Program
Mini course 1 - Computability Theory
Barbara Csima
Computability Theory is a branch of Mathematical Logic studying the relative computational strength of various objects (normally coded as sets of natural numbers). In this course we will introduce (or review) the notions of computability, relative computability, and the halting problem. We will see why computability theory lends itself to a proof technique known as “the priority method”, and work through some examples. We will discuss areas of research where computability theory is used, such as studying the structure of the Turing degrees, computable structure theory, reverse mathematics, and algorithmic randomness.
Mini course 2 - Computability in the context of continuous model theory
Bradd Hart
This series of three talks will focus on two reasonably recent developments in mathematical logic in general and model theory in particular. I will discuss the relatively new subject of continuous model theory - a version of model theory which is much more suited for many analytic settings such as metric spaces, C*-algebras and von Neumann algebras. In the first lecture, I will concentrate on the role of the ultraproduct in this setting and set the stage for the introduction of semantic approaches to problems of computability. The second lecture will necessarily concentrate on some of the critical technical parts of continuous model theory with particular emphasis on differences with classical logic. The key role of definable sets will be highlighted. The final lecture will look at recent interactions between continuous model theory and computability theory. The groundbreaking result MIP*=RE will be discussed and its consequences in continuous model theory, operator algebras and quantum mechanics will be described.
Mini course 3 - Treeability of equivalence relations via Stone duality and median graphs
Anush Tserunyan
The end goal of this tutorial is to prove a recent result of R. Chen, A. Poulin, R. Tao, and myself: if each connected component of a locally finite Borel graph 𝐺 is “tree-like” (e.g. quasi-isometric to a tree or of bounded treewidth), then there exists an acyclic Borel graph with the same connected components as 𝐺. This is an analogue, in the setting of countable Borel equivalence relations, of a well-known result from geometric group theory: if a Cayley graph of a finitely generated group is quasi-isometric to a tree, then the group is virtually free.
We will study treeable equivalence relations, their graphings, and basic descriptive set-theoretic tools for working with them. We will then focus on a locally finite connected graph and study its cuts and ends, and ultrafilters on its set of cuts (called orientations in this context). We will furthermore discuss median graphs (the 1-skeleta of CAT(0) cube complexes) and how the clopen ultrafilters form such a graph. Finally, we will learn a Borel algorithm for constructing a subtreeing of a median graph and put everything together to prove the main result.
Tentative Schedule
Wednesday
9:00-10:15 - Bradd Hart
10:15-10:45 - Coffee Break
10:45-12:00 - Anush Tserunyan
12:00-13:30 - Lunch
13:30-14:45 - Barbara Csima
Thursday
9:00-10:15 - Anush Tserunyan
10:15-10:45 - Coffee Break
10:45-12:00 - Barbara Csima
12:00-13:30 - Lunch
13:30-14:45 - Bradd Hart
Friday
9:00-10:15 - Barbara Csima
10:15-10:45 - Coffee Break
10:45-12:00 - Bradd Hart
12:00-13:30 - Lunch
13:30-14:45 - Anush Tserunyan