Abstract Algebra Lecture

[Karlotta Neifert]

Theweb is full of video lectures these days but, try as I might, I can find very little for Introduction to Group Theory. The closest I found was -learning-initiative/abstract-algebra . Are they any online introductory group theory lectures people would recommend?

Richard Borcherds has a set of online group theory lectures. In his own words, it's pitched at "first-year graduate or enthusiastic undergraduate" level, with an emphasis on examples rather than proofs.

I have recently found this crash course in group theory online.It is basically a good summary for anyone interested in learning the very basics of Group Theory or for reviewing some topics before going deeper into the theory.

I'd also recommend trying Professor Macauley's series on Group Theory. He explains things with a lot of detail, with examples and proofs, so the lessons can feel a bit long if you're only interested in getting the results/fundamentals, but they're quite good and useful, specially if you're seeing the topics for the first time.

Now, the reputation of this class precedes itself and it is very likely that you've heard about it from upperclassmen complaining about it. And you may be thinking to yourself: "Wait! I already took MATH 33A, which is Linear Algebra. Why do I have to take it again? In fact, MATH 33A is a pre-requisite for MATH 115A!"

In introductory courses like Math 33A, linear algebra often revolves around matrix studies. However, at the 115A level, the focus shifts to exploring vector spaces and their transformations. If you're unfamiliar with the concept of a vector space, don't worry - we'll delve into it soon. For starters, consider $\mathbbR^n$ - the set of all $n$-tuples of real numbers - as your introductory vector space. Typically, this is the sole vector space explored in elementary linear algebra courses. In 115A, however, we'll expand our horizons, exploring linear algebra in various other vector spaces, which proves to be incredibly beneficial.

Our approach involves starting from the very basics. It is perhaps helpful to momentarily set aside all your previous mathematical knowledge and treat 115A as a foundational course designed to systematically build a specific mathematical field from the ground up. This is our initial aim in 115A.

A noteworthy point regarding this goal is the following: You might be anticipating that exploring linear algebra in vector spaces beyond $R^n$ will be a radically different and exciting experience. However, I must clarify that abstract linear algebra in general vector spaces largely mirrors the linear algebra you've encountered in $\mathbbR^n$. The concepts of linear independence, transformations, kernels, images, eigenvectors, and diagonalization - all familiar topics within the realm of $\mathbbR^n$ - function similarly in 115A.

(2) Construct and Follow Abstract Mathematical Arguments and Statements

This goal extends beyond mere proof-writing. Upper-division mathematics, in contrast to lower-division studies, prioritizes the discovery and articulation of truths over computation. In 115A, every solution you formulate should be viewed as a mini technical essay, marking a departure from mere scratch work to determine problem solutions. Mastering the art of clear, logical, and effective communication of mathematical truths is a challenging yet essential skill to develop.

Here is a list of strategies and advice that I found useful in navigating this challenging yet rewarding course. Some of these advice are in retrospect (i.e. things I would do if I were to re-take the course).

3 years ago, I experimented with the 'weekly newsletter' approach and it was not only unsustainable, but also slightly spammy. My new model is as follows: I promise to only send an email to your inbox when I'm absolutely positive it is something you'll find interesting, and perhaps more importantly, actionable. Topics will include personal finance, productivity or general life insights.

There are weekly problem sets posted under "Assignments" in MyCourses. Each problem set consists of five problems. Students who attend an exercise session need to submit solutions only to three marked problems. Students who do not attend an exercise session need to submit solutions to all problems. Solutions can be uploaded as a single PDF file to MyCourses by Tuesday midnight. Since 5 best homework grades out of 6 are taken into account, the homework submission deadlines are strict and late submissions are not accepted.

Official announcements will be posted in MyCourses. The rest of communication for this course takes place in Zulip. The link to Zulip is the following: . Please be active asking your questions!

Lecture notes (possibly handwritten) are posted under "Materials" after each lecture. We will not follow one particular source, however, all essential material is included in any standard abstract algebra book or lecture notes, for example

This course is aimed at undergraduate mathematics and mathematics education majors. It is a first course in abstract algebra. In addition to being an important branch of mathematics in its own right, abstract algebra is now an essential tool in number theory, geometry, topology, and, to a lesser extent, analysis. Thus it is a core requirement for all mathematics majors. Outside of mathematics, algebra also has applications in cryptography, coding theory, quantum chemistry, and physics.

The chief prerequisite for this course is Math 290. In Math 290, students should learn basic logic, basic set theory, the division algorithm, Euclidean algorithm, and unique factorization theorem for integers, equivalence relations, functions, and mathematical induction. As these topics are of high importance in Math 371, it might be prudent for the instructor to review them at the beginning of the semester.

Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

These are full notes for all the advanced (graduate-level) courses I have taught since1986. Some of the notes give complete proofs (Group Theory,Fields and Galois Theory, Algebraic Number Theory, Class Field Theory,Algebraic Geometry), while others are more in the nature of introductoryoverviews to a topic. They have all been heavily revised from the originals. I am (slowly) in the process of producing final versions of them and publishing them.Please continue to send me corrections (especially significant mathematical corrections) and suggestions for improvements.

If the pdf files are placed in the same directory, some links will work between files (you may have to get the correct versionand rename it, e.g., get AG510.pdf and rename it AG.pdf).

The pdf files are formatted for printing on a4/letter paper.

The cropped files have had their margins cropped --- may be better for viewing on gadgets.

The eReader files are formatted for viewing on eReaders (they have double the number of pages).

At last count, the notes included over 2022 pages.

Algebraic Geometry

This is a basic first course. In contrast to most such accounts the notes study abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory.

Elliptic Curves

This course is an introductory overview of the topic including some of the workleading up to Wiles's proof of the Taniyama conjecture for most elliptic curvesand Fermat's Last Theorem. These notes have been rewritten and published.

Abelian Varieties

An introduction to both the geometry and the arithmetic of abelian varieties. It includes adiscussion of the theorems of Honda and Tate concerning abelian varieties over finite fieldsand the paper of Faltings in which he proves Mordell's Conjecture.

Lectures on Etale Cohomology

An introductory overview. In comparison with my book, the emphasis is onheuristics rather than formal proofs and on varieties rather than schemes, andit includes the proof of the Weil conjectures.

Class Field Theory

This is a course on Class Field Theory, roughly along the lines of the articlesof Serre and Tate in Cassels-Frhlich, except that the notes are moredetailed and cover more. The have been heavily revised and expanded from earlier versions.

Welcome to my class MATH 416 Abstract Linear Algebra! In this course we will study the structure of vector spaces and linear maps between vector spaces. Please see the Table of contents below for a detailed list of topics that we will cover.

Enrolled students: We will use Canvas for handing in homework and taking the quizzes before each lecture. Please make sure that you have access to the Canvas course website; otherwise, please contact me.

The class follows the "flipped classroom" paradigm: I will make pre-recorded lectures of the course material available on this website. You are supposed to watch them at home, and take a short quiz about the lecture before class starts. In class we will answer your questions about the material and solve additional problems in group work.

In order to accommodate unforeseen circumstances that might prevent you from keeping up with the class, I will drop the three lowest homework scores, the three lowest quiz scores, and the lowest of the three midterm exam scores.

I am dedicated to providing an inclusive and safe classroom experience for everyone, regardless of gender, gender identity and expression, sexual orientation, disability, physical appearance, body size, race, age or religion. I will not tolerate harassment and discriminating or disrespectful behavior between any classroom participants (including myself) in any form, whether in person or online. Violations of this code of conduct will be reported appropriately.(This code of conduct is based on a template provided by the Geek Feminism Wiki.)

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