Interests

Applied Probability Theory, Applied Representation Theory.

I am particularly interested in the fields of mathematics that are on the boundary of different theories. I love to see the Mathematics(in particular, discrete Math) playing the role of bridge connecting various people from diverse fields. I am currently interested in;

  • Application of Representation theory to Probability. Perci Diaconis's application of representation theory in probability theory. He, togather with Shahshahani, showed that the minimal number of shuffles required to make every configuration of n cards equiprobable is right around 1/2 n log n. The most exciting fact is that there is a cut-off time at which the total variation between the initial distribution and the uniform distribution collapses. I wonder what happens if I apply this methodology to the Random Walk on Braid Group, as the state converges to pseudo-anusov state. Braid Group is an interesting object that has application in fields like fluid dynamics...which makes it even more exciting!

  • Fast Fourier Transform on Groups. Please see below.

  • Chemical Dynamics Network.

  • I am also very interested in any other application of the probability and algebra to Biology. Biology is beautiful.


    • Publication

  • Irreducible graphs(Masanori Koyama, Michael Orrison, David Neel) Journal of Combinatorial Mathematics and Combinatorial Computing.

    • About FFT

  • Fast Fourier Transform with Tensor basis (FFT)
    with Mike Hansen, Michael Orrison , David Hemmer .

    Discrete Fourier Transform on finite group is a change of basis realized by the Wedderburn's isomorphism from the group ring CG to the block diagonal matrix ring. When the associated group is the cyclic group Cn, this becomes the familier DFT that is implemented in the Matlab. Fast Fouier Transform(FFT) is an efficient implementation of the DFT.

    Today, a very fast and easy implmentation of the DFT on commutative group ring is avaible. However, this has not been the case for non-commutative ring. It is a shame to leave this problem untouched, as the FFT for noncommutative group ring has applications in numerous interesting fields, such as in phylogenetics and in voting theory.Together with Micheal Orrison and Mike Hansen in the Applied representation theory group at Harvey Mudd College, I study the FFT on the Symmetric Group Sn. This FFT has applications in interesting fields like Voting theory and Genetics.

    The study of the FFT originally began as the study of "operators." The culmination of the FFT research in this direction is the FFT developped by David Maslen, which has a very fast asymptotic runtime O(n^2 n!). This algorithm, however, is extremely complicated, and requires very convoluted indexing scheme to implement.

    We suggest a change in paradigm here. By shifting our focus to the construction of the intermediate basis used in the algorithm, we can further exploit the hidden symmetries in the algebraic structures. In fact, using these symmetries, we can easily construct the sequence of orthogonal intermediate basis that yields a very efficient FFT (Simulation indicates that it is the fastest available today). Furthermore, our method might be applicable to the group algebra of general Weyl group. Following link is the preliminary work for this research done in my thesis.

    Fast Fourier Transform for Symmetric Group.