Translation Surfaces
Summer semester 2020

Important notice

Due to the coronavirus crisis, the lecture will be held online. It is therefore important that, if you are interested in attending the lecture you register on MÜSLI .

You should already have the link to the Zoom classrooms, they are two different links, one for the lectures, one for the exercise class, which will be valid for the whole semester. If you didn't get it, or you lost it, please send us an email

If you have security concerns about Zoom please write us an email. If you are unhappy about the fact that the lectures will be recorded, please write me an email.

Schedule

Main Lecture

• Thursday, 11:15-12:50, online
• Friday, 14:15-15:50, online

Exercise Session

• Wednesday, 2:15-3:45, online

Müsli and MaMpf

There is a page for this course on MÜSLI. Registration is open, please, register on MÜSLI.

There is also a page for this course on MaMpf. Where I'll post lecture notes and try to post video of lectures (this is not guaranteed, though!). If you have problems getting access, write me an email.

Lectures

Week 1	Introduction to Translation Surfaces Fundamental groups and covering spaces	[Fo, 1.3, 1.5]
Week 2	Riemann surfaces	[Fo, 1.1, 1.2]
Week 3	Branched coverings Abelian differentials	[Fo, 1.4]
Week 4	Threee definitions of translation surfaces	[Wr, 1.1]
Week 5	Examples of translation surfaces	[Wr, 1.2]
Week 6	Strata, period coordinates	[Wr, 1.3]
Week 7	No lecture!
Week 8	Dehn Twists, compactnes criterion	[FaMa, 3.1-3.3] [Wr, 1.3]
Week 9	The hyperelliptic locus, connected components, GL(2,R) action	[BC, 3.2] [Wr]
Week 10	Veech groups, Veech surfaces, cylinders	[Wr, 3.1 ] [Ra, ]
Week 11	Affine invariant manifolds, Masur-Veech measure, Straight line flow Periodicity vs minimality	[Wr, 3.2] [Wr, 4.1] [MaTa, 1.6]
Week 12	Masur's criterion for unique ergodicity Non-uniquely ergodic example	[MaTa, Thm 3.8] [MaTa, 3.1]
Week 13	Veech dichotomy, Compactification	[Vo, 3.2] [MaTa, 4.1]
Week 14	Cylinders everywhere, Back to billiards	[MaTa, 4.1] [MaTa, Th 1.10]

Final exam

The final exam will take place in the last week of the semester, on Wednesday, July 29 at 2.00 and on Friday, July 31 between 10 and 12.45. If you didn't fill out the doodle, and you want to take the exam, you should write me an email NOW.

The oral exam will be half an hour long for every student, the first question will be to discuss the solution of one of the exercises. You are welcome to bring your own solutions and have a quick look at them.

Contact

• JProf. Dr. Beatrice Pozzetti

• Dr. Anja Randecker

Office Hours

	Time	Office
JProf. Dr. Beatrice Pozzetti	Wednesday 11-12	online
Dr. Anja Randecker	Tuesday 10-11	online

Description

Translation surfaces are obtained by gluing together finitely many polygons in the Euclidean plane by identifying their sides via translations. Equivalently they can be described as the datum of a holomorphic differential on a Riemann surface or a flat metric on a surface with prescribed singularities. The equivalence of the three different perspectives gives rise to a rich theory, which is an active topic of research. This course is an introduction to this subject.

More precisely: we will talk about different equivalent definitions of translation surfaces and introduce their moduli spaces. We will then discuss some results about geodesics on translation surfaces. We will use, and explain, tools coming from surface topology, complex analysis, ergodic theory, homogeneous dynamics.

Prerequisites

This course is aimed at students who are interested in differential geometry. Students are expected to have a certain familiarity with topology (and fundamental groups), complex analysis and basic knowledge about manifolds. The course will be taught in English. Handwritten notes will be posted on MaMpf.

References

• [Wr] Alex Wright, Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surveys in Mathematical Sciences
• [Ha] Allen Hatcher: Algebraic topology.
• [Fo] Otto Forster: Lectures on Riemann surfaces.
• [FaMa] Benson Farb, Dan Margalit, A Primer on Mapping Class Groups.
• [BC] Andrei Bud, Dawei Chen, Moduli of differentials and Teichmüller Dynamics
• [MaTa] Howard Masur, Serge Tabachnikov, Rational Billiards and Flat Structures
• [Ra] Anja Randecker, Script zur Vortragsreihe Unendliche Translationsflächen
• [Zo] Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry, Volume I.
• [FM] Giovanni Forni and Carlos Matheus, Introduction to Teichmüller dynamics and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, Journal of Modern Dynamics.
• [Yo] Jean-Christophe Yoccoz, Interval exchange maps and translation surfaces, Clay Mathematics Proceedings.