Events

Michael Gregory, University of Waterloo

Basic Universal Algebra Aimed at Isomorphism Problems for c.e. Presentations

We begin with the notions of a universal algebra, homomorphism, congruence, and quotient algebra, and discuss the relationship between congruences and homomorphic images. We then introduce term algebras and varieties, culminating in a statement of Birkhoff's HSP Theorem. To prepare for later computability applications, we briefly review lattices and the congruence lattice of an algebra. Finally, we describe how finitely generated and computably enumerable algebras may be specified by presentations.

MC 5403

Julius Frizzell, University of Waterloo

Roth's Theorem

We will continue to discuss unitary transformations and generic measures and work towards a proof of Roth's theorem for arithmetic progressions.

MC 5417

Jules Ribolzi, University of Waterloo

Definable groups in the nonstandard model of CCM

We review the two main results about definable groups in the nonstandard model of CCM.

M3 4001

Faisal Romshoo, University of Waterloo

Anisotropic Calibrations

I aim to talk about some of the technical details in Tomasso Pacini and Kotaro Kawai’s paper ”Anisotropiccalibrations, adiabatic limits, and mirror symmetry” which Tomasso presented in the Geometry and Topology seminar last month. If time permits, I want to explore how we can generalize the notion of Smith maps using anisotropic calibrations.

MC 5417

Viktor Majewski, University of Waterloo

Generalized Seiberg-Witten Equations and where to find them

Everywhere.

MC 5417

Michael Gregory, University of Waterloo

The Complexity of the Isomorphism Problem for Finitely Generated Algebras

We review the arithmetic hierarchy and use it to analyze the isomorphism problem for finitely generated c.e. algebras. We introduce the ascending chain condition (ACC) on congruences and explain how it restricts the complexity of isomorphism. We show that any finitely generated c.e. algebra whose congruence lattice satisfies ACC has a \(\Pi_2\) isomorphism problem. Then, we prove that the class \(UF_2\) of algebras with two unary operations has \(\Sigma_3\)-complete isomorphism problem.

MC 5403

Anton Iliashenko, Beijing Institute of Mathematical Sciences and Applications

Deformation theory of associative and coassociative Smith maps

Associative and coassociative Smith maps are generalizations of pseudo-holomorphic curves in the \(G_2\) setting. We construct the right framework for the deformation theory using the spinorial formulation. This is enough to establish generic non-existence. Then we discuss where the theory goes from there.

MC 5403

Quang-Khai Nguyen, Universite de Lyon

Generating Series in Algebraic Dynamics

In this talk, we will discuss the generating series associated with a self-map of a projective variety. This series is important in understanding the dynamical degree and plays an important role in the recent construction of the transcendental dynamical degree by Bell, Diller, and Jonsson. This talk will focus on some analytic and algebraic properties of such a series. It turns out that in some cases, rationality is rather the exception.

MC 5403

Jack Jia, University of Waterloo

Categories of representations of groups are well-behaved

They are abelian (behave like module categories), symmetric monoidal (have tensor products), every object has a dual and is semi-simple, to name a few. A natural question to ask is whether every category that exhibits similar behaviour is a representation category. Deligne proved a remarkable theorem that shows every symmetric tensor category with some imposed growth condition is in fact a category of representations. Moreover, he constructed some symmetric tensor categories with faster-than-exponential growth-these are so-called Deligne categories, which can be interpreted as complex rank analogs of classical representation categories. In this talk, I will introduce the notion of symmetric tensor categories, state Deligne’s Theorem, and construct some of the Deligne categories.

MC 5403

Beining Mu, University of Waterloo

Degree of categoricity and treeable degrees

In this seminar, we will discuss treeable degrees and degree of categoricity. We will introduce past results on which degrees can or cannot be a degree of categoricity and when degrees of categoricity coincide with treeable degrees. We will also introduce a notion of \(\Pi^0_1\) singletons as an example of treeable degrees and their relation to degree of categoricity.

MC 5403