We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

Several studies have linked DNA methylation at individual CpG sites to aging and various diseases. Recent studies have also identified single CpGs whose methylation levels are associated with all-cause mortality. In this study, we perform an epigenome-wide study of the association between CpG methylation and mortality in a population of 435 monozygotic twin pairs from three Danish twin studies. The participants were aged 55–90 at the time of blood sampling and were followed for up to 20 years. We validated our results by comparison with results from a British and a Swedish cohort, as well as results from the literature. We identified 2806 CpG sites associated with mortality (false discovery rate ( $\mathrm{FDR})<0.05$ ), of which 24 had an association p-value below ${10}^{-7}$ . This was confirmed by intra-pair comparison controlling for confounding effects. Eight of the 24 top sites could be validated in independent datasets or confirmed by previous studies. For all these eight sites, hypomethylation was associated with poor survival prognosis, and seven showed monozygotic correlations above 35%, indicating a potential moderate to strong heritability, but leaving room for substantial shared or unique environmental effects. We also set up a predictor for mortality using least absolute shrinkage and selection operator (LASSO) regression. The predictor showed good performance on the Danish data under cross-validation, but did not perform very well in independent samples.

We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hörmander condition is satisfied such that the Brownian motions have smooth transition densities. We identify the most probable paths for the underlying Euclidean Brownian motion and discuss small time asymptotics of the transition densities on the manifold. The geometric setup yields an intrinsic approach to the estimation of mean and covariance in non-linear spaces.

Multiple epigenome-wide association studies have been performed to identify DNA methylation patterns regulated by aging or correlated with risk of death. However, the inter-relatedness of the epigenetic basis of aging and mortality has not been well investigated. Using genome-wide DNA methylation data from the Lothian Birth Cohorts, we conducted a genome-wide association analysis of all-cause mortality and compared this with age-associated methylation patterns reported on the same samples. Survival analysis using the Cox regression model identified 2552 CpG sites with genome-wide significance (false discovery rate < 0.05) for all-cause mortality. CpGs whose methylation levels are associated with increased mortality appear more distributed from the gene body to the intergenic regions whereas CpGs whose methylation levels are associated with decreased mortality is more concentrated at the promoter regions. In comparison with reported CpGs displaying significant age-dependent methylation patterns in the same samples, we observed a limited but highly significant overlap between mortality-associated and age-associated CpGs (p value 2.52e−06). Most importantly, the overlapping CpGs are dominated by those whose overall age-related methylation patterns reduce the risk of death. All-cause mortality is significantly associated with altered methylation at multiple genomic sites with differential distribution in gene regions for CpGs correlated with increased or decreased risk of death. The age-dependent methylation changes could reflect an active response to the aging process that contributes to maintain individual survival.

The classifying space of the embedded cobordism category has been identified in by Galatius, Tillmann, Madsen, and Weiss as the infinite loop space of a certain Thom spectrum. This identifies the set of path components with the classical cobordism group. In this paper, we give a geometric interpretation of the higher homotopy groups as certain cobordism groups where all manifolds are now equipped with a set of orthonormal sections in the tangent bundle. We also give a description of the fundamental group as a free group with a set of geometrically intuitive relations.Comment: 23 page