Arrow Contraction and Expansion in Tropical Diagrams.

Arrow contraction applied to a tropical diagram of probability spaces is a modification of the diagram, replacing one of the morphisms with an isomorphism while preserving other parts of the diagram. It is related to the rate regions introduced by Ahlswede and Körner. In a companion article, we use arrow contraction to derive information about the shape of the entropic cone. Arrow expansion is the inverse operation to the arrow contraction.

Conditioning in Tropical Probability Theory.

We define a natural operation of conditioning of tropical diagrams of probability spaces and show that it is Lipschitz continuous with respect to the asymptotic entropy distance.

The Complex Admixture History and Recent Southern Origins of Siberian Populations.

Although Siberia was inhabited by modern humans at an early stage, there is still debate over whether it remained habitable during the extreme cold of the Last Glacial Maximum or whether it was subsequently repopulated by peoples with recent shared ancestry. Previous studies of the genetic history of Siberian populations were hampered by the extensive admixture that appears to have taken place among these populations, because commonly used methods assume a tree-like population history and at most single admixture events. Here we analyze geogenetic maps and use other approaches to distinguish the effects of shared ancestry from prehistoric migrations and contact, and develop a new method based on the covariance of ancestry components, to investigate the potentially complex admixture history. We furthermore adapt a previously devised method of admixture dating for use with multiple events of gene flow, and apply these methods to whole-genome genotype data from over 500 individuals belonging to 20 different Siberian ethnolinguistic groups. The results of these analyses indicate that there have been multiple layers of admixture detectable in most of the Siberian populations, with considerable differences in the admixture histories of individual populations. Furthermore, most of the populations of Siberia included here, even those settled far to the north, appear to have a southern origin, with the northward expansions of different populations possibly being driven partly by the advent of pastoralism, especially reindeer domestication. These newly developed methods to analyze multiple admixture events should aid in the investigation of similarly complex population histories elsewhere.

Intrinsic Flat and Gromov-Hausdorff Convergence of Manifolds with Ricci Curvature Bounded Below.

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem.