Rigidity and homogeneity in combinatorial Banach spaces
The rigidity of an object is related to the existence of few automorphisms of it. The notion of homogeneity goes in the opposite direction, frequently allowing different "parts" of the object to be moved one to the other through automorphisms. Important homogeneity results in combinatorics come from Ramsey theory. A classical rigidity result in Banach space theory is the Banach-Stone theorem, which says that any linear bijective isometry between two C(K) spaces is induced by a homeomorphism between the compact spaces. In our talk, we will discuss these notions in the context of combinatorial Banach spaces, which are sequence spaces whose norm are induced by combinatorial families
Analytic functions on an ordered valued field
Model theory has had great success in establishing properties of sets defined by restricted analytic functions on an ordered field. Here "restricted" is defined using the ordering, and "analytic function" means that its power series is convergent in the sense of the ordering. Similar ideas have been used for the study of restricted analytic functions on a valued field, where "restricted" and "convergent" are now understood in the sense of the valuation. If a field has both an ordering and a valuation, which interact in a nice way, it is not so clear which of the relations should be considered primary in order to choose the functions to study. In this talk, I will explain the model-theoretic questions one might ask, review some of the past results, and discuss another collection of functions that one can study on an ordered valued field. I will do my best to explain all technical terms in this abstract (and more!).
A tour on ecumenical systems
Some questions naturally arise with respect to ecumenical systems: what (really) are ecumenical systems? What are they good for? Why should anyone be interested in ecumenical systems? What is the real motivation behind the definition and development of ecumenical systems?
Based on the specific case of the ecumenical system that puts classical logic and intuitionist logic coexisting in peace in the same codification, we would like to propose three possible motivations for the definition, study and development of ecumenical systems.
- Philosophical motivation.
Logical inferentialism, is the semantical approach according to which the meaning of the logical constants can be specified by the rules that determine their correct use. There are some natural (proof-theoretical) inferentialist requirements on admissible logical rules, such as harmony and separability. We will start by discussing such requirements in the view of Prawitz' ecumenical system.
- Mathematical/computational motivation.
Dowek has this very interesting remark:
``Which mathematical results have a classical formulation that can be proved from the axioms of constructive set theory or constructive type theory and which require a classical formulation of these axioms and a classical notion of entailment remains to be investigated.''
The second part of the talk is devoted to discuss ecumenical axiomatizations of mathematics.
- Logical motivation.
In a certain sense, the logical motivation naturally combines certain aspects of the philosophical motivation with certain aspects of the mathematical motivation. According to Prawitz, one can consider the so-called classical first order logic as ``an attempted codification of a fragment of inferences occurring in [our] actual deductive practice''. Given that there exist different and even divergent attempts to codify our (informal) deductive practice, it is more than natural to ask about what relations are entertained between these codifications.
Our claim is that ecumenical systems may help us to have a better understanding of the relation between classical logic and intuitionistic logic.
Maybe we can resume the logical motivation in the following (very simple) sentence:
Ecumenical systems constitute a new and promising instrument to study the nature of different (maybe divergent!) logics.
We will finish the talk by explaining how all this discussion can be lifted to the case of modal logics.
De Finetti's three-valued conditionals and Boolean algebras of conditionals: two sides of a same coin
Conditionals play a key role in different areas of logic and probabilistic reasoning, and they have been studied and formalised from different angles. Bruno de Finetti was one of the first who put forward an analysis of conditionals beyond the realm of conditional probability theory arguing that they cannot be described within the bounds of classical logic. He called them trievents: a conditional (a|b) is a basic object to be read "a given b" that turns out to be true if both a and b are true, false if a is false and b is true, and void if b is false. This approach, has been further developed by Gilio and Sanfilippo by interpreting conditionals as numerical random quantities with a betting-based semantics, and where the third value is a conditional probability.
On the other hand, following a more logico-algebraic approach, it has been recently shown that, in a finite setting, conditional events can be endowed with a structure of Boolean algebra and that a (unconditional) probability measure on the initial algebra of plain events can be canonically extended to a (unconditional) probability measure on the Boolean algebra of conditionals which is in fact a conditional probability.
In this talk we will show how that the apparent contradiction between the above two perspectives, one that looks at three-valued conditionals as random quantities and the Boolean algebraic perspective on conditionals, actually dissolves once we precisely set at which level the numerical and the symbolic representation intervene. In doing so, we pave the way to build a bridge between the long standing tradition of three-valued conditionals and the more recent proposal of looking at conditionals as elements from suitable Boolean algebras.
This is joint work with Tommaso Flaminio, Angelo Gilio, Hykel Hosni and Giuseppe Sanfilippo
To what extent do structural properties and computational properties coincide?
Given a countable structure $A$, we distinguish between two types of properties: structural properties and computational properties. Think of a structural property of $A$ as a property of the isomorphism type of $A$, for example, the sentences it satisfies, the types it realizes, or other properties such as (if $A$ is a group) being torsion-free. On the other hand a computational property of $A$ is a property of the different presentations (isomorphic copies with domain $\mathbb{N}$) of $A$. For example, it might be that every presentation of $A$ can compute a set $X$. Many of the most celebrated results of computable structure theory show the equivalence between a structural property and a computational property. I will talk about a few of these and also some interesting examples of computational properties which do not seem to be equivalent to any structural property.
Towards another vision of effectiveness at aleph_1
The first uncountable cardinal does not easily mend itself to methods inherited from the countable. We know this through a whole list of failures of properties such as compactness and Ramsey theorems. Similarly, the descriptive set theory at this level is very different from the classical descriptive set theory and does not really seem to give as much of an idea of effectiveness. We shall propose to look at the effectiveness at $\aleph_1$ from the point of view of automata theory and generalized decidability. In so doing, we shall introduce new classes of automata and consider MSO of trees.
Recent interactions between representation theory (of algebras) and model theory
In the last decade, a new crossroad between representation theory and model theory emerged. The specific programme (in representation theory) is called the Dixmier-Moeglin equivalence. This programme aims at characterizing the primitive ideals of an algebra over a field. Model theory had an appearance in the context of algebras of differential operators, and also in the context of Poisson algebras. In this talk, I will give a survey past and present results.