Set Theory: Compactness and Reflection properties as a motivation for set theoretical Investigations.
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel
Reflection principles come out naturally in many mathematical contexts in which one studies uncountable structures. Examples were studied in Topology, Algebra, Infinite combinatorics etc. Many times these principles and problems associated with them are important source for interesting Set Theoretical investigations.
A reflection principle for a property of mathematical structures of certain kind is the statement that a structure having the property has a small substructure with the property. "Small" typically means "having smaller cardinality" but there are other notions of "smallness". Few examples:
- Suppose that a collection of sets does not have a one to one choice function. Does there exists a smaller cardinality subfamily with the same property?
- Suppose that a topological space X is not metric. (In order to avoid trivialities assume that X satisfies some minimal conditions necessary for being metric. e.g. It is first countable.) Does the space has a smaller subspace which is not metric?
- Suppose that the Abelian group G is not free. Does it have a smaller subgroup which is not free?
- Stationary reflection: Let \kappa be a regular cardinal and S \subseteq \kappa is stationary. Does S reflect? Namely does there exists an ordinal \beta < \kappa such that S \cap \beta is stationary in \beta?
Compactness properties are dual properties to reflection properties. Typically they claim that if for a given structure, we have many small substructure with the property (e.g. for very smaller cardinality sub-structure), then the whole structure has this property. A reflection principle is equivalent to a compactness property for the negation of the property
and vice versa. Few examples:
- The tree property for a cardinal \kappa can be phrased as compactness property. Namely we have a \kappa tree (A tree with \kappa levels, each level of size less than \kappa). We know that every proper initial segment of the tree has a cofinal branch. The tree property is the statement that the whole tree has a cofinal branch.
- G is an Abelian group such that every smaller cardinality subgroup can be embedded into a direct product of copies of the integers: Z. Can G be so embedded?
- X is a topological space such that it is \lambda collection-wise Hausdorff. Namely very discrete closed set of cardinality \leq \lambda can be separated by a disjoint collection of open sets. Under what conditions can we infer that X is fully collection-wise Hausdorff. (Namely every discrete closed subset can be separated)?
Problems of this type are closely connected with the basic properties of the universe of Set Theory like the existence of large cardinals or combinatorial principles like \Box_\kappa There are interesting connections between different properties. Many times they give a natural motivation for additional axioms for Set Theory. In the tutorial we shall give few examples of such properties and their set theoretical connections. We are going to assume some familiarity with basic set theoretical concepts, but they will be rather elementary.
For instance we shall give the definition of the large cardinals we shall use. Forcing constructions will be used as black box.
Computability Theory: Computable Structure Theory
University of California, Berkeley, USA
We will introduce the basic concepts of Computable Structure Theory, give a taste of a few of the methods used, and end by explaining a few more recent results. Computable structure theory studies mathematical structures from a computational viewpoint. Our goal is to find interactions between computational properties and algebraic properties of structures, and also to understand to what extent those interactions are possible. Here are two basic questions we look at: How difficult is it to find a representation of a give structure? How difficult is it to find isomorphisms between structures? We will end mentioning some recent work relating the famous Vaught's conjecture from model theory to computability theoretic properties. (By computability here I mostly mean between non-computable objects, as in computability theory, and I'm not referring to time- or space-complexity as in computer science.)
Model Theory: Continuous model theory and applications
University of Illinois at Chicago, USA.
Abstract: Continuous logic is a relatively recent logic, introduced by I. Ben Yaacov and A. Usvyatsov and further developed by I. Ben Yaacov, A. Berenstein, C.W. Henson, and A. Usvyatsov, suited for studying structures from analysis that are based on metric spaces. The main novelty in this logic is that formulae are no longer true or false but can take on any value in a bounded interval of the real numbers. The model theory associated to this logic resembles classical model theory in that many of the familiar theorems (e.g. the compactness theorem, the omitting types theorem, the Beth Definability theorem) have continuous analogues that are, in fact, generalizations of their "discrete" counterparts.
In this tutorial, I will present the basics of continuous logic, defining the syntax and semantics of this logic as well as presenting some examples of continuous theories. I will also present the continuous ultraproduct construction and show how this yields the compactness theorem for continuous logic. I will then move on to two recent applications of continuous model theory. I will first discuss a result of I. Farah, B. Hart, and D. Sherman, where they show how to use model-theoretic (in)stability to answer questions about isomorphic ultrapowers asked by many people in the operator algebra community. Then I will discuss how I. Farah and S. Shelah use saturation of certain continuous reduced products to show, assuming the Continuum Hypothesis, that the Stone-Cech remainder of the real line has many nontrivial autohomeomorphisms.
Non Classical Logics: Dualities
Dipartimento di Matematica University of Salerno and Institue for Logic, Language, and Information - University of Amsterdam.
Abstract: Categorical dualities are pivotal tools in mathematical logic as well as in several other branches of mathematics. In logic, they are often used to give concrete semantics (usually based on geometric objects) to abstract syntax (usually encoded in algebraic terms).
Standard examples are Stone duality for Boolean algebras, Priestley duality for distributive lattices, the duality between finitely presented MV-algebras and rational polyhedra, and Gelfand duality for C*-algebras. The scope of the tutorial is to cover all these fundamental dualities with a unified approach stemming from the classical Galois connection between ideals of polynomials and zero-sets of polynomial functions. This will provide a general method that, given a class of structures, constructs a dual adjunction between the latter and a class of objects that can be naturally topologised. Thanks to this approach unexpected connections with Hilberts Nullstellensatz, Birkhoff subdirect representation, and Gel'fand transform will emerge during the tutorial.