We generalize the normalized combinatorial Laplace operator for graphs by defining two Laplace operators for hypergraphs that can be useful in the study of chemical reaction networks. We also investigate some properties of their spectra.

The Minkowski functional is a series of geometric quantities including the volume, the surface area, and the Euler characteristic. In this talk, we consider the Minkowski functional of the excursion set (sup-level set) of an isotropic smooth random field on arbitrary dimensional Euclidean space. Under the setting that the random field has weak non-Gaussianity, we provide the perturbation formula of the expected Minkowski functional. This result is a generalization of Matsubara (2003) who treated the 2- and 3-dimensional cases under weak skewness. The Minkowski functional is used in astronomy and cosmology as a test statistic for testing Gaussianity of the cosmic microwave background (CMB), and to characterize the large-scale structures of the universe. Besides, the expected Minkowski functional of the highest degree is the expected Euler-characteristic of the excursion set, which approximates the upper tail probability of the maximum of the random field. This methodology is used in multiple testing problems. We explain some applications of the perturbation formulas in these contexts. This talk is based on joint work with Takahiko Matsubara.

When studying survival data in the presence of right censoring, it often happens that a certain proportion of the individuals under study do not experience the event of interest and are considered as cured. The mixture cure model is one of the common models that take this feature into account. It depends on a model for the conditional probability of being cured (called the incidence) and a model for the conditional survival function of the uncured individuals (called the latency). This work considers a logistic model for the incidence and a semiparametric accelerated failure time model for the latency part. The estimation of this model is obtained via the maximization of the semiparametric likelihood, in which the unknown error density is replaced by a kernel estimator based on the Kaplan-Meier estimator of the error distribution. Asymptotic theory for consistency and asymptotic normality of the parameter estimators is provided. Moreover, the proposed estimation method is compared with a method proposed by Lu (2010), which uses a kernel approach based on the EM algorithm to estimate the model parameters. Finally, the new method is applied to data coming from a cancer clinical trial.

This talk aims at providing a gentle (and accessible) overview of abstract topological dynamics (biased by my personal preferences and work) in the regime of discrete spectrum. In order to avoid unnecessary abstraction, we will focus on some prominent examples: the point-set dynamics associated to mathematical quasicrystals, strange non-chaotic attractors and--if time allows--substitutive subshifts. Despite their seemingly unrelated nature, these systems show striking similarities and give an idea of how disorder/chaotic behaviour occurs in minimal systems.