Krull-gabriel dimension and the ziegler spectrum

We provide an introduction to the the Krull-Gabriel dimension of a ring, as well as many related ideas. In particular, we outline how Krull-Gabriel dimension relates to the Cantor-Bendixson rank of the Ziegler spectrum and with the Jacobson radical of the module category. We also include a list of examples of rings and categories where the Krull-Gabriel dimension has been calculated

Prismatic decompositions and rational GG-spectra

We study the tensor-triangular geometry of the category of rational $G$-spectra for a compact Lie group $G$. In particular, we prove that this category can be naturally decomposed into local factors supported on individual subgroups, each of which admits an algebraic model. This is an important step and strong evidence towards the third author's conjecture that the category of rational $G$-spectra admits an algebraic model for all compact Lie groups. To facilitate these results, we relate topological properties of the associated Balmer spectrum to structural features of the group $G$ and the category of rational $G$-spectra. A key ingredient is our presentation of the spectrum as a Priestley space, separating the Hausdorff topology on conjugacy classes of closed subgroups of $G$ from the cotoral ordering. We use this to prove that the telescope conjecture holds in general for rational $G$-spectra, and we determine exactly when the Balmer spectrum is Noetherian. In order to construct the desired decomposition of the category, we develop a general theory of `prismatic decompositions' of rigidly-compactly generated tensor-triangulated categories, which in favourable cases gives a series of recollements for reconstructing the category from local factors over individual points of the spectrum.Comment: 61 pages; all comments welcome

Rational GG-spectra for profinite GG

In this thesis we will investigate rational G-spectra for a profinite group G. We will provide an algebraic model for this model category whose injective dimension can be calculated in terms of the Cantor-Bendixson rank of the space of closed subgroups of G, denoted SG. The algebraic model we consider is chain complexes of Weyl-G-sheaves of rational vector spaces over the spaces. The key step in proving that this is an algebraic model for G-spectra is in proving that the category of rational G-Mackey functors is equivalent to Weyl-G-sheaves. In addition to the fact that this sheaf description utilises the topology of G and the closed subgroups of G in a more explicit way than Mackey functors do, we can also calculate the injective dimension. In the final part of the thesis we will see that the injective dimension of the category of Weyl-G-sheaves can be calculated in terms of the Cantor-Bendixson rank of SG, hence giving the injective dimension of the category of Mackey functors via the earlier equivalence.Comment: PhD thesis (2019

Computability Theory (hybrid meeting)

Over the last decade computability theory has seen many new and fascinating developments that have linked the subject much closer to other mathematical disciplines inside and outside of logic. This includes, for instance, work on enumeration degrees that has revealed deep and surprising relations to general topology, the work on algorithmic randomness that is closely tied to symbolic dynamics and geometric measure theory. Inside logic there are connections to model theory, set theory, effective descriptive set theory, computable analysis and reverse mathematics. In some of these cases the bridges to seemingly distant mathematical fields have yielded completely new proofs or even solutions of open problems in the respective fields. Thus, over the last decade, computability theory has formed vibrant and beneficial interactions with other mathematical fields. The goal of this workshop was to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work

Ranks and approximations for families of cubic theories

In this paper, we study the rank characteristics for families of cubic theories, as well as new properties of cubic theories as pseudofiniteness and smooth approximability. It is proved that in the family of cubic theories, any theory is a theory of finite structure or is approximated by theories of finite structures. The property of pseudofiniteness or smoothly approximability allows one to investigate finite objects instead of complex infinite ones, or vice versa, to produce more complex ones from simple structures

Ranks and approximations for families of cubic theories

In this paper, we study the rank characteristics for families of cubic theories, as well as new properties of cubic theories as pseudofiniteness and smooth approximability. It is proved that in the family of cubic theories, any theory is a theory of finite structure or is approximated by theories of finite structures. The property of pseudofiniteness or smoothly approximability allows one to investigate finite objects instead of complex infinite ones, or vice versa, to produce more complex ones from simple structures

Shelah's Main Gap and the generalized Borel-reducibility

We answer one of the main questions in generalized descriptive set theory, Friedman-Hyttinen-Kulikov conjecture on the Borel-reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel-reducibility notions of complexity. For any $\kappa$ satisfying $\kappa=\lambda^+=2^\lambda$ and $2^{\mathfrak{c}}\leq\lambda=\lambda^{\omega_1}$, we show that if $T$ is a classifiable theory and $T'$ not, then the isomorphism of models of $T'$ is strictly above the isomorphism of models of $T$ with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, $T$, the isomorphism of models of $T$ is either $\Delta^1_1$ or analytically-complete