Hello, I thought that working with Group Theory for my GSoC 2013 project would be ideal.
I have familiarized myself with the GAP library and their approach to group theory problems, and have a fairly extensive knowledge of group theory (at the undergraduate level, at any rate). The standard approach focuses heavily on representations in the form of permutations, but does not touch on matrix representations: either over the integers or Z/(2^n Z). This representation is more natural for defining, for example, Lie groups and reduces group multiplication to matrix multiplication. Group multiplication as matrix multiplication has the potential to be more efficient in terms of space and time than a permutation if certain precautions are taken. The other key advantage to a matrix representation is in regard to Lie Groups. The exponential map which defines the relationship between a Lie Group and its associated Lie Algebra can be computed using the power series expansion of a matrix in the Lie Group. Another area to focus on is the notion of algorithmic efficiency for tests for subgroups, normal supgroups, and the use of Sylow Theory. As defined in the GAP library, every group object is annotated with flags describing the group's properties, which are lazily evaluated. Many of these properties can be filled in purely by the group's construction. Using these properties, and weighing the relative algorithmic costs of testing them, could create an algorithm which adapts to take advantage of group structure. For instance, when testing whether a given subgroup N is normal to G, we can use either a straight test to insure gN = Ng for all g in G, or we can use Sylow's theorems. Sylow theory provides a far quicker test, assuming we already know N to be a Sylow p-subgroup of G. So, to choose which method to use, we must consider the amount of time it would take to confirm N is a Sylow p-subgroup. Finally, if this works out, I would like to implement support for character theory and the computation of dual groups, at least in the case of finite order. Any thoughts or criticisms would be much appreciated. -Tyler -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy?hl=en-US. For more options, visit https://groups.google.com/groups/opt_out.
