# RANDOM FIELDS AND THEIR GEOMETRY

## STATS 317 -- Stochastic Processes

 Instructors: Robert Adler, Jonathan Taylor Time: W 12:15pm-3:15pm Place: TBA Syllabus: The formal syllabus mentions: Semimartingales, stochastic integration, Ito's formula, Girsanov's theorem. Gaussian and related processes. Stationary and isotropic processes. Integral geometry and geometric probability. Maxima of random fields and applications to spatial statistics and imaging. What we shall really do: 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. While doing this, we will need to discuss integration with respect to random Gaussian measures, and using this shall give a brief introduction stochastic calculus, at the level of both stochastic differential equations and stochastic partial differential equations. In the second part we shall treat various (differential) 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. We will begin with a necessarily brief introduction to integral and differential geometry, which will be our "language of choice" for much of the rest of the course. After introducing some fundamental differential geometric concepts, we describe some of the fundamental formulae of integral geometry and their uses: Steiner-Weyl formulae for the volume of tubular neighbourhoods, the Gauss-Bonnet theorem and Chern-Federer's Kinematic Fundamental Formula. The final non-stochastic ingredient we cover is a small part of Morse theory, which relates the Euler-Poincare characteristic to the critical points of so-called Morse functions. Using Morse theory, we compute the expected Euler characteristic of the excursion set for a large class of smooth processes, which serves as an approximation to the distribution of the supremum of such processes. While going through this exercise, our earlier excursion to integral geometry pays off, revealing some nice connections to integral geometry. Finally, we conclude with some applications of suprema distributions to various statistical problems, some of which include likelihood ratio tests for signal detection in smooth Gaussian noise; control of Family Wise Error Rate in functional neuroimaging analyses; confidence bands in functional linear regression. More details are in the lecture outline. 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 suit 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. The main aim of the course is two-fold: On the theoretical front, it is aimed to try to blend some problems in probability and geometry. On the applied side, it is aimed at giving statisticians the theoretical tools for handling a variety of real problems related to random geometry. For some of these applications, including the areas of medical imaging and astrophysics, look here. For more medical imaging applications, take a look at some of Keith Worsley's papers found here. Textbook: See the "Course Outline" below for information on downloading what is needed. Grade: The final grade will depend on a combination of homework and presentations given by students at the end of the semester. The final structure will be determined by mutual agreement within the first two weeks of the quarter. Prerequisites: You will need a graduate course in Probability, and one in Stochastic Processes would not hurt but is less necessary. No background in Geometry is assumed. We shall develop what is needed during the course. Additional information: If you want extra information, you can reach us at robert@ieadler.technion.ac.il. and/or jtaylor@stat.stanford.edu Assignments: