Events

Kaleb Ruscitti, University of Waterloo

Building Kronecker Moduli Spaces

Kronecker moduli spaces are quiver moduli spaces and a generalization of Grassmannians. They parameterize n-tuples of matrices between two vector spaces, up to change of basis on both sides. In this seminar, I will describe how to construct them as GIT quotients and what properties we can prove about them from this construction.

MC 5417

Julius Frizzell, University of Waterloo

A Quick Introduction to Ergodic Theory

I will introduce the basic definitions and theorems (without proof) of ergodic theory that are needed to discuss Furstenberg's multiple recurrence theorem. The development will follow that in Chapter 1 of "An Introduction to Ergodic Theory" by Peter Walters and Chapter 3 of "Multiple Recurrence in Ergodic Theory and Combinatorial Number Theory" by Harry Furstenberg. Time allowing, I will also cover the statement of the Multiple recurrence theorem itself and its relationship to Szemerédi's theorem.

MC 5417

Benoit Charbonneau, University of Waterloo

Some homogeneous geometry on the manifold of full flags

I will be using the manifold of full flags of complex three-space (seen as the quotient of $\mathrm{SU(3)}$ byits torus) to illustrate how much geometry one can do with homogenous objects.

MC 5417

Facundo Camano, University of Waterloo

Boundary Conditions for Non-Euclidean Monopoles

In this talk, I will discuss the heuristic behind defining asymptotics for monopoles. Specifically, the asymptoticsshould be abelian solutions embedded into the gauge group. I will first go over this heuristic for Euclideanmonopoles and then move on to non-Euclidean situations such as hyperbolic and singly periodic.

MC 5417

Julius Frizzell, University of Waterloo

Szemerédi's Theorem and Multiple Recurrence

We will cover Szemerédi's Theorem and its equivalence to Furstenberg's multiple recurrence theorem, we will then begin to look at weak-mixing transformations in more detail.

MC 5417

Boris Li, University of Waterloo, Pure Mathematics

On the physics of bells

We shall discuss the acoustics and resonance of bells from the perspective of physics. After a short review ofnormal modes in musical instruments, we shall compare bells with one-dimensional systems such as strings andwinds. In contrast to these systems, bells exhibit strongly inharmonic partials due to its curved geometry. Wediscuss some consequences of this structure, including some prominent partials and the psychoacousticphenomenon of the strike note. Time permitting, we shall examine how bell geometry and thickness profilesaffect tuning.

Refreshments at 3:30pm. Talk starts at 4:00pm.

MC 5403

Spencer Kelly, University of Waterloo

Sobolev Spaces Over Compact Manifolds

The space of smooth sections of a vector bundle over a manifold is an infinite dimensional Fréchet Space, and thus many of the tools used in finite-dimensional geometry are rendered useless on this space. However, taking the completion of this space with respect to the Sobolev norm, we obtain a Banach space. What's even better is that in the $L^2$ case we obtain a Hilbert space. In this talk we will walk through different constructions of the$L^2$-Sobolev spaces of sections of a vector bundle over a compact manifold, and discuss the Sobolev embedding theorem. We will also work through some of the properties of differential operators on this space and, time permitting, we will finish with the Berger-Ebin decomposition for differential operators with injective symbol.

MC 5417

Viktor Majewski, University of Waterloo

Obstructions to Smooth Full Holonomy Cayley Fibrations

We study smooth fibrations of compact torsion-free $\textrm{Spin}(7)$-manifolds by Cayley submanifolds.Using geometric and topological constraints coming from the $\textrm{Spin}(7)$-structure, we show that only two topological configurations can arise. One of these is excluded by a spinnability criterion for fiber bundles,with the relevant hypothesis verified using gauge-theoretic input, while the remaining case is reduced to an open conjecture in $4$-manifold topology. In particular, this rules out smooth Cayley fibrations on all known examples of compact torsion-free $\textrm{Spin}(7)$-manifolds.

MC 5417

Anne Johnson, University of Waterloo

Twisted arcs on root stacks

We briefly introduce the theory of stacks via the stack of triangles using Kai Behrend’s exposition as a guide. We move on to Yasuda’s notion of the twisted arc space of a DM stack. As time permits, we take up the special case of twisted arcs on a root stack.

MC 5403

Barbara Csima, University of Waterloo

Priority Arguments in Computability Theory

This term, Computability Learning Seminar will focus on Priority Arguments. Priority Arguments are a common proof technique used in Computability Theory. A theorem is broken down to being equivalent to a list of requirements. These requirements are given a priority order, and a strategy is devised to meet all the requirements, making use of the priority order. In the early days of the subject, a big question (Post’s Problem -1944) was whether there were any non-computable computably enumerable (c.e.) sets that were not Turing equivalent to the halting set. The solution, from Friedberg (1957) and Muchnik (1956), was to construct a pair of Turing incomparable c.e. sets, using a finite injury priority argument. In this first talk, we will begin our examination of priority arguments by going through the proof of this theorem, introducing definitions and reviewing notions from Computability Theory as needed along the way.

MC 5403