Riemann-Hilbert problems
Welcome to Riemann-Hilbert problems, a course hosted by the Taught Course Center of Bath-Bristol-Imperial-Oxford and Warwick. This page summarizes all key information of the module and contains all course materials, most importantly our weekly schedule.
Key organizational information:
Instructor: Thomas Bothner
Office: 2A.15, Fry Building
Contact: thomas.bothner@bristol.ac.uk and 0117 428 4992
Meeting times: Thursdays 10:00am-noon starting April 28th, 2022 via Microsoft TEAMS
This course is suitable for students interested in Riemann-Hilbert problems (RHPs), in particular students interested in their rigorous mathematical theory and their applications to problems in modern mathematical physics. Topics will include
Informal appearances of RHPs (Painleve-II, mKdV, orthogonal polynomials on the real line)
Precise definition of a RHP (Cauchy operators, Carleson curves, Fredholm theory)
RHPs in applications (integrable operators, Wiener-Hopf method, nonlinear Schroedinger equation, Painleve-II)
Asymptotic analysis of RHPs (nonlinear steepest descent method)
There is no exam for this course. If you are taking the module for credit you must complete the posted exercises and email me your solutions in LaTeX typeset by the due date written on the exercise sheet.
This course is offered exclusively to PhD students and Postdocs at the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London. If you are eligible and interested in this course please sign up via tcc@maths.ox.ac.uk or contact me directly if there are any issues. The course will run for 8 consecutive weeks, see the weekly schedule below.
Course materials (click the links):
Welcome to Riemann-Hilbert problems - Introductory Slides
Lecture Notes (June 16th, 2022 version - let me know about typos)
Exercises (May 26th, 2022 version)
Weekly schedule:
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Week 1, April 28th, 2022 from 10:00am-noon Bristol time:
Topics: Overview, Informal appearances of RHPs
Materials: Lesson Notes
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Week 2, May 5th, 2022 from 10:00am-noon Bristol time:
Topics: Cauchy operators, Carleson curves
Materials: Lesson Notes
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Week 3, May 12th, 2022 from 10:00am-noon Bristol time:
Topics: The precise RHP
Materials: Lesson Notes
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Week 4, May 19th, 2022 from 10:00am-noon Bristol time:
Topics: Integrable operators, Wiener-Hopf method, nonlinear Schroedinger equation
Materials: Lesson Notes
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Week 5, May 26th, 2022 from 10:00am-noon Bristol time:
Topics: Nonlinear Schroedinger equation, Painleve-II
Materials: Lesson Notes
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Week 6, June 2nd, 2022 from 10:00am-noon Bristol time:
Topics: Painleve-II, generalized matrix factorizations
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Week 7, June 9th, 2022 from 10:00am-noon Bristol time:
Topics: Nonlinear steepest descent method (Szego's strong limit theorem)
Materials: Lesson Notes
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Week 8, June 16th, 2022 from 10:00am-noon Bristol time:
Topics: Nonlinear steepest descent method (long-time asymptotics for NLS, and large degree OPRL)
Materials: Lesson Notes