# Algebra

There have been 12 completed talks, 4 documents, and 19 topic suggestions tagged with algebra.

There are many subfields of algebra, including:

## Completed Talks

### Group Theoretic Attacks on the Enigma Cipher

Delivered by Laindon Burnett on Friday March 31, 2017

### 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

### Universal Property of Quotients

Delivered by Lirong Yang on Friday March 17, 2017

In this talk, we generalize universal property of quotients (UPQ) into arbitrary categories. UPQs in algebra and topology and an introduction to categories will be given before the abstraction. As in the discovery of any universal properties, the existence of quotients in the category of sets and that of groups will be presented.

If you have not yet been exposed to group theory, please read Monoids and Groups for an introduction.

A summary of this talk is available here.

### Inverse Galois Theory

Delivered by David Liu on Friday March 17, 2017

Almost 200 years after Évariste Galois's death, there is still one question about Galois groups — the symmetries of the roots of polynomials, that still remains unsolved. This is the Inverse Galois Problem — whether every group is a Galois group of a Galois extension of the rational numbers. In this talk, I will give an overview of the progress that has been made, the approaches that mathematicians are making, and directions for further research.

A note on the requirements for this talk: It is recommended that you be comfortable with groups, fields, field extensions, automorphisms, and basic Galois theory. This requirement can be met by any of the following suggested alternatives:

• having taken PMATH 347, and currently taking PMATH 348;

• having taken PMATH 347, and coming for the prerequisite knowledge presentation (see below);

• having equivalent knowledge to PMATH 347 and PMATH 348;

• otherwise, if you are currently taking or have taken MATH 146, this talk is still accessible, but it is highly recommended that you read An Introduction to Galois Theory by Dan Goodman, and watch this 18-minute lecture by Matthew Salomone (after reading the article) and optionally also come for the prerequisite knowledge presentation

A prerequisite knowledge presentation about Galois theory will be given at 17:00, prior to the beginning of the talk at 17:30. This presentation will last about 20 minutes. If you are unfamiliar with the material, it is recommended that you read some of the linked material above in addition to attending this presentation. Attending this presentation is optional. The reference material used for this presentation is the Galois Theory document.

### 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.

### Voting with Homomorphic Encryption

Delivered by Sidhant Saraogi on Friday December 2, 2016

In light of the recently concluded Elections or as John Oliver would call it “A horifying glimpse at Satan's Pinterest Board 2016”, “The One who must not be named” has repeatedly insinuated that the elections have been rigged. Our humble aim, present a voting scheme where:

• each voter casts exactly one ballot.

• voting is anonymous.

We delve into two areas on our way to prove our goal :

• Blind Signatures, which allow for anonymous voting

• Pallier Cryptosystem, which gives us the ability to sum up the votes even though they have been encrypted thus allowing the election to be “publically audited”.

We might also, if time permit, talk about more modern systems of enabling fair elections that have even been implemented in real life.

This talk is based off Ron Rivest’s lecture, of which a summary is available.

### 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.

### Basic Elliptic Curves

Delivered by Akshay Tiwary on Friday November 18, 2016

Elliptic Curves lie in the intersection of Algebra, Geometry, Number Theory and Complex Analysis. While my talk won't require any experience with complex analysis or algebraic geometry, I hope to expose you to this active area of research. Although I could tell you that the meat of Wiles' proof of Fermat's Last Theorem involved proving a special case conjecture previously known as the Taniyama Shimura conjecture involving elliptic curves, or that the Birch and Swinnerton-Dyer conjecture is an open millenium prize problem that talks about the relation between the arithmetic behaviour of elliptic curves and the analytic behaviour of an L-function, or that elliptic curves over finite fields is so useful for cryptography that there was a memo that recommended elliptic curves for federal government use in 1999, I don't have to because you're a math student who will attend this talk without needing to be wowed with cool applications, right?

