Past meetings: Fall 2021

December 6: Narmada Varadarajan (University of Toronto)

Local complexities and measurability for graph problems

Has anyone ever told you set theory is just abstract nonsense? Good news! Now you can tell them it has applications to algorithms. In this talk, I will sketch some recent results relating distributed algorithms to descriptive combinatorics. In particular, the complexity of a local solution corresponds to its measurability. I will sketch a proof that “non-playable” problems are not Borel and not O(log n). Only some basic graph theory and point-set topology will be assumed.

November 22: Peter Angelinos (University of Toronto)

Model Theory and Arithmetic Motivic Integration

Batyrev proved that birational Calabi-Yau manifolds over \mathbb{C} have the same Betti numbers using p-adic integration. Kontsevich reasoned that, since the choice of p was largely arbitrary, and presented his theory of motivic integration. To generalize it to positive characteristic, Denef and Loeser developed arithmetic motivic integration, which studies the formulas (in some language) which define certain subsets of algebraic varieties, rather than the varieties themselves. I will discuss the constructions (up to a point) necessary to define arithmetic motivic integration, and discuss a generalization of Ax-Kochen-Ershov, which Cluckers, Hales, and Loeser used to prove a transfer principle for the Fundamental Lemma (of the Langlands Program).

I will not prove anything.

November 15: Luciano Salvetti (University of Toronto)

Projections in the Calkin Algebra and P(N)/FIN: A dichotomy in the gap structure

Via Gelfand-Naimark duality / Stone duality we can view the Calkin Algebra as the "noncommutative analog" of P(N)/FIN. Similar consistency results hold for both objects, e.g., Shelah and Steprans proved (1988) that PFA implies that every automorphism of P(N)/FIN is trivial, while Farah showed (2011) that under OCA every automorphism of the Calkin Algebra is inner. However, the gap structure of these two objects may be very different. It is known (Todorčević, 1989) that there are no Analytic Hausdorff gaps in P(N)/FIN. In this talk we will compare the gap structure of the space of projections in the Calkin Algebra with the gap structure of P(N)/FIN. The main goal will be to show that the linear gap structure of the former can be strictly larger than the latter, more specifically (spoiler alert) we will construct an Analytic Hausdorff gap in the poset of projections of the Calkin Algebra. The content of this talk is mainly based on a paper by Beatriz Zamora-Aviles (2014).

November 8: Break

There was no meeting due to the University of Toronto reading week.

November 1: Clovis Hamel Ascanio (University of Toronto)

An outlook on Vaught’s conjecture, Morley’s theorem and recent results - Part II

This talk will (pretend to) be a gentle introduction to one of the most famous problems in Model Theory: Vaught’s conjecture. In that direction, we will discuss Morley’s theorem on the number of countable models and its absolute version. Then we will do a review of the two main theorems of a recent paper by Eagle, H., Muller and Tall (before the mice run loose).

No familiarity will be assumed, the talk will be as self-contained and introductory as possible.

October 25: Clovis Hamel Ascanio (University of Toronto)

An outlook on Vaught’s conjecture, Morley’s theorem and recent results

This talk will (pretend to) be a gentle introduction to one of the most famous problems in Model Theory: Vaught’s conjecture. In that direction, we will discuss Morley’s theorem on the number of countable models and its absolute version. Then we will do a review of the two main theorems of a recent paper by Eagle, H., Muller and Tall (before the mice run loose).

No familiarity will be assumed, the talk will be as self-contained and introductory as possible.

October 18: Keegan Dasilva Barbosa (University of Toronto)

An Introduction to Ramsey Degrees

The Kechris-Pestov-Todorcevic (KPT) correspondence has been a highly studied phenomena in recent years, due to the way in which it elegantly connects finite combinatorics with topological dynamics. In this talk, we will discuss the KPT correspondence, go over some examples of Fraïssé classes, and discuss Ramsey degrees (a measure of "Ramsey-ness"). This talk is designed for a generic audience with little to no prerequisites required beyond an interest in either combinatorics or set theory. The goal is to inspire young grad students and older veterans alike to consider working on problems in this (currently booming) field.