Fall 2023

Past talks:

November 23: Cesar Corral (York University)

Title: Ramsey convergence and high dimensional versions of sequential compactness

Abstract: The notion of n-sequential compactness was defined by Kubis and Szeptycki in order to generalize sequential compactness to any finite dimension n. To study these notions it is necessary to introduce a Ramsey-like convergence property on n dimensional sequences f : [omega]^n to X. We will talk about basic properties of these spaces, how to carry on these ideas to infinite dimensions, some applications to functional analysis and the interplay between Topology and the combinatorial notions that involve infinite dimensional sequentially compact spaces.

November 17: Postponed

November 10: Break

November 3: Break

October 26: Keegan Dasilva Barbosa (Fields Institute)

Title: Conant's Metric Spectra Problem

Abstract: structures are of high interest to Set Theorists, Model Theorists, and Combinatorists alike. One class of intriguing structures are the Urysohn spaces. Provided a subset of positive reals S satisfies some simple combinatorial properties, we can guarantee the existence of a Urysohn space. Hence, there is an ample amount of Urysohn spaces provided one is willing to look. Urysohn spaces were actually initially used to classify Banach spaces, until Gelfand and Naimark later showed a much more elegant theory existed (sorry Urysohn). Despite this, there is a wealth of seemingly avoided problems involving these metric spectra S  that are ignored on the basis of being too combinatorial. As a consequence, many open problems involving finite combinatorics (that could lead to infinitary results) go unresolved in favour of problems more in line with the average Set Theorists/Model Theorists interests. I think there has been a large oversight amongst set theorists to express these finitary problems to the combinatorial community. After all, an undergraduate student can make sense of these problems so why aren't we talking about them? In fact an undergraduate student has not only come to understand one of these problems posed by Conant, but has solved it up to a factor of 2. In this talk, I will go over a brief history of metric spectra. I will then proudly present the work of Veljko Tojić, an undergraduate student from Serbia I had the honour of supervising, and show how he used geometry and linear algebra to approach Conant's problem. This talk is for a general audience. Anyone with an interest in mathematics will be able to follow along.

October 19: Brinda Venkataramani (University of Toronto)

Title: Destroying Choice

Abstract: A longstanding technique for constructing models of set theory where choice fails has been through building permutation models. Results from these exotic models can then be lifted to ZF by various transfer principles. In 2016, Bruce and Blass introduced a new permutation model Vp and exhibited within it, the failure of choice for families of pairs, that Dedekind-finiteness does not imply finiteness, and that well-orderable families have choice functions. We will consider a natural extension of Vp called Vp+ and show that two of these three results hold in this new model, and discuss why studying such models is of interest.

October 12: Clement Yung (University of Toronto)

Title: Kastanas Game in Sequence Spaces

Abstract: The Kastanas game is a game introduced by Ilias Kastanas in 1983 to characterise the Ramsey property of subsets of reals. This characterisation allows us to understand Ramsey subsets of the reals from a game-theoretic perspective, which is helpful in alternative axiomatic systems, such as systems assuming the Axiom of Determinacy. We shall present a partial proof of this characterisation, and consider the same game in other spaces, such as the Hindman space and vector spaces over countable fields.

October 5: Luciano Salvetti (University of Toronto)

Title: The Cantor Game

Abstract: In 2006, Matt Baker introduced an infinite two-player game, now known as the Cantor Game, as a way to prove the uncountability of the real numbers. Fix a target subset of the reals S, then players I and II alternate turns picking numbers a1, b1, a2, b2,... respectively, such that a1<a2<...<b2<b1. We say that player I wins if and only if there is x in S such that an<x<bn for all n and otherwise player II wins. This game is similar to the well-known Perfect Set Game in the sense that it characterizes sets of reals with the perfect set property. However, the Cantor Game has a natural generalization to dense linear orders. In 2022, Will Brian and Steven Clontz proved that when played in the real numbers, player II has a winning strategy if and only if the target set is countable. They asked the question of whether this is true for other dense linear orders. In this talk we will discuss recent progress on this question. In particular, we will show that for any infinite cardinal k there is a dense linear order of size k for which player II always wins. The talk is aimed to grad and undergrad students with a little background on Set Theory. This is joint work in progress with Tona Matos. 

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