponentially-small phenomena, beyond-all-orders of regular perturbative expansions, dictate crucially important features of a physical problem. For example, such exponential asymptotics may govern the appearance of surface ripples in water. Or they may dictate the non-existence of solutions to a differential equation. This course will introduce you to a fascinating array of such problems arising in the physical sciences. It will be accessible to a wide range of students from different mathematical backgrounds; familiarity with asymptotic analysis is useful but not strictly necessary. 

==== Key details ====

All meetings are online and commence 7 October and run for 8 weeks.

Meeting times: 
  * Mondays 3-4pm
  * Wednesdays 3-4pm

Please feel free to email me at [[mailto:p.trinh@bath.ac.uk|p.trinh@bath.ac.uk]] for any enquiries.

==== Tentative list of topics ====

  - An introduction to the history of exponential asymptotics; initial review
  - Review of essential techniques in asymptotic analysis
  - The exponential integral and complementary error function
  - The Airy equation: from steepest descents to the Borel plane
  - The Pearcey equation and the higher-order Stokes phenomenon I
  - The Pearcey equation and the higher-order Stokes phenomenon II
  - On the universality of exponential asymptotics; methods for linear and nonlinear ODEs; dendritic crystal growth 
  - Linear eigenvalue problems and the quantum harmonic oscillator
  - Water waves at low speeds I
  - Water waves at low speeds II
  - Viscous fingering I
  - Viscous fingering II
  - PDEs I
  - PDEs II