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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

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

Gian Sanjaya, Department of Pure Mathematics, University of Waterloo

"Arithmetic Schemes"

We now look at examples of arithmetic schemes.

MC 5417

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory X"

We will conclude our series of talks on computable structure theory.

MC 5479

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

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

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:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Jakub Krásenský, Czech Technical University in Prague

"Criterion sets for quadratic forms over number fields"

By the celebrated 15 theorem of Conway and Schneeberger, a classical positive definite quadratic form over Z is universal if it represents each element of {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim and B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost. Since then, the question of uniqueness for particular situations has been studied by several authors.

We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. Here again, the existence of criterion sets for universality is known, and Lee determined the set for Q(sqrt5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Tianyi Zheng, UC San Diego

"Random walks on self-similar groups and conformal dimension"

Conformal dimension was introduced in the late 1980s by P. Pansu; it is a natural invariant in the study of the geometry of hyperbolic spaces and their boundaries. In this talk we will discuss how conformal geometry can be used to study random walks on iterated monodromy groups, in particular, random walk entropy bounds when the limit set has Ahlfors-regular conformal dimension strictly less than 2. Based on joint work with N. Matte Bon and V. Nekrashevych.

MC 5501