Gian Sanjaya, Department of Pure Mathematics, University of Waterloo
"Arithmetic Schemes"
We now look at examples of arithmetic schemes.
MC 5417
Franklin Tall, University of Toronto
"An undecidable extension of Morley’s theorem on the number of countable models"
We show that Morley’s theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, large cardinals, and Inner Model Theory.
MC 5479
Laindon Burnett, Department of Pure Mathematics, University of Waterloo
"Sturmian Sequence Decidability Does Not Generalize"
In 2022, it was shown that much like Presburger arithmetic itself, Presburger arithmetic along with a Sturmian sequence is a decidable theory. We give an overview of Konieczny's 2024 proof that this does not extend to generalized polynomials, themselves a generalization of Sturmian sequences.
MC 5479
Corey Jones, North Carolina State University
"Constructing actions of fusion categories on C*-algebras"
A fusion category is an algebraic object that simultaneously generalizes finite groups and their representation categories. Fusion categories can ``act" on C*-algebras by bimodules, extending the familiar concept of a group acting by automorphisms to a non-invertible setting. Building actions of specific fusion categories on specific C*-algebras is hard. In this talk, we will discuss a general method that allows for the construction of actions of fusion categories on interesting C*-algebras with minimal algebraic input. As an application, we construct actions of exotic fusion categories on noncommutative tori. Based on joint work with David Evans.
This seminar will be held both online and in person:
AJ Fong, Department of Pure Mathematics, University of Waterloo
"The mathematics of juggling (and perhaps a geometric application)"
Before videos could be easily transmitted over the internet, mathematical notation for juggling patterns was used by jugglers to share instructions and new patterns with each other. After introducing these, I will show that a mild generalisation of this gives a natural partial order on juggling patterns. If time permits, I will describe a relatively recent result that demonstrated that juggling patterns can be used to index a natural stratification of Grassmannians, which naturally arises in Poisson geometry, total positivity and cluster algebras.
MC 5417
Lucia Martin Merchan, Department of Pure Mathematics, University of Waterloo
"Hodge decomposition for Nearly Kähler manifolds"
Verbitsky proved that Nearly Kähler 6-dimensional manifolds satisfy Kähler-type identities. These lead to a Hodge decomposition in the compact case, and restrictions on their Hodge numbers. In this talk, we discuss a new proof for most of these results that is independent of the dimension. This is work in progress with Spiro, Michael and Aleks.
MC 5403
Aleksandar Milivojevic, Department of Pure Mathematics, University of Waterloo
"Obstructions to almost complex structures following Massey"
I will report on work in progress with Michael Albanese, in which we prove statements claimed by Massey in 1961 concerning the obstructions to finding an almost complex structure on an orientable manifold (or more generally, reducing the structure group of a real vector bundle over a CW complex to the unitary group). These obstructions involve the integral Stiefel-Whitney classes – which detect the existence of integral lifts of the mod 2 Stiefel-Whitney classes, namely putative Chern classes – and relations between the Pontryagin and Chern classes. A somewhat surprising aspect of these obstructions is that they are in fact generally proper fractional parts of what one might at first expect. For example, the obstruction in degree eleven is 1/24 of the eleventh integral Stiefel-Whitney class.
MC 5403
Anne Johnson, Department of Pure Mathematics, University of Waterloo
"Attributes and morphisms of schemes"
We start Chapter 3 of Eisenbud and Harris, discussing finiteness conditions, properness and separation. We discuss the construction of Proj S as time allows.
MC 5417
Andy Zucker, Department of Pure Mathematics, University of Waterloo
“Preliminaries on Polish Spaces”
This inaugural talk will introduce some of the background on Polish spaces that we will need in our study of Polish groups. We will mostly draw from the early chapters of Kechris's book ‘Classical Descriptive Set Theory’.
MC 5403
Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo
“The linear algebra of 2-forms in 4-dimensions”
I will present some important facts about the linear algebra of 2-forms in 4 dimensions, which everyone should know. We start with classical results about self-dual and anti-self dual 2-forms, and then proceed to discuss "hypersymplectic" structures in 4d à la Donaldson. Then we put all this on an oriented Riemannian 4-manifold.
MC 5417