Group theory

There have been 8 completed talks, 2 documents, and 3 topic suggestions tagged with group theory.

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Completed Talks

Group Theoretic Attacks on the Enigma Cipher

Delivered by Laindon Burnett on Friday March 31, 2017

Lie Groups and Special Relativity

Delivered by Mohamed El Mandouh on Friday March 24, 2017

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.

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.

Please read this basic overview of Lie Theory.

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?

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

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.

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.

Possible reference materials for this topic include

Quick links: Google search, arXiv.org search, propose to present a talk

algebra algorithm computational mathematics computer science group theory linear algebra open problem