Models and Sets Seminar

Location: MALL
Title: Higher independence at regular cardinals

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In the first part of this talk I will introduce the classical concept of a maximal independent family and its main properties. The second part of the talk will be devoted to deal with the generalisation of independence for regular uncountable cardinals. I will show the differences and similarities with the classical setting, as well as new lines of research that appear when dealing with this generalisation.

Finally, I will mention some recent results of Vera Fischer and myself regarding independence.

Location: MALL
Title: New irreducible generalised power series

A classical tool in the study of real closed fields are the fields $K((G))$ of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian group $G$. A fundamental result of Berarducci ensures the existence of irreducible series in the subring $K((G^{\le0}))$ of $K((G))$ consisting of the generalised power series with non-positive exponents. We generalize previous results and show that for certain order types almost all series are irreducible or irreducible up to a monomial.

Location: MALL
Title: Interdefinability and compatibility in certain o-minimal expansions of the real field
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The sets definable in an o-minimal expansion of the real field have a tame topological behaviour (uniform finiteness, good dimension theory, no pathological phenomena). Being able to tell if a certain real set or function is definable in a given o-minimal structure gives us information on how tame the geometry of that object is.

Let us say that a real function $f$ is o-minimal if the expansion $(R,f)$ of the real field $R$ by $f$ is o-minimal. A function $g$ is definable from $f$ if $g$ is definable in $(R,f)$. Two o-minimal functions $f$ and $g$ are compatible if $(R,f,g)$ is o-minimal. I will discuss the o-minimality, the interdefinability and the compatibility of two special functions, Euler’s Gamma and Riemann’s Zeta, restricted to the reals. Joint work with J.-P. Rolin and P. Speissegger.

Location: MALL
Title: The potentialist multiverse of classes
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Set-theoretic potentialism is the view that the universe of sets is never fully completed but is only given potentially. Tools from modal logic have been applied to understand the mathematics of potentialism. In recent work, Neil Barton and I extended this analysis to class-theoretic potentialism, the view that proper classes are given potentially (while the sets may or may not be fixed).

In this talk, I will survey some results from set-theoretic potentialism. After seeing how the tools apply in that context I will then discuss our work in the class-theoretic context

Title: Ccc without C, si? Si.
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What does the countable chain condition mean without the axiom of choice? We will discuss several possible definitions, all equivalent in ZFC, none equivalent in ZF(+DC). We will also present two "external" definitions (due to Bukovský and to Mekler) and see how they fit into this picture.

We will show that a ccc forcing can collapse ω1, and quite possibly be countably closed while doing so. On the other hand, with the "correct definition" of ccc, no cofinalities or cardinals are changed above ω1. Whether or not ω1can be collapsed is open, but we know that would require it to be singular.

This is a joint work with Noah Schweber.

Title: Dependence logic and its axiomatization problem
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Dependence logic, introduced by Väänänen (2007), is a non-classical logic for reasoning about dependence and independence. The logic extends first-order logic with a new type of atomic formulas, called dependence atoms, to specify explicitly the dependence relation between variables. Dependence logic adopts an innovative semantics, called team semantics (Hodges 1997), in which formulas are evaluated on a model with respect to sets of assignments (called teams), instead of single assignments. Teams are essentially relations on the model. For this reason, dependence logic is equi-expressive with existential second-order logic, and thus not fully axiomatizable. In this talk, I will give a concise introduction to dependence logic, and I will also survey recent developments in finding partial axiomatizations for the logic.

Title: The Interplay of Determinacy, Large Cardinals, and Inner Models
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The standard axioms of set theory, Zermelo-Fraenkel set theory with Choice (ZFC), do not suffice to answer all questions in mathematics. While this follows abstractly from Kurt Gödel's famous incompleteness theorems, we nowadays know numerous concrete examples for such questions. In addition to a large number of problems in set theory, even many problems outside of set theory have been showed to be unsolvable, meaning neither their truth nor their failure can be proven from ZFC. A major part of set theory is devoted to attacking this problem by studying various extensions of ZFC and their properties with the overall goal to identify the "right" axioms for mathematics that settle these problems.



Determinacy assumptions are canonical extensions of ZFC that postulate the existence of winning strategies in natural infinite two-player games. Such assumptions are known to enhance sets of real numbers with a great deal of canonical structure. Other natural and well-studied extensions of ZFC are given by the hierarchy of large cardinal axioms. Inner model theory provides canonical models for many large cardinal axioms. Determinacy assumptions, large cardinal axioms, and their consequences are widely used and have many fruitful implications in set theory and even in other areas of mathematics. Many applications, in particular, proofs of consistency strength lower bounds, exploit the interplay of determinacy axioms, large cardinals, and inner models. In this talk I will survey recent developments as well as my contribution to this flourishing area.

Title: The Sierpinski Carpet as a Final Coalgebra
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The background for this work includes Freyd's Theorem, in which the unit interval is viewed as a final coalgebra of a certain endofunctor in the category of bipointed sets. Leinster generalized this to a broad class of self-similar spaces in categories of sets, also characterizing them as topological spaces. Bhattacharya, Moss, Ratnayake, and Rose went in a different direction, working in categories of metric spaces, obtaining the unit interval and the Sierpinski Gasket as a final colagebras in the categories of bipointed and tripointed metric spaces respectively. To achieve this they used a Cauchy completion of an initial algebra to obtain the required final coalgebra. In their examples, the iterations of the fractals can be viewed as gluing together a finite number of scaled copies of some set at some finite set of points (e.g. corners of triangles). Here we will expand these ideas to apply to a broader class of fractals, in which copies of some set are glued along segments (e.g. sides of a square). We use the method of completing an initial algebra to obtain a final coalgebra which is Bilipschitz equivalent to the Sierpinski Carpet, and note that this requires substantially different machinery from previous results in order to handle the metric. Time permitting, we will expand on the Sierpinski Gasket results by considering different categories of metric spaces.

Joint work with Larry Moss.

Title: On large externally definable subsets in NIP
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Joint work with Martin Bays and Pierre Simon

Suppose that $M$ is a model of an NIP theory, and $X$ an externally definable subset: for some elementary extension $N$ of $M$, and some $c$ from $N$, $X = \{a \in M : \phi(a,c) \text{ holds}\}$.

How large should $X$ be to contain an infinite $M$-definable subset? Chernikov and Simon asked whether aleph1 is enough. I will discuss this question and relate it to questions in model theory and infinite combinatorics.

Title: On some generic structures
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This is a joint work with Yutaka Kuga, a student of mine.

The talk is about the generic structures produced by Hrushovski's predimension construction with a control function.

A predimension of a graph is the number of vertices minus the number of edges multiplied by some weight. With a predimension and some control function $f$, a class of finite graphs $\mathrm{K}_f$ is defined. Suppose $f$ is unbounded, $\mathrm{K}_f$ has the free amalgamation property and one point substructures are always closed in a sense defined by the predimension. Let $M$ be the generic structure of $\mathrm{K}_f$. Our result is that $\operatorname{Th}(M)$ is model complete if the weight of the predimension is a rational number. In the case that it is an irrational number, $\operatorname{Th}(M)$ is also model complete if $f$ satisfies some mild assumptions satisfied by all known examples of such $f$.

Using the techniques used in the proof of model completeness of $\operatorname{Th}(M)$, we can also show that $M$ is monodimensional in the case that the weight of the predimension is rational and $f$ is the function defined by Hrushovski in his original paper. Hence, the automorphism group of $M$ is a simple group (has no non-trivial normal groups) by a theorem of Evans, Ghadernezhad, and Tent. The same result is valid for most examples of such $f$.