Random Geometry and Topology A (22/23)
Faculteit  Science and Engineering 
Jaar  2022/23 
Vakcode  WMMA04105 
Vaknaam  Random Geometry and Topology A (22/23) 
Niveau(s)  master 
Voertaal  Engels 
Periode  semester I a 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Random Geometry and Topology A (tweejaarlijks 22/23)  
Leerdoelen  At the end of the course, the student is able to: 1. present a variety of random geometric problems and identify relevant tools to solve them. 2. present the definition of all concepts introduced in the course, and the relations between them. This includes concepts related to: polytopes, convex bodies, integral geometry, Poisson processes. 3. present and explain the theorems presented in this course (in particular SlivnyakMecke, kinematic and BlaschkePekanschin formulae) and is able to apply them to random or integral geometric problems. 4. understand the proofs of the most important theorems in this course and is able to reproduce them, as well as producing proofs of small variations of these theorems. 5. design a strategy to tackle an advanced random geometric problem (e.g. compute asymptotics of the expected vertex number of a random polytope), and carry out the computation. 

Omschrijving  The world surrounding us is essentially random. Random geometric structures have countless applications, both from the applied and theoretical world: material science, image analysis, signal processing, polytopal approximation, highdimensional asymptotic convex geometry... The aim of this course is to get an overview of the field of stochastic geometry and its interactions with convex geometry, integral geometry and Poisson process theory. We will consider a large variety of random geometric objects. In particular: 1) Random tessellations (a.k.a. mosaics) such as the Poisson hyperplane, Voronoi and Delaunay tessellations; 2) Random polytopes such as random convex hulls and the typical and zero cells of a tessellation; 3) Random Boolean models; 4) Random geometric graphs. The course is divided in three main parts. Each part starts with an introductory chapter following one of the three first sections of the book chapter "Some classical problems in random geometry" (Calka 2019) and is followed with indepth chapters. The introductory chapters start with a specific planar problem, which is treated in an elementary way. The natural extensions of these problems (to higher dimensional spaces for instance) motivate the subsequent chapters which provide a strong theoretical framework to study a large variety of random geometric objects. We will work in the Euclidean space. If time permits, we will also present the construction of some objects in spherical and hyperbolic geometry. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC), Opdracht (ASM), Werkcollege (T)  
Toetsvorm 
Opdracht (AST), Schriftelijk tentamen (WE)
(In order to pass the student needs to have 5.5 for each of the assignments (2 homeworks and 1 mini research project) and the exam. If the student fails an assignment, she/he gets one opportunity to revise her/his production before the next assignment. If the student fails the exam, she/he can resit once. The final grade is 0.5*Exam + 0.15*Homework_1 + 0.15*Homework_2 + 0.2*Mini_Research_Project.) 

Vaksoort  master  
Coördinator  Dr. G.F.Y. Bonnet  
Docent(en)  Dr. G.F.Y. Bonnet  
Verplichte literatuur 


Entreevoorwaarden  The course assumes knowledge of the following courses: Probability Theory, Probability and Measure, Linear Algebra 1&2, Calculus 1&2, Analysis  
Opmerkingen  Mandatory: Some classical problems in random geometry. Contributed chapter of Stochastic Geometry, edited by David Coupier, Lecture Notes in Math. 2237, Springer, Cham 2019. FREELY AVAILABLE at https://hal.archivesouvertes.fr/hal02377523  Pierre Calka (2019)  9783030135461 Recommended:  Chapters 5,7, 8 and 10 of Stochastic and integral geometry. Schneider and Weil, Probability and its Applications (New York). SpringerVerlag, Berlin, 2008. xii+693  Rolf Schneider and Wolfgang Weil (2008)  9783540788584  Chapters 14 of Lectures on Poisson Process. Last and Penrose. Cambridge University Press, 2018. FREELY AVAILABLE at https://www.math.kit.edu/stoch/~last/page/lectures_on_the_poisson_process/en  Guenter Last and Mathew Penrose (2018)  9781107458437 

Opgenomen in 
