====== 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]] 

==== Live stream ====

This course will be live streamed on Mondays and Wednesdays at 3pm on Twitch via https://www.twitch.tv/ptrinh

[[https://www.twitch.tv/ptrinh|{{::ea4sci:twitch.jpeg}}]]

==== Video archives ====

The videos can be found either on Twitch or on a separate [[https://www.youtube.com/playlist?list=PLILvrDLrWEryhUahC1nHMFgSHGVkehcWl|YouTube playlist access here]]. Note that it can take time for videos to be uploaded on YouTube but feel free to email me if there's one you need right away.



==== 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

==== Lecture notes ====

Please do not distribute.

  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap01_motivation.pdf|Chapter 1: motivation]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap02_review.pdf|Chapter 2: review ]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap03_MAE.pdf|Chapter 3: matched asymptotic expansions]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap04_integralapproximation.pdf|Chapter 4: integral approximations]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap05_steepestdescents.pdf|Chapter 5: the method of steepest descents]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap06_exponentialintegral.pdf|Chapter 6: the exponential integral]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap07_airy.pdf|Chapter 7: the Airy equation]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap08_pearcey.pdf|Chapter 8: the Pearcey equation (not covered in lectures)]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap09_rules.pdf|Chapter 9: rules of exponential asymptotics]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap10_crystal.pdf|Chapter 10: dendritic crystal growth]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap11_lineareig.pdf|Chapter 11: linear eigenvalue problems]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap12_lowfroude.pdf|Chapter 12: low Froude water waves]] 
  - [[http://www.ptrinh.com/trinh_papers/EA4Sci_chap13_fingering.pdf|Chapter 13: Saffman-Taylor viscous fingering]]