Geometric and probabilistic limit theorems in topological data analysis

Gauge Theory in Two Dimensions: Topological, Geometric and Probabilistic Aspects

Two dimensional Yang-Mills theory has proved to be a surprisingly rich model, despite, or possibly because of, its simplicity and tractability both in classical and quantum forms. The purpose of this article is to give a largely self-contained introduction to classical and quantum Yang-Mills theory on the plane and on compact surfaces, along with its relationship to ChernSimons theory, illustra...

Quantum algorithms for topological and geometric analysis of data

Extracting useful information from large data sets can be a daunting task. Topological methods for analysing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying topological features and for determining how such features persist as the data is viewed at different scales. Here we present quantum machine learning algor...

Recent Advances on Topological and Geometric Data Analysis

Limit Theorems for Multifractal Products of Geometric Stationary Processes

Abstract: We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein-Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We present the general conditions for the Lq convergence of cumulative processes to the limiting processes and investigate their q-th order moments and R...

Limit theorems for geometric functionals of Gibbs point processes

Observations are made on a point process Ξ in R in a window Qλ of volume λ. The observation, or ‘score’ at a point x, here denoted ξ(x, Ξ), is a function of the points within a random distance of x. When the input Ξ is a Poisson or binomial point process, the large λ limit theory for the total score ∑ x∈Ξ∩Qλ ξ(x, Ξ ∩Qλ), when properly scaled and centered, is well understood. In this paper we es...