### 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.

### Groups

Delivered by Fengyang Wang on Wednesday October 12, 2016

This talk will cover the basics of group theory. There are no official prerequisites for this talk, but MATH 145 and MATH 146 are an asset. The group theory part of the talk will mostly be based on Alekseev, 2004, specifically sections 1.1 (motivation), 1.2 (transformation groups), 1.5, 1.9, and 1.13 (various morphisms), and 1.11 (quotient groups).

Due to time constraints, it is inevitable that some content must be rushed. I will not go over proofs and derivations in full rigour, and I highly advise studying the reference material after the talk to catch up on what you may have missed. Please message me in advance if there are any other subjects in particular that you would like me to discuss.

A summary of this talk is available here.

## Documents

### Galois Theory

This is a reference document on Galois Theory by Akshay Tiwary and Fengyang Wang. It covers a subset of the material of PMATH 348.

### Lie Theory

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

### Monoids and Groups

This is a reference document on Monoids and Groups by Fengyang Wang. It covers a subset of the material of PMATH 347.

### 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

### Almost Integers and Pisot Numbers

Sometimes some non-integers like $e^{π} - π = 19.9990999791...$ are curiously close to integers. While often this is just a numerical coincidence, sometimes there is a clear mathematical reason for certain numbers to be almost integers. Pisot numbers, which are special types of algebraic integers, provide a systematic means to construct almost integers that approximate integers at an exponential rate. Of course, Pisot numbers are interesting on their own right and an interested speaker can also venture into the connections to Beta expansions (see the 2nd and 3rd references below), or fractals (see the 4th reference below).

Required Background: Basic analysis at the level of 147 and 138, algebra at the level of 145. Any other background depends on the direction the speaker takes with this topic.

Possible reference materials for this topic include

### 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.

Possible reference materials for this topic include

### Compressed Sensing

Compressed sensing is about minimizing the information gathered and stored by sensors, reducing the need for file compression later on for transmission. This can reduce costs for certain applications, such as non-visible wavelength cameras.

Possible reference materials for this topic include

### Continued Fractions and Hyperbolic Geometry

This topic studies the beautiful connection between continued fractions and the famous Farey tessekation of the hyperbolic plane.

Possible reference materials for this topic include

### Costas Arrays

Permutation matrices for which all $\frac{n(n-1)}{2}$ displacements between all pairs of 1s are distinct are known as Costas arrays, and these have important applications in sonar. The Welch construction of Costas arrays uses concepts from number theory, whereas the Lempel-Golomb construction uses concepts from field theory.

Possible reference materials for this topic include

### Down with Determinants

Possible reference materials for this topic include

### Galois Field Arithmetic

A Galois field is a finite field and are used in a variety of applications, including in classical coding theory and cryptography algorithms. This topic studies how to efficiently optimize arithmetic in such fields.

Possible reference materials for this topic include

### Lattices, Linear Codes, and Invariants

This topic studies sphere packing and the connection to lattices and linear codes.

Possible reference materials for this topic include

### Lindstrom-Gessel-Viennot Lemma

This topic studeis an interesting connection between combinatorics and determinants.

Possible reference materials for this topic include

### Markov Equation and Irrational Numbers

Possible reference materials for this topic include

### Monsky's Theorem

This topic studies the proof of Monsky's theorem that it is not possible to dissect a square into an odd number of triangles of equal area.

Possible reference materials for this topic include

### Quivers and Platonic Solids

Possible reference materials for this topic include

### Representation Theory and Voting Theory

Possible reference materials for this topic include

### Sparse Matrices

Sparse matrices are frequently used in scientific and numerical computation. How do they work? What new findings have there been? Speakers are encouraged to specialize this broad topic to a particular subfield of interest.

Possible reference materials for this topic include

### Sphere Packing, Lattices, and Groups

Possible reference materials for this topic include

### Symmetric Groups, Young Tableaux and Representation Theory

Possible reference materials for this topic include

• Bruce Sagan : The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions