Lunedì 11 maggio ore 14:00 aula S
Palazzo Campana [via Carlo Alberto 10]
Alberto Marcone (Università di Udine)
Le funzioni ordinali, quali ad esempio quelle della gerarchia di Veblen, sono strumenti importanti in teoria della dimostrazione. In questo seminario le considereremo invece dal punto di vista della teoria della computabilità e della reverse mathematics. Sia f una funzione ordinale della gerarchia di Veblen. Allora esiste un ordine lineare X tale che f(X) ha una successione discendente. computabile, ma qualunque successione discendente in X è fortemente non-computabile. Dal punto di vista della reverse mathematics studiamo la forza assiomatica dell'enunciato: se X è un buon-ordine allora f(X) è un buon ordine. Lavoro in collaborazione con Antonio Montalbán.
Palazzo Campana [via Carlo Alberto 10]
Quando Veblen incontra Turing
Alberto Marcone (Università di Udine)
Le funzioni ordinali, quali ad esempio quelle della gerarchia di Veblen, sono strumenti importanti in teoria della dimostrazione. In questo seminario le considereremo invece dal punto di vista della teoria della computabilità e della reverse mathematics. Sia f una funzione ordinale della gerarchia di Veblen. Allora esiste un ordine lineare X tale che f(X) ha una successione discendente. computabile, ma qualunque successione discendente in X è fortemente non-computabile. Dal punto di vista della reverse mathematics studiamo la forza assiomatica dell'enunciato: se X è un buon-ordine allora f(X) è un buon ordine. Lavoro in collaborazione con Antonio Montalbán.
Venerdì 28 novembre ore 14:00 aula S
Palazzo Campana [via Carlo Alberto 10]
Luca Motto Ros (KGRC Vienna)
The analysis of the structure of analytic equivalence relations (ER for short) under Borel-reducibility has been one of the most relevant subject in the recent history of Descriptive Set Theory. Among ERs, a special place is occupied by the isomorphism relation on countable models of a certain L_{ω_1,ω}-sentence φ. However, isomorphism relations are just a special case, as there are many example of ERs which are even not Borel-reducible to an isomorphism relation. On the contrary, we will show in this talk that the bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) relation is able to capture the whole complexity of the ER-structure: for every ER E there is an L_{ω_1,ω}-sentence φ such that E is Borel-equivalent to bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) on the collection of countable models of φ. This is joint work with Sy D. Friedman.
Palazzo Campana [via Carlo Alberto 10]
Analytic equivalence relations and bi-embeddability
Luca Motto Ros (KGRC Vienna)
The analysis of the structure of analytic equivalence relations (ER for short) under Borel-reducibility has been one of the most relevant subject in the recent history of Descriptive Set Theory. Among ERs, a special place is occupied by the isomorphism relation on countable models of a certain L_{ω_1,ω}-sentence φ. However, isomorphism relations are just a special case, as there are many example of ERs which are even not Borel-reducible to an isomorphism relation. On the contrary, we will show in this talk that the bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) relation is able to capture the whole complexity of the ER-structure: for every ER E there is an L_{ω_1,ω}-sentence φ such that E is Borel-equivalent to bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) on the collection of countable models of φ. This is joint work with Sy D. Friedman.
Mercoledì 12 novembre ore 11:00 aula Monod
Palazzo Campana [via Carlo Alberto 10]
I will give an introduction to the work of Sela on the elementary theory of free groups, and algebraic geometry on that groups. Then I will discuss some model theoretic properties of torsion-free hyperbolic groups and superstable groups acting on trees.
Palazzo Campana [via Carlo Alberto 10]
On model theory of groups acting on trees.
Abderezak Ould Houcine (Lione)I will give an introduction to the work of Sela on the elementary theory of free groups, and algebraic geometry on that groups. Then I will discuss some model theoretic properties of torsion-free hyperbolic groups and superstable groups acting on trees.
Mercoledì 11 novembre ore 14:00 aula S
Palazzo Campana [via Carlo Alberto 10]
Palazzo Campana [via Carlo Alberto 10]
On finitely generated models.
Abderezak Ould Houcine (Lione)
Mercoledì 10 settembre ore 11:30-12:30 aula C
Palazzo Campana [via Carlo Alberto 10]
I have introduced Polish structures in order to apply model theoretic ideas in the studies of purely descriptive set theoretic and topological objects such as Polish G-spaces or, more generally, Borel G-spaces. In particular, Polish structures generalize profinite structures introduced by Newelski. Polish structures allow us to apply ideas and techniques from model theory, descriptive set theory, topology and the theory of profinite groups.
