I will survey on the cluster structure of random geometric graphs in a regime that is less discussed in the literature. The statistics of interest include the number of k-components, the number of components, the number of vertices in the giant component, and the connectivity threshold. We show LLN and normal/Poisson approximation by Stein's method. Based on recent joint works with Mathew Penrose (Bath).

TBC

Multi-parameter persistence is a main research topic in topological data analysis. Major questions involve the computation and the structural properties
of persistence modules. In this context, I will sketch two very recent results:

(1) We define a natural bifiltration called the localized union-of-balls bifiltration that contains filtrations studied in the context of local persistent homology as slices. This bifiltration is not k-critical for any finite k. Still, we show that a representation of it (involving algebraic curves of low degree) can be computed exactly and efficiently. This is joint work with Matthias Soels (TU Graz).

(2) Every persistence modules permits a unique decomposition into indecomposable summands. Intervals are the simplest type of summands, but more complicated indecomposables can appear, and usually do appear in examples. We prove that for homology-dimension 0 and density-Rips bifiltration, at least a quarter of the indecomposables are intervals in expectation for a rather general class of point samples. Moreover, these intervals can be ``peeled off'' the module efficiently. This is joint work with Angel Alonso (TU Graz).

A central topic of study in analytic number theory is the behaviour of the Riemann zeta function. Many theorems and conjectures in this area are closely connected to concepts from probability theory. In this talk, we will discuss several results on the typical size of the zeta function on the critical line, over different scales. Along the way, we will see the role that is played by some probabilistic phenomena, such as the central limit theorem and multiplicative chaos.

It is known that there exists certain randomness in the values of the Moebius function. It is widely believed that this randomness predicts significant cancellations in the summation of the Moebius function times any 'reasonable' sequence. This rather vague principle is known as an instance of the 'Moebius randomness principle'. Sarnak made this principle precise by identifying the notion 'reasonable' as deterministic. More precisely, Sarnak's Moebius Disjointness Conjecture predicts the disjointness of the Moebius function from any arithmetic functions realized in any topological dynamical systems of zero topological entropy. In this talk, I will firstly introduce some background and progress on this conjecture. Secondly, I will talk about some of my work on this. Thirdly, I will talk some related problems to this conjecture.