Welcome to Painleve differential equations in the complex plane, a course hosted by the Taught Course Center of Bath-Bristol-Imperial-Oxford-Warwick and Swansea. 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 +44 117 455 2070
Meeting times: Thursdays 12:00-14:00, starting April 27th, 2023 via Microsoft TEAMS
This course is suitable for students interested in Painleve differential equations, in particular students interested in their rigorous complex analytic theory and their analysis from a Nevanlinna theoretic viewpoint. Topics will include
Meromorphic nature of solutions (proof arguments of Hukuhura, Hinkkanen and Laine)
Growth of Painleve transcendents (finite growth order for Painleve-I, Painleve-II and Painleve-IV)
Value distribution of Painleve transcendents (deficiencies and ramification indices, second main theorem)
Selected topics for Painleve-I and Painleve-II (nonexistence of first integrals, solution representation as quotient of entire functions, higher order analogues, rational and classical function solutions)
There is no exam for this course. If you are taking the module for credit you must complete the posted reading assignment and email me your report in LaTeX typeset by the due date written on the assignment sheet.
This course is offered exclusively to PhD students and Postdocs at the Universities of Bath, Bristol, Oxford, Warwick, Swansea and Imperial College London. If you are eligible and interested in this course please sign up via tcc@maths.ox.ac.uk with your MS Teams email address 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 Painleve differential equations in the complex plane - Introductory Slides
Lecture Notes (June 11th version - let me know about typos)
Reading assignments (May 18th version)
Weekly schedule:
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Week 1, April 27th, 2023 from 12:00-14:00 Bristol time:
Topics: Overview, Preliminaries, Meromorphic nature of solutions
Materials: Lesson Notes
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Week 2, May 4th, 2023 from 12:00-14:00 Bristol time:
Topics: Meromorphic nature of solutions (continued)
Materials: Lesson Notes
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Week 3, May 11th, 2023 from 12:00-14:00 Bristol time:
Topics: Basics of Nevanlinna theory (part 1)
Materials: Lesson Notes
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Week 4, May 18th, 2023 from 12:00-14:00 Bristol time:
Topics: Growth of Painleve transcendents
Materials: Lesson Notes
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Week 5, May 25th, 2023 from 12:00-14:00 Bristol time:
Topics: Basics of Nevanlinna theory (part 2), Value distribution of Painleve transcendents
Materials: Lesson Notes
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Week 6, June 1st, 2023 from 12:00-14:00 Bristol time:
Topics: Deficiencies and ramification indices, the SMT for Painleve transcendents
Materials: Lesson Notes
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Week 7, June 8th, 2023 from 12:00-14:00 Bristol time:
Topics: Selected topics for Painleve-I
Materials: Lesson Notes
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Week 8, June 15th, 2023 from 12:00-14:00 Bristol time:
Topics: Selected topics for Painleve-II
Materials: Lesson Notes