Welcome to William (Riley) Casper's webpage.
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Posts of recent research activity are included below. Hopefully these will be moved to a blog form some day.
I've put my quasi-geostrophic paper on the arXiv, in the hopes of getting some feedback from the community. this is usually about the point where I notice a bunch of typos, etc, so we'll see how it goes.
My paper connecting orthogonal polynomials to the linear stability of a shear cosine profile has been updated to contain proofs of the stated theorems, as well as proofs of appropriately modified well-known related theorems taken from the context of the inviscid Navier-Stokes equation. In this paper, Rayleigh's equation is a bit different, due to the presence of rotation, and this has significant impacts on some properties of Rayleigh's original equation. For example, our version is no longer invariant under the choice of an inertial reference frame. Furthermore, certain instability criteria associated with this equation are modified.
In this update of my bispectral algebras paper, I define a matrix version of Sato's Tau function and use it to prove a structural result for the bispectral triples having rank 1 bispectral algebras. The Tau function is originally defined by taking the Fredholm determinant of a certain Fredholm operator on a closed subspace of the Hilbert space of square-integrable functions on the unit circle. For the matrix Tau function, we adopt the same idea, but use a certain quasideterminant of the linear operator. When we do this, multiplicativity of the determinant is no longer true in general. However, by a small miracle, we have the right notion of multiplicativity at the right time, in terms of a twisted matrix Tau function which we also define.
An updated draft of my paper on the structure of noncommutative bispectral algebras of matrix differential operators is now available. It firms up some of the ideas presented in the previous draft. One of the sticking points here is what it should mean for an algebra of matrix differential operators to be ``rank 1", ie. how to extend the definition from the scalar case of commuting differential operators.
This quarter, Sara Billey's WXML students, the Graphletes, are working on developing a database of graphs, akin Sloane's Online Encyclopedia of Integer Sequences (OEIS). Part of the difficulty in this project is extracting graphs from the literature in an accurate and expedient fashion. Luckily for us, graphs in pdf files compiled from latex source code without images tend to be stored in a vector format, which can then be interpreted after converting to an EPS file format with pdftocairo. Using this idea, I've created a python code which attempts to find the graphs in a pdf file and write them out in terms of node and edge lists in JSON format. The code may be found on my github repo.
