Archive: all, 2023, 2022, 2021, 2020

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Title: Geometric triviality in differentially closed fields

Speaker's homepageSlides

In this talk we revisit the problem of describing the 'finer' structure of geometrically trivial strongly minimal sets in $DCF_0$. In particular, I will explain how recent work joint with Guy Casale and James Freitag on Fuchsian groups (discrete subgroup of $SL_2(\mathbb{R})$) and automorphic functions, has lead to intriguing questions around the $\omega$-categoricity conjecture of Daniel Lascar. This conjecture was disproved in its full generality by James Freitag and Tom Scanlon using the modular group $SL_2(\mathbb{Z})$ and its automorphic uniformizer (the $j$-function). I will explain how their counter-example fits into the larger context of arithmetic Fuchsian groups and has allowed us to 'propose' refinements to the original conjecture.

Title: Type spaces, Hrushovski constructions and giraffes

The KPT correspondence established a connection between extreme amenability of automorphism groups of first-order structures and Ramsey theory. In this talk, I will consider automorphism groups $\operatorname{Aut}(M)$ which fix points on type spaces $S_n(M)$ via a natural action. We will explore a few examples from a combinatorial perspective, and building on work of Evans, Hubicka and Nesetril, we will see that there is an omega-categorical structure M which does not have any omega-categorical expansion $M'$ with $\operatorname{Aut}(M')$ fixing points on type spaces.

Title: Proving o-minimality of real exponentiation with restricted analytic functions (Part 2)

Speaker's homepageSlides

The o-minimality of real exponentiation with restricted analytic functions is one of the most applied model theoretic results. I'll discuss the key steps of the van den Dries-Macintyre-Marker proof, based on (1) quantifier elimination for restricted analytic functions, (2) the interplay between analytic functions and the Archimedean valuation, and (3) Hardy fields.

The talk will be self-contained, but it's also meant to be a conclusion to our summer reading of van den Dries-Macintyre-Marker.

Title: Realizability and the Axiom of Choice

Speaker's homepageSlides

Realizability aims at extracting the computational content of mathematical proofs. Introduced in 1945 by Kleene as part of a broader program in constructive mathematics, realizability has later evolved to include classical logic and even set theory. Recent methods that generalize the technique of Forcing led to define realizability models for the theory ZF, but realizing the Axiom of Choice remains problematic. After a brief presentation of these methods, we will discuss the major obstacles for realizing the Axiom of Choice and I will present my recent joint work with Guillaume Geoffroy that led to realize weak versions of the Axiom of Choice.

Title: Building countable generic structures

Speaker's homepageSlides

In this talk I will discuss a new method of building countable generic structures with the algebraic closure property. This method generalises the well-known methods of Fraïssé and Hrushovski pre-dimension construction. I will start with an overview of the construction method of Fraïssé-Hrushovski and then as an application of the new method I will construct a generic non-sparse graph that its automorphism group is not amenable. This method is particularly useful for constructing non-simple generic structures. Time permitting I will explain how to construct non-simple structures with $TP_2$ and $NSOP1$.

Title: An introduction to large cardinal axioms

Speaker's homepage

Large cardinal axioms are axioms that extend the standard ZFC axioms for set theory in a strong way - they allow you to prove the consistency of ZFC and the large cardinals that came below. I will give a brief survey of these axioms.

Title: Around stability theory (Part 2)

Speaker's homepage

Model-theoretic stability theory was developed in the 1970s, with Shelah in a lead role, as providing a notion of 'tameness' for first order theories. In particular, uncountably categorical theories are stable, and on the other hand a complete unstable theory over a countable language has $2^\kappa$ nonisomorphic models of size $\kappa$ for any uncountable cardinality $\kappa$. Stability can be characterised in many different ways, and stability provides a powerful notion of independence between subsets of a model.

I will give a very informal overview of stability theory, and of some of the generalisations of stability which have been developed more recently (in particular simplicity, and NIP).

Title: Algebraic flows on tori: an application of model theory

Speaker's homepage

A complex torus $\mathbb{T}$ is a Lie group which is obtained as a quotient of a finite dimensional complex vector space, $\mathbb{C}^g$, by a lattice; so there is a canonical projection map $\pi$ from $\mathbb{C}^g$ into $\mathbb{T}$. If we consider an algebraic subvariety $V$ of $\mathbb{C}^g$, then we can ask what the image of $V$ under $T$ looks like: Ullmo and Yafaev proved that if $V$ has dimension 1, then the closure of $\pi(V)$ in the Euclidean topology is given by a finite union of translates of closed subgroups of $\mathbb{T}$, and conjectured that this should hold in higher dimensions. Using model theoretic methods, Peterzil and Starchenko showed that this conjecture isn't quite true, but that a similar, slightly more complicated statement holds, describing the closure of $\pi(V)$ in terms of finitely many closed subgroups of $\mathbb{T}$. In this talk, I'll introduce the problem and describe the main ingredients of the Peterzil-Starchenko proof.

Title: Split Principles and Large Cardinals

Speaker's homepageSlides

The original split principle is an equivalent formulation of a cardinal failing to satisfy the combinatorial essence of weak compactness. Gunter Fuchs and I expanded the notion in order to characterize the negation of other large cardinal properties. These split principles give rise to seemingly new large cardinals. In this talk I plan to introduce split principles and potentially compare them with flipping properties, which are another way to characterize various large cardinal properties.