November 1:
November 15: Jorge Cruz (UNAM)
November 22: Daniel Calderón (University of Toronto)
November 29: TBA
December 6: TBA
October 25: Keegan Dasilva Barbosa (Fields Institute)
Abstract: In this talk, we will discuss the nature of combinatorics when statements about colourings become quantified over Borel sets instead of any set. We will motivate this talk by considering a very simple graph, the unit circle where two points are adjacent if they are \alpha apart in arc length for some fixed irrational alpha. This graph is rather intriguing, as it arises from a minimal flow. Morover, any colouring corresponds to a type of "anti"-flow decomposition. We will first show this graph is two colourable by proving a fundamental fact from graph theory, the De Brujin-Erdos theorem. We will then show the story changes when we require our partitions be Borel (as an aside, this inadvertently provides a proof that there are subsets of the Reals which are not Borel using transfinite combinatorics). This will be an incredibly elementary/undergrad friendly talk where all terms and definitions will be provided, and little pre-requisites are required outside of some topology and rudimentary probability theory (yes, probability theory 😊).
October 18: Jing Zhang (University of Toronto)
Abstract: I will give a few examples of how to establish combinatorial independences by forcing over models of ZF that are not models of the axiom of choice, including a few historical ones. In particular, I will explain why people do that. The goal is to give a very remote/gentle introduction to the recent breakthrough in set theory that MM++ implies Woodin's (*) axiom. It'll be as self-contained as possible.
October 11: Tona (University of Toronto)
Abstract: This talk is aimed at undergrads. We give a brief overview of different ways we can make sense of the problem of distinguishing 'big' sets from 'small' sets and why such a problem is interesting. We start by considering the different perspectives: set-theoretical, topological, analytical, etc. and their relationship. We will mention the Baire category theorem and might discuss some small cardinals (e.g. ideal additivity) informally. If there is time, we will talk about large cardinals (e.g. whose existence is undecidable) and what they have to say about these problems as well.
Abstract: The study of C*-algebras under a model theoretic framework has led to interesting notions that compress common properties of a given C*-algebra (model). One of these notions is the degree-1 saturation, a weak analogue of 'saturated model' in the classical sense. For example, it is well known that the corona of a sigma-unital, non-unital C*-algebra is countably degree-1 saturated (Farah, Hart, 2013). This leads to the (perhaps idle) question: what about uncountable degree-1 saturation? I will prove that uncountable degree-1 saturation fails very badly for certain coronas, meaning it fails at the first uncountable cardinal. No Model Theory / C*-algebra background will be assumed.