Abstract Algebra by Benedict Gross, PhD, George Vasmer Leverett Professor of Mathematics, Harvard University.
From the webpage:
Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields.
Videos, notes, problem sets from the Harvard Extension School.
The relationship between these videos and those found on YouTube isn’t clear.
The text book for the class was Algebra by Michael Artin. (There is a 2nd edition now.)
There are two comments that may motivate you to pursue these lectures:
First, Gross remarks in the first session that there are numerous homework assignments because you are learning a language. Which makes me curious why math isn’t taught like a language?
Second, the Wikipedia article on abstract algebra observes in part:
Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties. This creates a false impression that in algebra axioms had come first and then served as a motivation and as a basis of further study. The true order of historical development was almost exactly the opposite. For example, the hypercomplex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. Most theories that are now recognized as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. An archetypical example of this progressive synthesis can be seen in the history of group theory.
Interesting that techniques are developed for quite practical reasons but later justified with greater formality.
Suggests that semantic integration should focus on practical results and leave formal justification for later.
Yes?
I first saw this in a tweet by Steven Strogatz.