“...in the opinion of many people, it is a major problem in model theory”
[D. Lascar]
leading experts (e.g. Barwise, Makkai, Morley, Scott, Shelah) worked on it
VC is “the CH for the number of countable models of a first order theory”
Our plan is to confirm the conjecture for new classes of theories
Embeddings: Forcing by copies and embedding monoids
A classification of structures is induced by the forcing equivalence of posets
of their copies
The existing results are surprisingly uniform
Our plan is to obtain general results (in particular, for Fraïssé limits)
Condensations: Ehrenfeucht-Fraïssé games and reversibility
A classification of structures is induced by condensations between them
The existing results were obtained using combinatorial methods
We plan to modify and apply model-theoretic methods (e.g. EF games)
Categorical methods in Ramsey theory, big Ramsey degrees and the dual Ramsey property
The unique synergy between these two apparently unrelated areas of modern mathematics will bring completely new and mostly unexpected results.
By applying the machinery of category theory we expect to obtain new results concerning dual big Ramsey degrees, and new insights into the behavior of the dual Ramsey property in the infinite.
Saturation of iterated nonstandard models
High degree of saturation enables construction of various important objects in a model
Iteration of the star operator gives canonical tensor pairs
Our plan is to combine these two ideas and apply them to get further results about ultrafilters
Modifications of topologies
Forcing related properties of several convergence structures on Boolean algebras will be explored
Topologies generated by the local closure function in ideal topological spaces (introduced by Kuratowski 1933) and its cover properties will be investigated
