Location: MALL 1
Title: Model Theory and Combinatorics or (The Unexpected Virtue of Tameness)
Hrushovski once called model theory the "geography of tame mathematics"; a quote which may make no sense if you're not a model theorist (or if you are, in fact, a geographer). I will try to explain this point of view, by discussing how model theorists draw "dividing lines" separating the tame theories from the wild ones and illustrating this through examples from combinatorics.
The aim of this talk is to provide a gentle introduction to some recent developments in the area. Perhaps ambitiously, I will try to (briefly and informally) discuss three very different topics: (i) Szemerédi Regularity in stable (due to Malliaris-Shelah), NIP (due to Fox, Pach and Suk), and distal structures (due to Chernikov and Starchenko); (ii) Algorithmic tameness in hereditary classes of relational structures (joint work with S. Braunfeld, A. Dawar and I. Eleftheriadis); and (iii) Zarankiewicz's problem in semilinear (due to Basit, Chernikov, Starchenko, Tao and Tran) and semibounded (joint work with P. Eleftheriou) o-minimal structures.
Location: MALL
Title: A proof theorist's job, part II: into harmonious systems
This talk will be about ordinal analysis, a technique at the core of a proof theorist's toolchain. Using Tait calculus as the formal proof system for arithmetic, we will replicate Gentzen's famous consistency proof as the seemingly useless statement "enough of set theory ⊢ Con(PA)". Hopefully, I will convince the audience that the left-hand side of the turnstile can be weakened to a theory of transfinite induction formalised in primitive recursive arithmetic, and this proof thus measures the proof-theoretic strength of a formal theory using the linear ordering in a given ordinal representation system.
Location: MALL
Title: A proof theorist's job: transfinite induction in a finitary context
This talk will be about ordinal analysis, a technique at the core of a proof theorist's toolchain. Using Tait calculus as the formal proof system for arithmetic, we will replicate Gentzen's famous consistency proof as the seemingly useless statement "enough of set theory ⊢ Con(PA)". Hopefully, I will convince the audience that the left-hand side of the turnstile can be weakened to a theory of transfinite induction formalised in primitive recursive arithmetic, and this proof thus measures the proof-theoretic strength of a formal theory using the linear ordering in a given ordinal representation system.
Location: MALL
Title: The Axiom of Determinacy and Large Cardinals — Part 2: Not So Large After All
This is the second and final talk in this survey of the Axiom of Determinacy (AD) and its implications on large cardinals. In the first talk, we looked at some first properties of AD and defined the cardinal Θ, which will play a prominent role in the rest of this series. In this talk, we will begin by defining some large cardinal properties — properties that cannot be provably exhibited by cardinals under ZFC — measurability, strong compactness, and supercompactness. We will then state results by Solovay which show that under ZF + AD (or its stronger counterpart ZF + ADR), ω1 satisfies these properties at least partially. We will sketch proofs for some (time permitting, all) of these results.
Although this talk builds on the first one, we will recall all relevant definitions and results from there, so that this talk will be accessible even if the audience has not attended the first one.
Location: MALL
Title: The Axiom of Determinacy and Large Cardinals - Part 1: Strange Beginnings
The Axiom of Determinacy (AD) was proposed by Mycielski and Steinhaus in 1962. By then, it was already known that the Axiom of Choice (AC) implies the falsehood of AD, causing AD to be sidelined by many set-theorists. However, in 1967, Solovay showed that AD implies the measurability of ω1, and the new-found realisation that large cardinal properties are exhibited by small cardinals under AD contributed to the induction of AD into mainstream set-theoretic research.
This is a two-talk series that will culminate in Solovay’s results on some large cardinal properties exhibited by ω1 under AD (and its stronger counterpart ADR). In the first talk, we will lay the groundwork by introducing infinite games and defining AD, then discussing some of its pop-culture properties, such as its annihilation of the Banach-Tarski Paradox as well as its surprising implications on the real numbers (including the Continuum Hypothesis). If time permits, we will define a cardinal Θ that equals the successor of the continuum under AC, but can be much larger under AD.
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: 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: Cardinal characteristics of the continuum
When speaking about subsets of the real number line, we may define some idea of smallness. When we do so, it is often the case that a countable union of small sets is not enough to cover the whole real line. Any list of countably many real numbers can be extended, so the shortest inextensible list of real numbers must be uncountable. Any countable collection of nowhere dense sets also does not cover the real line, so we need uncountably many of those to do that. The same holds for Lebesgue measure zero sets, strongly null sets, nowhere everywhere sets, and more. If aleph_0 isn't good enough then how many do we need?
Location: MALL
Title: Functional interpretations and the contraction rule therein
Functional interpretations are generalisations of Goedel's Dialectica interpretation of Heyting arithmetic into a primitive-recursive functional system, which arose from a disbelief in universal quantifiers and integrated the ideas of functionals and proofs; the Dialectica interpretation shows the consistency of Heyting arithmetic (hence that of Peano arithmetic) with respect to that primitive-recursive functional system. We give an introduction to the idea of functional interpretations, and in the course, we will see how a seemingly innocuous inference rule called contraction (from A\vee A, deduce A) interferes with the interpretations.
