# Bram Petri

## Random Methods in Geometry

### Practical information

• Semester: 2016/2017 - summer
• Course code: V5D4 - Selected Topics in Geometry
• Location: Endenicher Allee 60 - 0.008
• Time: Thursday 10:00 - 12:00
• Exam:
• First opportunity:
27 July 2017
14:00-16:00
GHS
• Second opportunity:
18 September 2017
9:00 - 11:00
KHS

### Contents

Random methods provide a way to answer questions of the form: "what does a typical [insert your favorite object here] look like?" A classical example is the theory of random graphs. Random graphs have not only been used to study the typical behavior of large graphs but also to provide existence proofs. The latter application is often called "the probabilistic method". The idea of this method is that it is sometimes easier to show that certain behavior appears with a non-zero probability (in a suitable model) than to explicitly construct objects that exhibit this behavior.

The goal of this course will be to discuss applications of these methods in geometry. Subjects will include:
• Some basic probability theory, in particular Poisson approximation.
• Random graphs: counting the number of regular graphs and expansion.
• Random hyperbolic surfaces: The genus distribution and pants decompositions.
The list above is however subject to change depending on the audience.

### Prerequisites

• Necessary: Point-set topology (topological spaces, continuity, etc.), analysis, linear algebra.
• Useful: Differential geometry (manifolds, curvature, etc.), basic probability theory (probability measures, independece, expected value, etc.), representation theory of finite groups.

### Lecture notes

I will post my notes here after each lecture. The exercises for every week can be found on the last pages of these notes.
DISCLAIMER: I do not guarantee in any way that these notes are correct. I will be happy to hear of any mistakes that are found.

The complete set of notes can be found here.

The (less up to date) versions that were posted weekly are here:

### Exam material

Here is a note that gives an overview of what you are expected to know for the exam. As a general rule, you should not worry if you have understood the material and are able to do the exercises.

### Literature

Probability theory:
• Santosh S. Venkatesh. The theory of probability. Explorations and applications. Cambridge: Cambridge University Press, 2013.
- Chapters XV and XVIII.
Graph Theory:
• Bela Bollob‌as. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin., 1(4): 311 - 316, 1980.
• Bela Bollobas. Random graphs. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985.
- Chapter 2.4.
• Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4): 439 - 561, 2006.
- Sections 2 and 7.
• N. C. Wormald. Models of random regular graphs. In Surveys in combinatorics, 1999 (Canterbury), volume 267 of London Math. Soc. Lecture Note Ser., pages 239 - 298. Cambridge Univ. Press, Cambridge, 1999
Hyperbolic geometry:
• Alan F. Beardon. The geometry of discrete groups Springer-Verlag, New York, 1995.
- Chapter 7.
• Peter Buser. Geometry and spectra of compact Riemann surfaces. Modern Birkh ̈auser Classics. Birkh ̈auser Boston, Inc., Boston, MA,
- Chapters 1 and 3
Random surfaces:
• Larry Guth, Hugo Parlier, and Robert Young. Pants decompositions of random surfaces. Geom. Funct. Anal., 21(5):1069 - 1090, 2011.
• Robert Brooks and Eran Makover. Random construction of Riemann surfaces. J. Differential Geom., 68(1):121 - 157, 2004.