# > BUC-XIV 20.08.18 - 24.08.18

#### Probability challenges

The BUC will centre around a collection of short courses earlier in the week. During the short courses, some tractable problems will be offered to PhD students who will be given on the Tuesday. Students have until Friday to solve in groups and present, thereby becoming better familiar with the material.

## Organizers

Andreas Kyprianou (University of Bath, UK)
Juan Carlos Pardo (CIMAT, México)

## Courses

 Course Title A Mariana Olvera-Cravioto (Berkeley) Branching Distributional Equations and their Applications This course will be centered around the theory and applications of distributional equations of the form: $R \stackrel{D}{=} \Phi( N, Q, {C_i}, {R_i})$ where the $\{R_i\}$ are i.i.d. copies of $R$, independent of the vector $(N, Q, {C_i})$, and \Phi is a deterministic map. Such equations arise in a wide range of settings, ranging from the analysis of computer algorithms (e.g. Quicksort and PageRank) and the study of queueing networks with synchronization, to applications in biology and statistical physics. Although the solutions to these equations are not in general unique, we are often interested in one particular solution, the so-called attracting endogenous solution. These special solutions can be constructed on a mathematical structure known as a weighted branching process, and their asymptotic behavior can be explicitly computed in many important cases. To complement the theory, we will also introduce a Monte Carlo algorithm based on the bootstrap that can be effectively used for most choices of \Phi.   In terms of applications, the course will focus on two specific cases, the linear equation (known as the smoothing transform) and the maximum equation (known as the high-order Lindley equation). The former appears in the analysis of Quicksort and Google’s PageRank, while the latter in the analysis of queueing networks with synchronization. The PageRank example in particular will allow us to explore in more detail the connection between branching distributional equations and complex networks. In particular, we will discuss two different types of random graph models that, due to their local tree-like structure, allow us to make the connection between the graph and a (weighted) branching process precise. The coupling techniques involved lie at the core of much or random graph theory, and are therefore of independent interest. B Andreas Kyprianou (University of Bath) Self-similar Markov processes C Elie Aidekon (Paris VI) CLE fragmentation

## Timetable

 Monday 08:30-09:30 Breakfast 09:30-10:30 A 10:30-11:00 Coffee 11:00-12:00 B 12:00-12:30 Coffee 12:30-13:00 PhD presentation 13:00-13:30 PhD presentation 13:30-15:00 Lunch 15:00-17:00 Work sessions 17:00-18:00 C 18:00-20:00 Welcome on the balcony

 Tuesday 08:30-09:30 Breakfast 09:30-10:30 B 10:30-11:00 Coffee 11:00-12:00 C 12:00-12:30 Coffee 12:30-13:30 Cecile Mailler (University of Bath): 13:30-15:00 Lunch 15:00-17:00 Work sessions 17:00-18:00 A

 Wednesday 08:30-09:30 Breakfast 09:30-10:30 C 10:30-11:00 Coffee 11:00-12:00 A 12:00-12:30 Coffee 12:30-13:30 Alex Watson (Manchester): 13:30-15:00 Lunch 15:00-17:00 Work sessions 17:00-18:00 B

 Thursday 08:30-09:30 Breakfast 09:30-10:30 C 10:30-11:00 Coffee 11:00-12:00 A 12:00-12:30 Coffee 12:30-13:30 Cecile Mailler (University of Bath): 12:30-13:00 13:00-15:00 Lunch 15:00-17:00 Work sessions 17:00-18:00 B

 Friday 08:30-09:30 Breakfast 09:30-10:30 A 10:30-11:00 Coffee 11:00-12:00 B 12:00-12:30 Coffee 12:30-13:30 Soluton presentations 13:30-15:00 Lunch 15:00-17:00 Solution presentations 17:00-18:00 C

## Participants

Bath CIMAT UNAM Other
Andreas Kyprianou
Sandra Palau
Emma Horton
Dorottya Fekete
Minmin Wang
Isaac Gonzalez
Cecile Mailler
Alice Callgero
Juan Carlos Pardo
Victor Rivero
Hélene Leman
Natalia Cardona Tobón
Camilo González
Jose Luis Perez
Antonio Murillo

Arno Siri-Jegousse
Lizbeth Peñaloza Velasco
Alejandro Hernández Wences
Pablo Jorge Hernández Hernández
Miriam Ramírez García
Osvaldo Angtuncio Hernández

Alex Watson
Elie Aidekon
Mariana Olvera-Cravioto
Kei Noba