Objectives:
Theories of dense and sparse graph limits are one of the most important recent tools of discrete mathematics. Their emergence and development have led to many breakthroughs on old problems in extremal graph theory and random graph theory, and especially have linked discrete mathematics to areas such as probability theory, functional analysis or group theory in a profound way. Recognitions related to the development of the field include the 2012 Fulkerson Prize, the 2013 Coxeter-James Prize, and the 2013 David P. Robbins Prize.
The project will study the theories of dense a sparse graph limits as well as the related theory of inhomogeneous random graphs. Specific problems in the area of inhomogeneous random graphs include questions on key graph parameters such as the chromatic number or the independence number. In the theory of sparse graph limits our main goal is to extend our understanding of local-global convergence. A further goal is to create a comprehensive theory of limits of subgraphs of hypercubes.
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
grants
