Abstract Nonsense

Crushing one theorem at a time

Current ‘Schedule’


So, once again, I need to update this to let (those interested) know what I am planning on blogging about. Considering the courses I’m taking this term (see my ‘about me’ if you are interested in more detail) I am going to be blogging mostly about algebra and differential geometry of curves and surfaces. In particular, I’d like to (this semester) get through ring theory and module theory for ring theory and get to at least the Theorema Egregium for differential geometry (but in a perfect world I’d get to Gauss-Bonnet).

The books I’ll be primarily using are:

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Geometry

  • Carmo, Manfredo Perdigão Do. Differential Geometry of Curves and Surfaces. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.
  • Bloch, Ethan D. A First Course in Geometric Topology and Differential Geometry. Boston: Birkhäuser, 1997. Print.
  • Montiel, Sebastián, A. Ros, and Donald G. Babbitt. Curves and Surfaces. Providence, RI: American Mathematical Society, 2009. Print.
  • O’Neill, Barrett. Elementary Differential Geometry. Amsterdam: Elsevier Academic, 2006. Print.
  • Pressley, Andrew. Elementary Differential Geometry. London: Springer, 2001. Print

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The last book is the course book, whose progression of material is my favorite, but the level of rigor is…eh…not enough. So, I will be mostly following the last book but supplementing it with the other books plus misc. research to phrase it in more theoretical, mature language.

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Algebra

  • Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
  • Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print
  • Reid, Miles. Undergraduate Commutative Algebra. Cambridge: Cambridge UP, 1995. Print.
  • Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.
  • Lang, Serge. Algebra. New York: Springer, 2002. Print.
  • Grillet, Pierre A. Algebra. New York: Wiley, 1999. Print.
  • Hungerford, Thomas W. Algebra. New York: Rinehart and Winston, 1974. Print.
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May 22, 2011 - Posted by | Uncategorized

5 Comments »

  1. Can you give a link to a page explaining how to navigate wordpress blogs efficiently? I find it difficult to look up topics like “Repesentation Theory” in the blog because the way the results are presented they take up several pages.

    (and if you have any ideas on how representation theory could be applied to this idle musing, let me know: http://www.sosmath.com/CBB/viewtopic.php?f=22&t=55561 )

    Comment by Stephen Tashiro | September 17, 2011 | Reply

  2. Stephen, have you tried using the ‘categories’ dropdown bar located among the ‘stuff’ on the right hand side of every page? If not, you can just drop it down and select ‘representation theory’ from where you will get only topics which I have labeled representation theory. If you are looking for something more specific you could use the search bar located at the top of the ‘stuff’ on the left.

    I’m sorry, I’m having a difficult time understanding precisely what these ‘contingency tables’ are in the link you posted.

    Best,
    Alex

    Comment by Alex Youcis | September 18, 2011 | Reply

  3. I tried the categories menu. The problem is that it lists the most recent things first, so it’s time consuming to get to the beginning of a topic. However, I eventually found the first post on representation theory, which was my immediate goal.

    The table is to show whether different subjects (e.g. people) have different responses (e.g. “amused”, “shocked”,”awed”,”indifferent”) to different treatments (e.g. watching different video clips). My calling it a “contingency table” may be an error.

    I don’t have specific data in mind, and I’m only mildly interested in statistics. My thought is that worrying about how to do statistics on the table leads to an interesting mathematical problem.
    In statistics, defining a “statistic” means to define a real valued function of the data. ( So a statistic is not a single number like 48.3 as a layman thinks of a statistic). Assuming there is no known chonology for the order in which the subjects were tested, a desireable property for a statistic defined on such data tables is that it be invariant under pairwise swaps of entire rows and pairwise swaps off entire columns. So it should be invariant under certain subgroups of the group of all possible permutations of the entries in the table.

    For a group acting on a set of variables (e.g. x1,x2,x3) the symmetic functions ( e.g. x1 x2 + x1 x3 + x2 x3) form a basis for the set of all multinomial functions that are invariant under permutations of the variables. In the case of the data table, the entries (e.g. “amused”) aren’t naturally regarded as numerical values. So I don’t see that using the symmetric functions provides insight into what good statistics can be defined. To me the natural statistics would involve patterns in the table that were invariant under row and column swamps. My example of a statistic was to count the number of rectangluar shapes in the table that have the same result at each vertex of the rectangle. I don’t know any mathematics that tells me how to find a simple set of basis functions for functions defined in terms of patterns. And I don’t see how to rephrase the problem of finding such a basis in terms that make it fit the use of symmetric functions.

    Comment by Stephen Tashiro | September 20, 2011 | Reply

    • Stephen, is there perhaps an e-mail address I could reach you at? I don’t know how much I’d be able to help with your question, but I think anything I can do would be better be done through that.

      Comment by Alex Youcis | September 21, 2011 | Reply

  4. My email is tashiro at zianet.com. You needn’t give the question much priority. I’m a retired guy who has time for idle speculation.

    Comment by Stephen Tashiro | September 21, 2011 | Reply


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