Final Report Summary - MODSTABBAN (Model theoretic stability for Banach spaces)
The goal of the project was to study and classify non-separably categorical, and, more generally, stable classes of Banach structures. Categoricity means that the first order continuous theory essentially fully describes the structure - at least up to cardinality (it is a well known consequence of the Compactness Theorem that it is impossible to determine cardinality of an infinite dimensional space by its first order theory). This is perhaps the strongest “structure” property of an elementary class. Stability is a weaker property (every uncountably categorical class is stable, but not the other way around). It has many equivalent definitions. One way to view stability is via the existence of “nice” regular patters in the structure. Alternatively, one can say that stable structures do not admit complex combinatorial behaviour.
It was our thesis that categoricity in Banach spaces is essentially explained by the existence of the “nicest” possible pattern in this context. Specifically, we have conjectured that a categorical class, the norm in every structure is determined by an inner product. In other words, every structure in such a class is determined by an underline Hilbert space. Similarly, we have conjectured that a stable Banach structure has the local structure of the “second nicest” kind in this context, that is, of an Lp space. In addition, we have asked to what extent the isometric theory can be extended/generalized to the isomorphic concept.
The first problem, that is, the structure of an uncountably categorical Banach structure, has been almost fully solved. Specifically, we have proven a structure theorem, while a team of our collaborators have constructed new examples that show that one can not make a significant improvement in the statement of our result.
We have also essentially solved the second problem completely. Specifically, we have studied the locally stable behaviour and showed that even under the weakest local assumption, that is, in a generically stable type, there exists an ell-p type in the span of a spreading model. Consequently, any sequence approximating a generically stable type, contains an ell-p space almost isometrically.
The last question and the most challenging question (the isomorphic theory) has been addressed as well. We have shown that weak (isomorphic) categoricity implies weak (isomorphic) stability, and have developed the local and global theory of weak stability, analogous to classical (isometric) stability.
It was our thesis that categoricity in Banach spaces is essentially explained by the existence of the “nicest” possible pattern in this context. Specifically, we have conjectured that a categorical class, the norm in every structure is determined by an inner product. In other words, every structure in such a class is determined by an underline Hilbert space. Similarly, we have conjectured that a stable Banach structure has the local structure of the “second nicest” kind in this context, that is, of an Lp space. In addition, we have asked to what extent the isometric theory can be extended/generalized to the isomorphic concept.
The first problem, that is, the structure of an uncountably categorical Banach structure, has been almost fully solved. Specifically, we have proven a structure theorem, while a team of our collaborators have constructed new examples that show that one can not make a significant improvement in the statement of our result.
We have also essentially solved the second problem completely. Specifically, we have studied the locally stable behaviour and showed that even under the weakest local assumption, that is, in a generically stable type, there exists an ell-p type in the span of a spreading model. Consequently, any sequence approximating a generically stable type, contains an ell-p space almost isometrically.
The last question and the most challenging question (the isomorphic theory) has been addressed as well. We have shown that weak (isomorphic) categoricity implies weak (isomorphic) stability, and have developed the local and global theory of weak stability, analogous to classical (isometric) stability.