Impossibility results on stability of phylogenetic consensus methods

We answer two questions raised by Bryant, Francis and Steel in their work on consensus methods in phylogenetics. Consensus methods apply to every practical instance where it is desired to aggregate a set of given phylogenetic trees (say, gene evolution trees) into a resulting, "consensus" tree (say, a species tree). Various stability criteria have been explored in this context, seeking to model desirable consistency properties of consensus methods as the experimental data is updated (e.g., more taxa, or more trees, are mapped). However, such stability conditions can be incompatible with some basic regularity properties that are widely accepted to be essential in any meaningful consensus method. Here, we prove that such an incompatibility does arise in the case of extension stability on binary trees and in the case of associative stability. Our methods combine general theoretical considerations with the use of computer programs tailored to the given stability requirements

Connessioni minime: il problema classico di Steiner e generalizzazioni

The classical Steiner problem is the problem of nding the shortest graph connecting a given finite set of points. In this seminar we review the classical problem and introduce a new, generalized formulation, which extends the original one to infinite sets in metric spaces.Il problema classico di Steiner richiede di determinare il grafo di lunghezza minimica che connette un dato insieme infinito di punti. In questo seminario rivedremo il problema classico e introdurremo una nuova formulazione piuù generale che estende il problema originario a insiemi anche infiniti in spazi metrici

Structure of metric cycles and normal one-dimensional currents

We prove that every one-dimensional real Ambrosio-Kirchheim normal current in a Polish (i.e. complete separable metric) space can be naturally represented as an integral of simpler currents associated to Lipschitz curves. As a consequence a representation of every such current with zero boundary (i.e. a cycle) as an integral of so-called elementary solenoids (which are, very roughly speaking, more or less the same as asymptotic cycles introduced by S. Schwartzman) is obtained. The latter result on cycles is in fact a generalization of the analogous result proven by S. Smirnov for classical Whitney currents in a Euclidean space. The same results are true for every complete metric space under suitable set-theoretic assumptions

Flows of measures generated by vector fields

We show that for a large class of measurable vector fields in the sense of N. Weaver (i.e. derivations over the algebra of Lipschitz functions), the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure flows along'' the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense

The Steiner problem for infinitely many points

Let AA be a given compact subset of the euclidean space. We consider the problem of finding a compact connected set SS of minimal 11-dimensional Hausdorff measure, among all compact connected sets containing AA. We prove that when AA is a finite set any minimizer is a finite tree with straight edges, thus recovery the classical Steiner Problem. Analogously, in the case when AA is countable, we prove that every minimizer is a (possibly) countable union of straight segments

Optimal transportation networks as flat chains

We provide a model of optimization of transportation networks (e.g. urban traffic lines, subway or railway networks) in a geographical area (e.g. a city) with given density of population and that of services and/or workplaces, the latter being the destinations of everyday movements of the former. The model is formulated in terms of the Federer‐Fleming theory of currents, and allows us to get both the position and the necessary capacity of the optimal network. Existence and some qualitative properties of solutions to the relevant optimization problem are studied. Also, in an important particular case it is shown that the model proposed is equivalent to another known model of optimization of a transportation network, the latter not using the language of currents