Instructor: Fabrizio Bianchi

E-mail: fbianchi at imperial.ac.uk

Office: Huxley 614

Office hours: Friday 2:00-3:00. Feel free to write me a mail to arrange a different appointment if this time is not convenient for you.

Student representative for the course: Hamaad Yousaf (hry14 at ic.ac.uk)

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If you are following this class, or anyway (vaguely) plan
to take this exam, please email me. I need this to know how many people are interested in order to ask for the recording and possibly move the class to a more convenient time for you.
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Lectures
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Monday 10:00-11:00, room Huxley 342 (room 658 for the last week)

Wednesday 9:00-10:00, room Huxley 342 (room 658 for the last week)

Friday 3:00-4:00, room Huxley 139

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Description of the course
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This elementary course starts with introducing surfaces that come from special group actions (Fuchsian/Kleinian groups). It turns out that on such surfaces one can develop a beautiful and powerful theory of iterations of conformal maps, related to the famous Julia and Mandelbrot sets. In this theory many parts of modern mathematics come together: geometry, analysis and combinatorics.

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Topics we cover in the course
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Part 1) Hyperbolic metrics and Moebius transformations, Fuchsian groups, Riemann surfaces

Part 2) Conformal dynamics on C: Fatou and Julia set, periodic points, local normal forms, global dynamics

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Prerequisites
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The course will try to be as elementary and self contained as possible, defining all the necessary notions from topology, complex analysis and dynamics. In particular, the class
M3P60 (Geometric complex analysis) is not required.

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Lecture notes
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Here you can download the last version of the lecture notes. I will put them online
as soon as the course advances.

I prefer to post non-definitive versions as soon as possible in order to help students that cannot attend some class. Please contact me if you find any error!

For the first part of the course, I will be following (a subset of) the notes "Hyperbolic geometry" by Charles Walkden, that you can find
here. From Chapter 3, I will be mainly following part of Milnorâ€™s book "Dynamics in one complex variable".

Chapter 1:
Basics and Moebius transformations.

Chapter 2:
Fuchsian groups

Chapter 3:
Riemann surfaces

Chapter 4:
Dynamics on Riemann surfaces: generalities

Chapter 5:
Dynamics on Riemann surfaces: local dynamics

Chapter 6:
Dynamics on Riemann surfaces: global dynamics

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Assessment
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You will be required to hand in two sets of homeworks. These will form 10 percent of the final mark for the module.

The first problem sheet is posted
here
on Friday (Jan 27). The due date is February 16. Please submit your work by 16:00 on February 16 to the Undergraduate office on floor 6 in the Huxley building.

The second problem sheet is posted
here
on Wednesday (Mar 01). The due date is March 16. Please submit your work by 16:00 on March 16 to the Undergraduate office on floor 6 in the Huxley building.

The remaining 90 percent is determined by a two-hour written examination at the end of the year.

A mock exam will be posted here soon.

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References for background/further reading
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Kleinian Groups, by Berbard Maskit,

The Geometry of Discrete Groups, by Alan F. Beardon,

Dynamics in one complex variable, by John Milnor

Riemann surfaces, dynamics and geometry, lecture notes by Curtis McMullen

Hyperbolic Geometry, lecture notes by Charles Walkden