Location: MALL
Title: How far is almost strong compactness from strong compactness.
Almost strong compactness of $\kappa$ can be characterized as follows: for every $\delta < \kappa < \lambda$, there is an elementary embedding $j_{\delta,\lambda}: V \rightarrow M$ with critical point $\geq \delta$, so that $j_{\delta,\lambda}" \lambda \subseteq D \in M$ and $M \vDash |D|< j_{\delta,\lambda}(\kappa)$. Boney and Brooke-Taylor were wondering whether almost strong compactness is essentially the same as strong compactness. Recently, Goldberg showed that if $\kappa$ is of uncountable cofinality and SCH holds from below then these two closely related concepts are the same. In this joint work with Zhixing You, we show that these two can be different in general cases.
Location: MALL 1
Title: Differentially Large Fields and Taylor Morphisms
Differential largeness is a generalisation of the notion of largeness for pure fields, introduced by Leon-Sanchez and Tressl. This class of differential fields contains many of the model-theoretically tame classes, such as differentially closed fields, closed ordered differential fields, etc. One of the tools that have been developed to study such fields is known as the `twisted Taylor morphism’, which essentially transforms ring homomorphisms into differential ring homomorphisms into the ring of power series in a uniform way. We generalise this notion, and show that differential largeness can also be characterised in terms of generalised Taylor morphisms. If time allows, we will talk about the structure of these generalised Taylor morphisms.
Location: MALL
Title: Definable refinements of classical algebraic invariants
In this talk I will explain how methods from logic allow one to construct refinements of classical algebraic invariants that are endowed with additional topological and descriptive set-theoretic information. This approach brings to fruition initial insights due to Eilenberg, Mac Lane, and Moore (among others) with the additional ingredient of recent advanced tools from logic. I will then present applications of this viewpoint to invariants from a number of areas in mathematics, including operator algebras, group theory, algebraic topology, and homological algebra.
Location: MALL
Title: Is this talk going to be about the partite method (Nesetril-Rödl, 1989) which shows that the class of all finite ordered graphs is Ramsey?
Location: MALL
Title: Absolute Model Companionship, the AMC-spectrum of set theory, and the continuum problem
We introduce a classification tool for mathematical theories based on Robinson's notion of model companionship; roughly the idea is to attach to a mathematical theory $T$ those signatures $L$ such that $T$ as axiomatized in $L$ admits a model companion. We also introduce a slight strengthening of model companionship (absolute model companionship - AMC) which characterize those model companionable $L$-theories $T$ whose model companion is axiomatized by the $\Pi_2$-sentences for $L$ which are consistent with the universal and existential theory of any $L$-model of $T$. We use the above to analyze set theory, and we show that the above classification tools can be used to extract (surprising?) information on the continuum problem. Slides
Location: MALL
Title: Summability in Algebras of Generalized Power Series
Spaces of generalized power series have been important objects in asymptotic analysis and in the algebra and model theory of valued structures ever since the introduction of the first instances of them by Levi-Civita and Hahn. A space of generalized series can be understood to be built up from a triple (Γ , < , F), where (Γ, <) is an ordered set and F is an ideal of Noetherian subsets of Γ, as the k-vector space k(Γ,F) of functions f : Γ → k whose support Supp f = {γ ∈ Γ : f(γ)= 0} lies in F.A prominent feature of these objects is the notion of formal infinite sum, or rather more appropriately, of infinite k-linear combination: a family (f_i)_{i \in I} such that the union fo the Supp(f_i) is yet in F and for every γ ∈ Γ the set {i ∈ I : γ ∈ Supp(f_i)} is finite, is said to be summable. Given such a family it is possible to associate to any I-indexed family of scalars (k_i)_{i \in I} the infinite linear combination of the f_i with the coefficienti k_i in the obvious way. This feature is referred to, throughout literature, as a strong linear structure on the vector space k(Γ,F) and maps F : k(Γ,F) → k(∆,G) preserving it, are referred to as strong(ly) linear maps. In this talk, after some history and motivation, I will argue about a suitable category-theoretical framework for the study of the above outlined notion of strong linearity, in particular I will justify a notion of reasonable category of strong k-vector spaces generalizing the above setting and prove that up to equivalence there is a unique universal category ΣVect with the property that every reasonable category of strong vector spaces has a fully faithful functor to into it.ΣVect can be defined as an orthogonal subcategory of Ind-(Vect^op) and the canonical monoidal closed struture on Ind-(Vect^op) restricts to ΣVect.Time permitting, the relation ΣVect has with another orthogonal subcategory of Ind-(Vectop) equivalent to the category of linearly topologized vector spaces that are colimits of linearly compact spaces will be also described. Finally I will present some open questions of combinatorial nature in this setting.
Location: MALL
Title: Non-linearities in the analytical hierarchy
It is commonly known that there exist true $\Pi^0_1$ sentences which are mutually independent over theories such as PA. The analogue of this fact for $\Pi^1_1$ fails: there do not exist true $\Pi^1_1$ sentences which are mutually independent over theories such as $PA$ + all true $\Sigma^1_1$ sentences. We study the corresponding situation for $\Sigma^1_n$ and $\Pi^1_n$ under assumptions such as $V = L$ or large cardinals. This is joint work with F. Pakhomov.
Location: MALL 1
Title: Foundationless geology and a Foundation conservativity result
It is well-understood that the Axiom of Foundation has no "mathematical consequences" over ZFC - Foundation, since every mathematical structure is isomorphic to one whose universe is an ordinal by the well-ordering theorem. Over ZF - Foundation, there are mathematical consequences to adding Foundation, e.g. the sentence "if all orderable sets are well-orderable, then every set is well-orderable." In joint work with Asaf Karagila, we identify a precise sense in which there is no simpler consequence of adding Foundation. In particular, for any $\varphi$ a sentence in second-order logic, adding Foundation does not refute the existence of a set model of $\varphi.$ This talk will focus on applying techniques of set-theoretic geology in a context without Choice or Foundation, which is a key ingredient in the proof of this theorem. Slides
Location: MALL 1
Title: O-minimal tame set-theoretic topology
We give a positive answer in the o-minimal setting to a conjecture in set-theoretic topology and explore similar open problems in topology from the point of view of o-minimality. Slides
