Linear algebra

There have been 6 completed talks, 2 documents, and 6 topic suggestions tagged with linear algebra.

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Completed Talks

Metric embeddings and dimensionality reduction

Delivered by Frieda Rong on Friday March 31, 2017

In this talk, we consider embeddings which preserve the pairwise distances of a set of points. It is often useful to find mappings from one high dimensional space to a lower dimensional space that preserve the geometry of the points. One source of applications is in streaming large amounts of data, for which storage is costly and/or impractical. However, the study of such embeddings has also inspired developments in the design of approximation algorithms and compressed sensing.

At the crux of the talk is the remarkable Johnson-Lindenstrauss lemma. This fundamental result shows that for Euclidean spaces, it is possible to achieve significant dimensionality reduction of a set of points while approximately preserving the pairwise distances. An elementary proof will be given, along with subsequent speed improvements with sparse projections and an interesting use of the Fourier transform. We will also discuss applications of the lemma to the fields mentioned above.

Lie Groups and Special Relativity

Delivered by Mohamed El Mandouh on Friday March 24, 2017

Tensor Products

Delivered by Felix Bauckholt on Friday January 27, 2017

What is a tensor product? How are they useful? This talk will elaborate on this broad, widely applicable topic.

Quandles: The Algebra of Knots

Delivered by Brennen Creighton-Young on Friday November 25, 2016

One of the major goals of knot theory is to determine whether or knot two knots can be continuously deformed into one another. This idea can be fully captured algebraically — and leads us to algebraic structures that not only capture the desired notion of knot deformations, but reveal themselves as truly fascinating mathematical objects. We will use only linear algebra and elementary abstract algebra.

Though continuous deformation is a straight forward idea, the rigorous definition of this, the notion of ambient isotopy, is practically unusable. This talk will provide a quick overview of basic knot theory and will work towards developing numerous algebraic strategies to identify when two knots are ambient isotopic. We will focus mostly on providing motivation for keis, an algebraic representation of knots, as well as their generalization, quandles. Quandles will prove to be not only helpful with respect to the goal of the classification of knots, but also as rich algebraic structures. The talk will involve small amounts of group theory, linear algebra and module theory.

Naïve Lie Theory

Delivered by Aidan Patterson on Friday November 25, 2016

In 1870, Sophus Lie was studying the symmetries of differential equations, which generally form “continuous” groups. The analogous problem for polynomials was solved by Galois previously, so there was incentive to solve the related problem for these new kinds of groups. We’ll motivate such continuous groups by looking at matrices, assuming only a little linear algebra.

Lie started a study of simplicity in these continuous groups. Lie understood these groups as groups generated by infinitesimal elements, which led him to believe that a group $G$ should be generalised to consider infinitesimal elements.

Today we separate the infinitesimal elements of a group $G$ to form a Lie algebra $g$, which captures most of the important structures of $G$, but is easier to handle. This talk will focus on motivating Lie's definitions, and provide some techniques used to prove simplicity for Lie groups. As well, some specific examples such as $\mathrm{O}(n)$, $\mathrm{SO}(n)$, $\mathrm{U}(n)$, $\mathrm{SU}(n)$, and $\mathrm{Sp}(n)$ will be mentioned to illustrate the concepts presented.

Please read this basic overview of Lie Theory.

Hypercomplex Numbers

Delivered by Fengyang Wang on Friday November 18, 2016

There are exactly three distinct two-dimensional unital algebras over the reals, up to isomorphism. Each of these algebras corresponds to a unique geometry, with applications. This talk will develop the concepts needed to understand two-dimensional algebras over the reals, starting from the definitions of key concepts. We will rediscover the familiar complex numbers and generalize its construction to find the other hypercomplex number systems. We will then prove the result that these are the unique hypercomplex number systems, up to isomorphism. Finally, we will discuss possible generalizations to $n$ dimensions. Please ensure that you have a good understanding of fundamental concepts of two-dimensional linear algebra.

We will use Catoni, F., Cannata, R., Catoni, V., & Zampetti, P. (2004) as a reference.

A summary of this talk is available here.

Documents

Lie Theory

This is a reference document on Lie Theory by Aidan Patterson. It covers a subset of the material of PMATH 763.

Two-Dimensional Linear Algebra

This is a reference document on Two-Dimensional Linear Algebra by Fengyang Wang. It covers a subset of the material of MATH 136 or MATH 146.

Talk Suggestions

Complexity of Matrix Multiplication

Volker Strassen showed that $n^3$ matrix multiplication was not optimal in 1969. Since then, new algorithms such as Coppersmith-Winograd have further improved the time complexity of matrix multiplication. It is conjectured that matrix multiplication is possible in $O(n^{2+ɛ})$ for any $ɛ>0$, however small. This is one of the few remaining open problems in finite-dimensional linear algebra.