Joey Lakerdas-Gayle, University of Waterloo
Fundamentals of Computability Theory 3
We will continue learning about priority constructions, now using the finite injury method, following Robert Soare's textbook.
MC 5403
Paul Cusson, University of Waterloo
The Kodaira embedding theorem and background material
The Kodaira embedding theorem is a crucial result in complex geometry that forms a nice bridge between differential and algebraic geometry, giving a necessary and sufficient condition for a compact complex manifold to be a smooth projective variety, that is, a complex submanifold of a complex projective space. The material and proof will follow the exposition in Griffiths & Harris's classic textbook.
MC 5479
Nikolay Bogachev, University of Toronto
Commensurability classes and quasi-arithmeticity of hyperbolic reflection groups
In the first part of the talk I will give an intro to the theory of hyperbolic reflection groups initiated by Vinberg in 1967. Namely, we will discuss the old remarkable and fundamental theorems and open problems from that time. The second part will be devoted to recent results regarding commensurability classes of finite-covolume reflection groups in the hyperbolic space H^n. We will also discuss the notion of quasi-arithmeticity (introduced by Vinberg in 1967) of hyperbolic lattices, which has recently become a subject of active research. The talk is partially based on a joint paper with S. Douba and J. Raimbault.
MC 5417
Matthew Harrison-Trainor, University of Illinois at Chicago
Back-and-forth games to characterize countable structures
Given two countable structures A and B of the same type, such as graphs, linear orders, or groups, two players Spoiler and Copier can play a back-and-forth games as follows. Spoiler begins by playing a tuple from A, to which Copier responds by playing a tuple of the same size from B. Spoiler then plays a tuple from B (adding it to the tuple from B already played by Copier), and Copier responds by playing a tuple from B (adding it to the tuple already played by Spoiler). They continue in this way, alternating between the two structures. Copier loses if at any point the tuples from A and B look different, e.g., if A and B are linear orders then the two tuples must be ordered in the same way. If Copier can keep copying forever, they win. A and B are isomorphic if and only if Copier has a winning strategy for this game. Even if Copier does not have a winning strategy, they may be able to avoid losing for some (ordinal) amount of time. This gives a measure of similarity between A and B. A classical theorem of Scott says that for every structure A, there is an α such that if B is any countable structure, A is isomorphic to B if and only if Copier can avoid losing for α steps of the back-and-forth game, that is, when A is involved we only need to play the back-and-forth game for α many steps rather than the full infinite game. This gives a measure of complexity for A, called the Scott rank. I will introduce these ideas and talk about some recent results.
MC 5501
David McKinnon, University of Waterloo
How crowded can rational solutions be?
Say you've got the equation x^2-2y^2=z^4-1. Lots of rational solutions there, like (1,1,0). How are those solutions distributed in 3-space? In particular, how close can they get to (1,1,0)? This abstract has the questions, but the talk has the answers. Well, some of 'em.
MC 5479
Cynthia Dai, University of Waterloo
Height Modulis on Toric Stacks
In this talk we will go through Matt’s work on height modulis on weighted projective space, mainly its construction, and then its application. If time permits, I will talk about generalizations of this construction to toric stacks.
MC 5403
Matthew Harrison-Trainor, University of Illinois Chicago
Scott analysis of linear orders
The Scott analysis measures the complexity of describing a structure up to isomorphism, and equivalently the complexity of describing its automorphism orbits, and of computing isomorphisms between different copies. I will introduce the Scott analysis in general and talk about the Scott analysis of linear orders in particular. Linear orders have a few special properties which makes their behaviour quite interesting and sometimes different from structures in general.
MC 5479
Joey Lakerdas-Gayle, University of Waterloo
Symmetrically indivisible and elementarily indivisible structures
A first order structure M is indivisible if for every colouring of M into two colours, there is a monochromatic substructure N of M that is isomorphic to M. We will consider two stronger properties: M is symmetrically indivisible if N can be chosen so that every automorphism of N extends to an automorphism of M; and M is elementarily indivisible if N can be chosen to be an elementary substructure of M. We will discuss Model-Theoretic methods developed by Kojman and Geschke (2008), Hasson, Kojman, and Onshuus (2009), and Meir (2019) to study the relationships between these notions.
MC 5403
Jacques Van Wyk, University of Waterloo
“The” Generalised Levi-Civita Connection
I will discuss the notions of generalised metrics and generalised connections in generalised geometry. A generalised connection has an associated torsion tensor, so one may ask, if given a generalised metric G, whether there is a torsion-free connection D compatible with G; this is the analogue of the Levi-Civita connection. We will see that there are infinitely many such connections D, that is, there is no unique “generalised Levi-Civita connection,” a striking difference from the situation for Riemannian geometry.
MC 5479
Adina Goldberg, University of Waterloo
Synchronous Quantum Games
We recast nonlocal games using string diagrams, allowing for a natural extension to quantum games (with bipartite question and answer states). We define strategies in this setting and show that synchronous quantum games require synchronized players to win. We give examples of some quantum games on quantum graphs and see that these require quantum homo/isomorphisms to win. (The talk is based on a preprint ``Quantum games and synchronicity'' (https://arxiv.org/abs/2408.15444). This work is inspired by Musto, Reutter, and Verdon's paper ``A compositional approach to quantum functions'', and relies heavily on the reference ``Categories for Quantum Theory'' by Heunen and Vicary for string diagrams in quantum information.)
MC 5417