**Instructor:**Robert J. Adler**Time:**Tuesdays, 09:30-11:30.(Tentative)**Place:**Meyer (EE) Building, Room 354.**Language of instruction:**To be decided on at the first lecture. Choices are between Hebrish (bad Hebrew) and Strine. (For details on the latter see here or here .)**Content:**The course will, essentially, be in two parts.In the first (most of the semester) 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 interested in signal and/or image processing, and are on the more theoretical side of the subject, you will find a lot here that will open your eyes to topics not normally covered in standard courses in these areas.If you are interested in geometry then you should find here an interesting application of geometry in a random setting that is quite different to the usual deterministic one.**Textbook:**The main source will be Topological complexity of smooth random functions , a set of lecture notes with Jonathan Taylor, that you can also download from here. But I shall also use the unfinished book*Applications of Random Fields and Geometry, Foundations and Case Studies*with Jonathan Taylor and Keith Worsley for examples and homework exercises. A backup source, at a higher mathematical level than we shall (probably) reach in the course, is Random Fields and Geometry**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:**This is the $64,000 question: There are two levels at which this course can be given, MSc or advanced PhD. The MSc level requires undergraduate probability and stochastic processes, along with a decent level of mathematical maturity. The PhD level requires some background in measure theoretical probability, at the level of Foundations of Random Processes (048868) , with corresponding mathematical maturity. Somewhere during the first lecture of the semester we shall decide, together, at exactly what level the course will be give. (Perhaps even both, with the MSc material as the basic course and some extra lectures for the highly motivated.)

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