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-Forstneri\v{c} 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 1930's, 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 article we present 3 classes of properties: 1. density property, 2. flexibility 3. Oka-Forstneri\v{c}. 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.Comment: thanks added, minor correction

A strong Oka principle for embeddings of some planar domains into CxC*

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into ℂ×ℂ∗, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to ℂ×ℂ∗ of recent results of Wold and Forstnerič on the long-standing problem of properly embedding open Riemann surfaces into ℂ2, with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson’s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.Tyson Ritte

Reconstructing Distances among Objects from Their Discriminability

continuous stimulus space, discrete stimulus space, discrimination, Fechnerian Scaling of Discrete Object Sets (FSDOS), Multidimensional Fechnerian Scaling (MDFS), Nonconstant Self-Dissimilarity, Regular Minimality, psychometric function, same-different judgments, subjective distance,