Objectives:
An object can be called generic if it occurs typically, in the sense that its copies can be found in every residual set in an appropriate space of objects. The aim of the project is to study generic objects appearing in several areas of mathematics, finding new tools for constructing and detecting such objects, and exploring their combinatorial structure. Well-known examples of generic objects in model theory are Fraisse limits. Generic structures occur also naturally in topology (e.g. the Cantor set) and various areas of mathematical analysis (e.g. generic Banach spaces). Cohen's set-theoretic forcing offers a strong tool of constructing generic objects, usually needed for specific purposes and not related to Fraisse limits. One of our objectives is to explore links between the set-theoretic method of forcing and model-theoretic methods for constructing universal homogeneous structures.
Objectives:
We plan to investigate problems concerning uniformly continuous and Lipschitz mappings between Banach spaces and their possible applications in other areas of mathematics, such as theoretical computer science, differential equations etc. This project is devoted to the following aspects of the subject:
a) Uniformly continuous and coarse mappings
b) Lipschitz isomorphism
c) Lipschitz free spaces
d) Linear and descriptive properties