Seminars

In the talk, we discuss existing approaches, known from the literature, to detection of stochastic ordering of the two survival curves as well as pose and solve the novel testing problem on it. Specifically, the null hypothesis asserts the lack of the ordering, while the alternative expresses its existence. An introduced test statistic is a functional of the standardized two-sample Kaplan-Meier process sampling in a randomly selected number of the random points being the observed survival times in the pooled sample and exploits the information contained in a specially defined one-sided weighted log-rank statistic. It automatically weighs the magnitude and sign of their components becoming a sensible procedure in the considered testing problem. As a result, the corresponding test asymptoticly controls the errors of both kinds at the specified significance level α. The conducted simulation study shows that the errors are also satisfactorily controlled when sample sizes are finite. Furthermore, in the comparison to the best and most popular tests, the new solution turns out to be a promising procedure which improves them upon. A real data analysis confirms that findings.

The general motivation standing behind this research is to understand relationships between dynamical and model-theoretic properties of definable [topological] groups and between dynamical properties of groups of automorphisms of first order structures and model-theoretic properties of the underlying theories. More specifically, our goal is to understand model-theoretic consequences of various notions of amenability.

Among the notions of amenability that we are interested in are: definable amenability of a definable group, classical amenability of a topological group, and, more generally, [weak] definable topological amenability of a definable topological group. We also introduce and study amenable theories.

The consequences of amenability that we obtain are the appropriate versions of G-compactness: for first order theories this is the equality of Lascar strong types and Kim-Pillay strong types; for definable [topological] groups this is the equality of suitably defined connected components $G^{000}$ and $G^{00}$ of the group $G$ in question.

Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions.

My series of talks will be based on my preprint “Amenability and definability” joint with Ehud Hrushovski and Anand Pillay. In the first series of talks, I will focus on the context of definable [topological] groups; the second series will be devoted to our new notion of amenable theory.

The notion of inverse limits was generalized by Ingram and Mahavier to multivalued settings. We investigate topological properties that are preserved by those generalized inverse limits. We have +theorems about local connectedness, trivial shape, arc-likeness, tree-likeness, dimension etc. The talk is illustrated by many examples.