**Instructor: **Robert J. Adler

**Teaching Assistant: **Sreekar Vadlamani

**Times and places: **

Lectures: Thursday, 10:30-12:30, Bloomfield Building (IE&M) 310.

Exercise class: To be determined

**Place: **Bloomfield Building (IE&M) Room 310.

**Content: **The course will, essentially, be in two parts.

In the first we shall treat the general theory of random (mainly Gaussian) processes on quite arbitrary parameter spaces: e.g. high dimensional Euclidean space, general metric spaces, and manifolds. Most of this theory (e.g. continuity and boundedness issues) is independent on the geometry of the parameter space.

In the second part we shall treat various geometric problems related to random fields. Much of this (e.g. global behaviour) is geometry specific, and makes for a nice blend of Probability and Geometry.

**Who might be interested in this course: **If you are interested in probability or stochastic processes, then the material of this course is as essential to your education as courses in Markov processes, martingales, diffusions etc.

If you are primarily a statistician, but want to also understand the stochastic processes background to topics like empirical measures (essentially histograms for multivariate or other non-ordered data) or to understand the basis for many (Kolmogorov-Smirnov like) statistical tests, then this course may suite you.

If you are interested in geometry (esp. integral or differential) then you should find here an interesting application of geometry in a random setting that is quite different to the usual deterministic one.

**Textbook: **A new book I am writing with Jonathan Taylor, entitled *Random Fields and Geometry *, and available as a free download. I strongly recommend bringing a copy to lectures.

**Grade: **The final grade will (probably) depend on a combination of homework and either presentations given by students at the end of the semester or a take home exam. The final structure will be determined by mutual agreement within the first two weeks of the semester.

**Prerequisites: **Students will need basic knowledge of stochastic processes at the level of Stochastic Processes 98413 (IE&M) or 106429 (Mathematics) as well as basic measure theoretic probability at the level of IE&M’s Theory of Probability 098416 or Math’s Advanced Probability 106349. Some basic analysis, as in Math’s Elements of Modern Analysis for Electrical Engineering 108324 is also needed.

## Course Outline

The following outline is not cast in stone: I will probably change it depending on who takes the course and what we turn out to be interested in, but it should give you an idea of where I am heading.

Weeks | Topic |
---|---|

1 |
Gaussian processes. The Brownian family of processes |

2 |
Borell and Slepian inequalities and their variations. Isoperimetric inequalties. |

3 |
Zero-one laws for Gaussian processes. Karhunen-Loeve expansions. |

4-5 |
Majorising measures, entropy and boundedness/continuity. |

6-7 |
Suprema distributions, Tube formulae. |

8-9 |
Generalised Rice formulae for Gaussian processes from R^d to R^d. Number of critical points, etc. |

10-11 |
Global geometrical structure of Gaussian processes, including excursions into Integral Geometry and Differential Topology. ( Not assumed prerequisites.) |

12 |
Back to tube formulae. |

13-14 |
Real valued Gaussian and non-Gaussian processes on differentiable manifolds. |