Workshop 5

The fifth workshop of the scientific network cumulants, concentration and superconcentration will take place on September 31st  to October 2nd 2019  at University of Osnabrück.

We are looking forward to courses by our guests Christian Hirsch, Ivan Nourdin & Giovanni Peccati.

Monday, 30.9.
all day                Welcome and time to work
13:00 – 14:00  members of the scientific network: “Abschlussbericht und Zukunftsperspektiven”
14:30 – 15:30  Functional CLT for persistent Betti numbers on point processes 1 by Christian Hirsch

Tuesday, 1.10.
09:30 – 10:30   Functional CLT for persistent Betti numbers on point processes 2 by Christian Hirsch
10:30 – 11:30   group work and coffee
11:30 – 12:30   Limit theorems for functionals of random measures: combinatorial aspects and integration by parts 1 by Giovanni Peccati
12:30 – 14:00   lunch break
14:00 – 15:00   Limit theorems for functionals of random measures: combinatorial aspects and integration by parts 2 by  Giovanni Peccati
15:00 – 15:30   coffee break
15:30 – 16:30   Around the Breuer-Major theorem 1 by Ivan Nourdin
16:30 – 17:30   group work

Wednesday, 2.10.
09:30 – 10:30  Around the Breuer-Major theorem 2 by Ivan Nourdin
10:30 – 12:00   group work and coffee
12:00 – 12:30   The volume of simplices in high-dimensional Poisson-Delaunay tessellations by Anna Gusakova
12:30 – 14:00   lunch break
14:00 – 16:00   group work and coffee

The workshop poster can be downloaded here.

University of Osnabrück
Institute for Mathematics,
Albrechtstraße 28a, 49076 Osnabrück
Building No. 69
All talks will take place in the first floor, room 69/125.


Christian Hirsch: Functional CLT for persistent Betti numbers on point processes
The persistence diagram and persistent Betti numbers form one of the key tools in topological data analysis, an emerging methodology to leverage invariants from algebraic topology to extract insights from data. The two lectures are devoted to proving a functional CLT for the persistence diagram on point processes via cumulative methods. In the first lecture, we discuss in detail the role of persistent Betti Numbers in topological data analysis. We start with the intuition, give the precise definition in terms of homology of simplicial complexes and conclude with a selection of current computation algorithms. The second lecture describes how to approach the CLT by combining the Bickel-Wichura machinery with cumulant measures associated with point processes. This program works in the the planar case and for bounded features. We conclude by discussing challenges in higher dimensions and possibly unbounded features. This talk is based on joint work with Christophe Biscio, Nicolas Chenavier and Anne Marie Svane.

Ivan Noudin: Around the Breuer-Major theorem
Let X be a centered, unit-variance and stationary Gaussian sequence with covariance function r. Let f be a square integrable function with respect to the standard Gaussian measure on the real line. Consider the sequence F_n of partial sums associated with f(X_n). The celebrated Breuer-Major theorem provides sufficient conditions on the covariance r and of the Hermite index of f in order for F_n to exhibit Gaussian fluctuations. It has far-reaching applications in many different areas, such as mathematical statistics, signal processing or geometry of random nodal sets.
The goal of this series of of two talks is to introduce the audience to this beautiful result. We will then review some recent applications and extensions. All the needed material will be introduced progressively; no specific knowledge is required, beyond the definitions and very basic properties of Gaussian processes (mostly Brownian motion).

Giovanni Peccati: Limit theorems for functionals of random measures: combinatorial aspects and integration by parts
I will explore some parts of a recent body of work, focussing on limit theorems for (Gaussian or point) random measures obtained by using methods of infinite-dimensional stochastic analysis, such as Malliavin calculus and integration by parts formulae. I will try to highlight some of the combinatorial implications of our main results (mostly related to the so-called Rota-Wallstrom Theory), and also to develop and illustrate some simple geometric applications.