Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

Impact of range expansions on current human genomic diversity

The patterns of population genetic diversity depend to a large extent on past demographic history. Most human populations are known to have gone recently through a series of range expansions within and out of Africa, but these spatial expansions are rarely taken into account when interpreting observed genomic diversity, possibly because they are difficult to model. Here we review available evidence in favour of range expansions out of Africa, and we discuss several of their consequences on neutral and selected diversity, including some recent observations on an excess of rare neutral and selected variants in large samples. We further show that in spatially subdivided populations, the sampling strategy can severely impact the resulting genetic diversity and be confounded by past demography. We conclude that ignoring the spatial structure of human population can lead to some misinterpretations of extant genetic diversity.

Models of hybridization during range expansions and their application to recent human evolution

Several lines of genetic, archeological and paleontological evidence suggest that anatomically modern humans (Homo sapiens) colonized the world in the last 60,000 years by a series of migrations originating from Africa (e.g. Liu et al., 2006; Handley et al., 2007; Prugnolle, Manica, and Balloux, 2005; Ramachandran et al. 2005; Li et al. 2008; Deshpande et al. 2009; Mellars, 2006a, b; Lahr and Foley, 1998; Gravel et al., 2011; Rasmussen et al., 2011). With the progress of ancient DNA analysis, it has been shown that archaic humans hybridized with modern humans outside Africa. Recent direct analyses of fossil nuclear DNA have revealed that 1–4 percent of the genome of Eurasian has been likely introgressed by Neanderthal genes (Green et al., 2010; Reich et al., 2010; Vernot and Akey, 2014; Sankararaman et al., 2014; Prufer et al., 2014; Wall et al., 2013), with Papua New Guineans and Australians showing even larger levels of admixture with Denisovans (Reich et al., 2010; Skoglund and Jakobsson, 2011; Reich et al., 2011; Rasmussen et al., 2011). It thus appears that the past history of our species has been more complex than previously anticipated (Alves et al., 2012), and that modern humans hybridized several times with local hominins during their expansion out of Africa, but the exact mode, time and location of these hybridizations remain to be clarifi ed (Ibid.; Wall et al., 2013). In this context, we review here a general model of admixture during range expansion, which lead to some predictions about expected patterns of introgression that are relevant to modern human evolution.

The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.