Elenco seminari del ciclo di seminari
“LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR”

A deep result of Gao-Jackson is that orbit equivalence relations induced by Borel actions of countable discrete abelian groups on Polish spaces are hyperfinite. Hjorth asked if indeed any orbit equivalence relation induced by a Borel action of an abelian Polish group on a Polish space, which is also essentially countable, must be essentially hyperfinite. We show that any countable Borel equivalence relation (CBER) which is treeable must be classifiable by an abelian Polish group (such as $\ell_1$). As the free part of the Bernoulli shift action of $F_2$ is a treeable CBER, and not hyperfinite, this answers Hjorth's question in the negative. On the other hand, for certain abelian Polish groups such as $\matbb{R}^\omega$, Hjorth's question has a positive answer. Indeed, we show that any orbit equivalence relation induced by a Borel action of a countable product of locally compact abelian Polish groups which is also potentially $\BPi^0_3$ must be Borel-reducible to $E_0^\omega$. By a dichotomy result of Hjorth-Kechris, this implies that essentially countable such orbit equivalence relations are hyperfinite. This uses a result of Cotton that locally compact abelian Polish groups yield essentially hyperfinite orbit equivalence relations, as well as the Hjorth analysis of Polish group actions.

It was asked in [J. Math. Log. , Vol. 22, No. 01, (2022)] if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups.We prove that under the assumption of weak elimination of imaginaries this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of Aut(GL(V)) for any vector space of dimension $\aleph_0$ over a finite field, in affinity with the classical description for finite dimensional spaces due to Schreier and van der Waerden.

This talk consists of two parts. In the first part, we give an overview of the theory of model categories. This provides a framework which axiomatizes the notion of homotopy which is familiar from the setting of topological spaces. Originally developed by Quillen in the 1960s, these ideas allowed for a formalization of the similarities between homotopy theory and homological algebra. In particular, there are important connections between topological spaces, simplicial sets and chain complexes. We will see that the structure of a model category allows for the construction of a categorical localization at the so-called class of weak equivalences. For example, this can be applied to the model category of chain complexes, giving rise to the derived category of an abelian category. In the second part, we concentrate on the category of simplicial presheaves. In 1987, it was shown by Jardine that the category of simplicial presheaves can be endowed with the structure of a model category. This makes it possible to consider the homotopy theory of presheaves. In recent years, these ideas have received renewed interest, as they can be used in the construction of different flavors of derived geometry. For example, it has been shown that those simplicial presheaves which properly encode a notion of homotopy can be characterized by a descent condition in terms of hypercovers. In turn, this descent condition can be interpreted as a formulation of the classical sheaf axioms 'up to homotopy'.

With the solution of Hilbert’s fifth problem, our understanding of connected locally compact groups has significantly increased. Therefore, the contemporary structure problem on locally compact groups concerns the class of totally disconnected locally compact (= t.d.l.c.) groups. The investigation of the class of t.d.l.c. groups can be made more manageable by dividing the infinity of objects under investigation into classes of types with “similar structure”. To this end we introduce the rational discrete cohomology for t.d.l.c. groups and discuss some of the invariants that it produces. For example, the rational discrete cohomological dimension, the number of ends, finiteness properties FP_n and F_n, and the Euler-Poincaré characteristic.

Knots are very familiar and tangible objects in everyday life and play an important role in modern mathematics. A mathematical knot is a homeomorphic copy of S_1 embedded in S_3. Proper arcs are intuitively obtained by cutting a knot and are defined as copies of the unit interval embedded in a closed ball. Following an earlier paper by Weinstein (then called Kulikov), we use discrete objects, such as linear and circular orders, to gain insights into arcs and knots. To this end we study in detail the relation of convex embeddability between countable linear and circular orders. This leads to results about the combinatorial and descriptive set theoretic complexity of natural subarc and subknot relations. We point out that knot theory usually considers only tame knots, while we are dealing mainly with wild knots. Joint work with Martina Iannella, Luca Motto Ros and Vadim Weinstein

Let L be a first-order 2-sorted language. Let X be some fixed structure. A standard structure is an L-structure of the form ⟨M,X⟩. When X is a compact topological space (and L meets a few additional requirements) it is possible to adapt a significant part of model theory to the class of standard structures. This has been noticed by Henson and Iovino in the case of Banach spaces (and metric structures in general). However, in the last 20 years the most popular approach to the model theory of metric structures uses real-valued logic (Ben Yaacov, Berenstein, Henson, Usvyatsov). Arguably, this is neither natural nor general enough. We show that a few adaptations of Henson and Iovino's approach suffices for a natural and powerful theory. This is based on three facts: - every standard structure has a positive elementary extension that is standard and realizes all positive types that are finitely consistent. - in a sufficiently saturated structure, the negation of a positive formula is an infinite disjunction of positive formulas. - there is a pure model theoretic notion that corresponds to Cauchy completeness. To exemplify how this setting applies to model theory we discuss ω-categoricity and (local) stability.

I will introduce the main algebraic and model theoretic properties of the rings M/nM where M is a model of Peano Arithmetic. I will explain the role played by the classical result due to Feferman and Vaught in this analysis.