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

Spencer Kelly, University of Waterloo

The Sobolev Embedding Theorem and The Berger-Ebin Decomposition Theorem

Serving as a part two of my previous talk, in this talk we will begin with briefly talking about the Sobolev Embedding Theorem for \(L^2\)-based Sobolev spaces. We will then move on to deriving the Berger-Ebin decomposition theorem for differential operators with injective symbol.

MC 5417

Alexander Teeter, University of Waterloo

Freedman’s Results and Donaldson’s Exclusions: Classification of 4-manifolds by intersection forms

We look at Serre's algebraic classification of symmetric integral unimodular forms, and then Freedman's topological classification of topological 4-manifolds by their intersection forms. After, we take a turn to looking at Donaldson's exclusions, finding that smooth 4-manifolds are not as well behaved. Time permitting, we will see how Donaldson's exclusions naturally lead to the existence of exotic \(\mathbb{R}^4\)s.

MC 5417