Events

Paul Cusson, University of Waterloo

Monopoles with rotational symmetry

We will first look at SU(2)-monopoles invariant under the action of a circle subgroup of SO(3) about the z-axis.The polynomials cutting out their spectral curves in TP^1 will be derived, and these will be used to describe thespectral curves of S^1-invariant SU(N)-monopoles for arbitrary N.

MC 5403

Jennifer Zhu, University of Waterloo

Morphisms of Quantum Confusability Graphs

It would be unrealistic to have an information channel — quantum or classical — that always sends information with absolute accuracy; that is, we must expect a channel to have noise. In 1956, Shannon introduced the notion of zero-error capacity of a noisy (classical) channel using the confusability graph of this channel. In 2010, Duan, Severini, and Winter developed the analogous notion (quantum confusability graphs) for quantum channels and show that one can recover various types of zero-error capacities of quantum channels. In the first half of this talk, we will see how these quantum confusability graphs are derived and how they subsume Shannon's notion of classical confusability graphs.

QNC 1201

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 5403

Catherine St-Pierre, University of Waterloo

Organizational meeting

We will organize the seminars for the summer.

MC 5403

Jennifer Zhu, University of Waterloo

Categorical Limits of Quantum Graphs and Possibilities Induced by QuantumPseudometrics

Quantum graphs and quantum pseudometrics as defined by Kuperberg and Weaver have roots in quantum errorcorrection but have since been developed as subjects in their own right. The motivation for the first half of thisthesis is to build an infinite quantum graph from finite quantum graphs. The latter have been subjected to fargreater scrutiny due to their connections to categorical quantum theory, while the former have been somewhatneglected. To be precise, we define and take the categorical (co)limit of quantum graphs by developing a newnotion of morphism compatible with previous notions but carrying less baggage. The inspiration for the secondhalf follows from the (unpublished) theorem that pure states on a von Neumann algebra \mathcal{M} are givenby maximal filters in the projection lattice of \mathcal{M}. Upon the observation that points in a metric space$(X, d)$ with topology $\tau$ are also given by maximal filters $\tau$ and that quantum pseudometrics provide anotion of distance between projections in $B(\ell^2) \overline{\otimes} \mathcal M$, we are led to a notion ofdistance $f$ between pure states induced by these quantum pseudometrics. Also this function $f$ does not satisfythe triangle inequality, we make some parallels between it and David Lewis's conception of ``possible worlds.''

MC 2009

Yash Singh, University of Waterloo

Vector bundles on toric stacks

This thesis is concerned with a generalization of Klyachko's classification of toric vector bundles to toric stacks.The work of Klyachko gave an elegant method of studying toric vector bundles through filtrations of a vectorspace. We extend these techniques to vector bundles on a toric stack and generalize the aspects of Klyachko'swork to a more geometric setting. In particular, we show that the category of reflexive sheaves on a toric stack isequivalent to a category of filtered reflexives sheaves of its largest Deligne-Mumford substack. We then combinethis with an equivariant version of Gubeladze's result on the splitting of vector bundles on toric varieties to provea classification theorem for vector bundles on toric stacks.

MC 5403

Paul SkoufranisYork University

Non-Commutative Majorization

The maps that send a self-adjoint matrix $A$ to $U^*AU$ where $U$ is a unitary matrix are essential inQuantum Information Theory as these maps transmit quantum information in a reversible way. When convexcombinations of such maps are taken, one obtains what are known as the mixed unitary quantum channels, whichare essential models for how quantum information can be transmitted when noise is present. Just as the unitaryconjugates of a self-adjoint matrix can be determined via spectral data, so too can the image of a self-adjointmatrix under all possible mixed unitary quantum channels. Since this is equivalent to characterizing the convexhull of the unitary orbit of a self-adjoint matrix, this problem has a well-known solution from operator theoryinvolving the notion of matrix majorization of one self-adjoint operator by another. In this talk, we will examinehow we can extend the notion of matrix majorization to non-commutative contexts. In particular, we will discussa notion of non-commutative majorization that characterizes the potential outputs under all quantum channels ofany non-commutative tuple of matrices. This is based on joint work with Matt Kennedy.

MC 5417 or Join on Zoom

Spiro Karigiannis, University of Waterloo

Decomposition of the Riemann Curvature Tensor

The Riemann curvature tensor R of a Riemannian metric decomposes into three orthogonal components: thescalar curvature, the traceless Ricci curvature tensor, and the Weyl curvature tensor. I will explain in detail therepresentation theory and linear algebra underlying this decomposition. Moreover, we will see that in the specialcases of dimensions 2, 3, 4 one can say more. As an application, I will discuss the Singer-Thorpe Theoremcharacterizing Einstein metrics in 4 dimensions in terms of this decomposition. If time permits (and it may wellpermit, as this will be a 2.5 hour talk with a break midway), I will briefly discuss a generalization of these ideasto G2-geometry in 7 dimensions.

MC 5417

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