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Contents 1 Connections on Riemann Surfaces and the Stokes Phenomenon 1.1 Outline 1.2 Student work 1.3 References 1.4 Lecture notes

Connections on Riemann Surfaces and the Stokes Phenomenon

Outline

Meromorphic connections on Riemann surfaces appear in many parts of mathematics: they played a major role in the 19th century, in the form of ordinary differential equations with singularities in the complex domain. The Stokes phenomenon was discovered in this context, as an asymptotic discontinuity in the behaviour of solutions, such as the Airy functions in geometric optics or the Gauss confluent hypergeometric functions. One highlight of this work was the discovery in 1900 of the Painlevé transcendants and the statement of Hilbert's 16th problem concerning the monodromy map.

In the 20th century, there was a focus on the appearance of these objects in completely integrable systems: the isomonodromic deformation theory provides a multitude of integrable systems which underlie a vast number of previously-known systems, such as reductions of the Yang-Mills equations. Most recently, meromorphic connections have appeared in two new contexts: first in understanding the behaviour of stability conditions on Abelian categories in algebraic geometry, and second in providing a new interpretation of the Feynman integral of quantum field theory.

In this course, we shall explore a small part of the vast literature on meromorphic connections, leading to an investigation of the most recent work of Bridgland-(Toledano-Laredo) and Witten which provides spectacular applications of the subject. We hope to cover the following topics:

• ODEs on Riemann surfaces
• Meromorphic connections
• The Riemann-Hilbert correspondence
• The Stokes Phenomenon and Stokes matrices, solution asymptotics
• The Stokes sheaf
• The Isomonodromic integrable system and Frobenius manifolds
• The Hitchin moduli space and the Geometric Langlands programme
• Relation to the Feynman integral, in particular for Chern-Simons theory

Student work

This will be essentially a reading course + seminar for students. Evaluation will be based on general participation as well as a final project based on reading a paper in the field. Also, there is a possibility that a group of students may assist in the taking of notes for the class, as a substitute for the final project.

References

1. Books
   1. Wasow's textbook on asymptotic methods in ODE
   2. Deligne's text on ODE
   3. Hitchin on Riemann surfaces and bundles
   4. Olver on Asymptotics
   5. Iwasaki-Kimura-Shimomura-Yoshida from Gauss to Painlevé special functions
   6. Dingle's book on Asymptotics From Berry's website.
   7. Coddington and Levinson on ODE
   8. Simpson's text on Asymptotics of ODE
   9. Sabbah on Isomonodrom
   10. Yakovenko-Ilyashenko on Analytic ODE
   11. Valee-Soares on Airy functions and applications to physics
   12. Gray treatise on history of linear ODE
   13. Singer and van der Put on Galois theory of Linear ODE
   14. Forster's text on Riemann Surfaces
   15. Incredible monograph by Zoladek covering all the above
2. Theses
   1. Philip Boalch
   2. Michael Wong on Hecke modifications and isomonodromy via Tyurin (related to the Hitchin, Krichever, Hurtubise papers below)
3. Articles
   1. Kapustin-Witten on geometric Langlands
   2. Witten on gauge theory wild ramification
   3. Gukov-Witten on wild ramification
   4. Witten on analytic continuation of Chern-Simons
   5. Witten on the path integral
   6. Bridgeland-(Toledano-Laredo) on Stokes vs stability conditions
   7. Hitchin on Schlesinger's equations
   8. Woodhouse on isomonodromy extending Hitchin's work
   9. Biquard-Boalch on wild ramification
   10. Krichever on Isomonodromy
   11. Berry on smoothing Victorians
   12. Hurtubise on geometry of isomonodromy
   13. Jimbo-Miwa-Ueno foundational paper on Stokes
   14. Balser-Jurkat-Lutz foundational paper on solving irregular systems and Birkhoff factorization
   15. Boalch on quasi-Hamiltonian geometry
   16. Boalch article based on thesis
   17. Fission - new work by Boalch
   18. Boalch on Klein solution to Painleve
   19. Boalch Isomonodromy for G-bundles and Weyl groups - general groups
   20. Nitsure on logarithmic D-modules
   21. Sabbah on the work of Bolibrukh
   22. Alekseev-Malkin-Meinrenken on Q-Hamiltonian reduction
   23. Alekseev-Bursztyn-Meinrenken on Dirac geometry on Lie groups
   24. Lambert-Rousseau on Stokes phenomenon for confluent hypergeometric equations- a new approach
   25. Martinet and Ramis on the wild fundamental group
4. Sites
   1. Rainbows
   2. Resources on Picard-Fuchs equations and the Gauss-Manin connection

Lecture notes