Beginnings

The Narrow Road draws its title from Oku no Hosomichi (The Narrow Road to the Interior), the famous travel diary of Matsuo Basho as he journeyed into northern Japan. My aim is to follow a similar wandering journey, but instead travelling into the abstract highlands of pure mathematics, pausing to admire the beauty and sights along the way, much as Basho did. That means we have a long way to travel: from the basics of abstract or pure mathematics, through topology, manifolds, group theory and abstract algebra, category theory, and more. There may well be some detours along the way as well. It is going to take a long time to get to where we are going, but along the way we’ll see plenty of things that make the trip worthwhile. Indeed, as is so often the case, the journey means more than the destination.

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11 Responses to “Beginnings”

  1. Walt Says:

    What is a pro-finite Lie ring?

  2. lmcinnes Says:

    A pro-finite Lie ring is a compact Hausdorff topological Lie ring such that the open ideals of finite index form a neighbourhood base of 0. Alternatively you can view it as an inverse limit of a system of finite Lie rings. As you can see, explaining them decently to a lay audience requires a bit of unpacking, and at least touching on a wide variety of topics. In practical terms I am studying them because Lie theoretic methods have proved productive in pro-finite group theory, and in particular pro-p group theory; for example Zelmanov’s solution to the Restricted Burnside Problem makes significant use of pro-p groups and Lie theoretic methods.

  3. Daniel Collins Says:

    Welcome to the club, old mate!

  4. Corporate Serf Says:

    In your “Why abstract mathematics matters” page, you might want to mention the work of Herlihy, who used algebraic topological methods to establish the unsolvability of some distributed co-ordination problems back in the nineties.

    I (vaguely) know what a lie group/algebra is (from Jacobson’s intro published by dover). What is a lie ring?

  5. lmcinnes Says:

    I’m not familiar with Herlihy’s work, so I’ll have to look into that. As to what a Lie ring is: the quick and easy approach is an axiomatic definition; a Lie ring is non-associative ring that is anti-symmetric and satisfies the Jacobi identity. That is, it is an Abelian group L with an operation [·,·]:L→L such that for all x,y,z ∈ L
    [x,y] = -[y,x]
    [x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0

  6. Cristobal Says:

    Good site! I found in google.com

  7. Ricardo Says:

    I’m sad to say that I was part of that huge crowd of not-so-mathematical-savy people that went through high school and college math classes fully committed to the anti-math prejudice. Now, I’m starting to see the error of my ways since I’m at a stage of my career where I realize that a deeper, more confortable handling of abstract mathematics could help me go a long way.

    I hope mine is not a lost cause. I’ll be your avid reader.

  8. Margus Says:

    Your blog has thus far been extremely enlightening. It was really fun to come upon a mathematician with similar interests to myself (Im currently a Bachelors student in CS, with a thesis topic in Provable security but with main interest in the foundations of both mathematics and computer science). Thus far I haven’t come across a mathematician with an interest in Zen (although I have read a few books by a few). I was wondering what actually brought you to it and weather you had any good books or reccommendations for a young person beginning his journey in both math and eastern philosophy.

  9. Leland McInnes Says:

    Margus: Interestingly I came to Buddhist philosophy via a rather roundabout route via works such as those by Daniel Dennett, Susuan Blackmore, Daneil Wegman, and Douglas Hofstadter. Which is to say, I came to these perspectives from a more science based approach to questions of consciousness. I’m not sure whether that is terribly helpful to you, but I can certainly say that the general area, and the authors mentioned, are both quite enlightening.

  10. Andrew Briscoe Says:

    If you havn’t already check out Godel, Escher, Bach and http://scienceblogs.com/goodmath/

  11. Oliver Says:

    I don’t know how I got here, and I am a frightened stranger in a strange land. I’ll post anyway.

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