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Introduction: Charles Clingen

Call me … Charlie. I’m long retired after a long career in software development management preceded by a long-underused degree in Physics. Over the past several years I’ve been studying gauge theory as a way of keeping mentally active. My goal is not to “learn about” gauge theory, but to actually learn enough to do some simple calculations using all the underlying disciplines. As my roadmap, I’ve been using the first portion of “Gauge Fields, Knots and Gravity” by John Baez & Javier P. Munian. My journey is extremely exciting and rewarding – I’m learning (small bits of) various fields of math that I had never even heard of before. It is a huge landscape! Amazing! But it is also frustrating – I have no fellow students to study with and I don’t dare impose on the pros with my all-to-frequent dumb questions. Fortunately the boundlessly patient Prof. Baez has rescued me from time to time, for which I am deeply grateful.

I am taking this class because I want to understand the occasional references to category theory that I have encountered, and also because achieving even a glimpse of understanding of category theory will be exciting and challenging. And for the first time I will have the opportunity to interact with others and ask lots of questions and maybe someday, even be able to answer a few question myself.

Based on my studies of the fields underlying gauge theory, I’ve noted some basic common ideas that seem to form a kind of mathematical framework. Organizing and understanding those ideas early on might make it easier to teach complicated, abstract mathematical theories, like gauge theory, even to high school students. And a bit of category theory might facilitate that organization and understanding. I hope that this course will help me to test that (some would say overly-optimistic) hypothesis.

Finally, I’m taking this class because the world of education is rapidly changing and maybe this class points to one of the future methods of effective teaching and learning. I hope so.

Comments

  • 1.
    edited April 2018

    Hi, Charlie! Good to see you here! I hope you dive in, start reading the Lectures and the comments on them, try some of the Exercises, ask questions about them, chat with people, and generally soak up some ideas. This intro to category theory is a bit odd in that it starts out talking about preorders and posets. But these are very simple sorts of categories, which may make it easier to get a feeling for the key concepts of "left adjoint" and "right adjoint".

    Comment Source:Hi, Charlie! Good to see you here! I hope you dive in, start reading the [Lectures](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Lectures) and the comments on them, try some of the [Exercises](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Exercises), ask questions about them, [chat with people](https://forum.azimuthproject.org/categories/Chat), and generally soak up some ideas. This intro to category theory is a bit odd in that it starts out talking about preorders and posets. But these are very simple sorts of categories, which may make it easier to get a feeling for the key concepts of "left adjoint" and "right adjoint".
  • 2.

    Hi John! Good to be here! I'm just getting started, but already I find the whole class idea -- the format, the ability to chat with people, the exercises and puzzles, and the content -- very exciting and motivating. This is a very interesting experiment!! Ahh... there's my old nemesis, the "adjoint", again! I assume it's not closely related to the adjoint in differential geometry -- I'm looking forward to finding out!

    Comment Source:Hi John! Good to be here! I'm just getting started, but already I find the whole class idea -- the format, the ability to chat with people, the exercises and puzzles, and the content -- very exciting and motivating. This is a very interesting experiment!! Ahh... there's my old nemesis, the "adjoint", again! I assume it's not closely related to the adjoint in differential geometry -- I'm looking forward to finding out!
  • 3.
    edited April 2018

    No, adjoint functors are not related to the adjoint action of a group on its Lie algebra. They're related to another sort of "adjoint". Every linear map between vector spaces \(T : V \to W \) gives rise to a linear map going the other way between the dual vector spaces, \( T^{\ast} : W^{\ast} \to V^{\ast} \), and we call \( T^{\ast} \) the adjoint of \(T\).

    Anyway, learn about left and right adjoints here and your world will expand in a very interesting way. All the stuff we're talking about here is much more generally applicable than differential geometry.

    Comment Source:No, adjoint functors are not related to the adjoint action of a group on its Lie algebra. They're related to another sort of "adjoint". Every linear map between vector spaces \\(T : V \to W \\) gives rise to a linear map going the other way between the dual vector spaces, \\( T^{\ast} : W^{\ast} \to V^{\ast} \\), and we call \\( T^{\ast} \\) the **adjoint** of \\(T\\). Anyway, learn about left and right adjoints here and your world will expand in a very interesting way. All the stuff we're talking about here is much more generally applicable than differential geometry.
  • 4.

    John - OK. Left and right adjoints it is! My next major goal.

    Comment Source:John - OK. Left and right adjoints it is! My next major goal.
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