A Polish structure is a pair (X,G) where G is a Polish group acting (faithfully) on a set X so that the stabilizers of all points are closed subgroups of G. We say that (X,G) is small if for every natural number n there are countably many orbits on X^n under G.
A simple non-profinite example of a small Polish structure is the unit circle with the full group of homeomorphisms. In fact, most natural examples of compact metric spaces with the full group of homeomorphisms are small Polish structures. More complicated examples are Hilbert cube and the pseudo-arc with the full group of homeomorphisms.
I will start my talk with some basic things about profinite structures. Then I will concentrate on Polish structures. I will discuss a purely topological notion of independence, called non-meager independence, that satisfies some nice properties (e.g. symmetry, transitivity, existence of independent extensions) in small Polish structures, and so allows us to introduce basic stability-theoretic concepts and to prove fundamental results about them (e.g. Lascar inequalities). In profinite structures this notion of independence coincides with m-independence introduced by Newelski.
If time permits, in the second part of my talk I will concentrate on the structure of small compact G-groups, i.e. small Polish structures (X,G) where X is a compact group and G acts continuously on X as a group of automorphisms. I will present an example of such a group which is not solvable-by-finite. On the other hand, under a natural model theoretic assumption of 'superstability' with respect to non-meager independence, each such group is solvable-by-finite, and assuming finiteness of the underlying rank, it is even nilpotent-by-finite.
I will finish with some open questions.
Palazzo Campana [via Carlo Alberto 10]
Some model theory of Polish structures
Krzysztof Krupinski (Wroclaw)I have introduced Polish structures in order to apply model theoretic ideas in the studies of purely descriptive set theoretic and topological objects such as Polish G-spaces or, more generally, Borel G-spaces. In particular, Polish structures generalize profinite structures introduced by Newelski. Polish structures allow us to apply ideas and techniques from model theory, descriptive set theory, topology and the theory of profinite groups.
A Polish structure is a pair (X,G) where G is a Polish group acting (faithfully) on a set X so that the stabilizers of all points are closed subgroups of G. We say that (X,G) is small if for every natural number n there are countably many orbits on X^n under G.
A simple non-profinite example of a small Polish structure is the unit circle with the full group of homeomorphisms. In fact, most natural examples of compact metric spaces with the full group of homeomorphisms are small Polish structures. More complicated examples are Hilbert cube and the pseudo-arc with the full group of homeomorphisms.
I will start my talk with some basic things about profinite structures. Then I will concentrate on Polish structures. I will discuss a purely topological notion of independence, called non-meager independence, that satisfies some nice properties (e.g. symmetry, transitivity, existence of independent extensions) in small Polish structures, and so allows us to introduce basic stability-theoretic concepts and to prove fundamental results about them (e.g. Lascar inequalities). In profinite structures this notion of independence coincides with m-independence introduced by Newelski.
If time permits, in the second part of my talk I will concentrate on the structure of small compact G-groups, i.e. small Polish structures (X,G) where X is a compact group and G acts continuously on X as a group of automorphisms. I will present an example of such a group which is not solvable-by-finite. On the other hand, under a natural model theoretic assumption of 'superstability' with respect to non-meager independence, each such group is solvable-by-finite, and assuming finiteness of the underlying rank, it is even nilpotent-by-finite.
I will finish with some open questions.
Mercoledì 10 settembre ore 10:00-11:00 aula C
Palazzo Campana [via Carlo Alberto 10]
In this talk I'll try to explain how ideas from logic can be used to study problems in metric geometry. First I'll introduce the space of Katetov maps on a metric space and discuss some of its properties, then I'll try to explain how they can be used to solve an old problem about Urysohn's universal metric space. If time permits I'll also discuss an application to isometric embeddings of metric spaces into Banach spaces.
Palazzo Campana [via Carlo Alberto 10]
Katetov maps and some applications to metric geometry.
Julien Melleray (Lione)In this talk I'll try to explain how ideas from logic can be used to study problems in metric geometry. First I'll introduce the space of Katetov maps on a metric space and discuss some of its properties, then I'll try to explain how they can be used to solve an old problem about Urysohn's universal metric space. If time permits I'll also discuss an application to isometric embeddings of metric spaces into Banach spaces.