LIST OF SOME MATH 116 PROJECT IDEAS ...may evolve during course

* Implement Kress's (Martensen-Kussmaul) spectral quadrature for log
  times analytic kernels, use to get spectral convergence in the
  DLP+SLP exterior scattering problem.

* Implement Alpert quadrature for SLP+DLP in Helmholtz, check convergence
  rates, review uses in literature.

* Implement boundary integral method for other linear PDEs from heat flow or
  fluid dynamics (modified Helmholtz equation, heat equation, Stokes),
  review applications in the literature.

* Implement transmission scattering (scattering from a dielectric
  material with interior wavenumber differing from the exterior). Use
  hypersingular-cancelling layer representations inside and outside
  (as in Rokhlin 1983), with a singular quadr scheme such as Kapur-Rokhlin,
  Alpert, or Kress.

* Use adaptive Gaussian quadrature on the Nystrom interpolant to make
  an interior Laplace BVP solver that evaluates accurately at any
  points up to the boundary. See Helsing's work on this, review some
  of the literature.

* Implement anything from the course in a smooth 3D domain,
  demonstrating convergence rate and discussing scaling of
  computational effort. You will need to do quadrature in 2D for this,
  and possibly handle singularities in order to get high-order 
  convergence.

* Try MPS or MFS for wave scattering or interior BVPs, study
  convergence, and efficiency, relative to that of boundary integral methods.

* Study the convergence rate of the MPS with a single singular corner,
  and relate to complex analytic properties of the solution, based on
  T. Betcke's thesis and publications.

* Corners in boundary integral methods: read about and test some
  quadrature rules adapted for corner singularities (eg, Atkinson,
  Kress books, Kress 1991 review), and compare against piecewise
  (dyadic) Gaussian quadrature in the style of Rokhlin/Greengard.

* Compute the logarithmic capacity of the unit square to machine 
  precision. (Will involve either boundary integral or MPS type 
  methods, careful consideration of quadrature or basis sets).

* Investigate numerically `creeping waves' or other high-frequency wave
  phenomena beyond the geometric optics approximation. (Eg, diffraction).

* Code a 1-level Fast Multipole Method which achieves O(N^(3/2)),
  or better O(N^(4/3)) for the self-interaction of N points in R^2 with 
  the Laplace kernel. For the former a src-to-multipole expansion is 
  needed for points falling in boxes; for the latter scaling you also
  need translation operators and local expansions.

* Study statistics of Laplacian eigenmodes and compare against random
  matrix theory predictions (involves a little reading about quantum chaos).

* Compute Laplacian eigenmodes on a polyhedron by matching value and
  derivatives on the edges, as in the MPS.

* Present Vekua's theory of PDEs in the complex plane (Henrici review).

* Make concave objects with pockets to trap scatt waves, look for
  narrow resonances, match with WKB prediction for lifetimes (using
  gaussian cross-section, geom optics description).

* Inverse problem: use the map from incident to far field pattern to
  invert for the first few Fourier modes of a smooth shape.

* Hyperbolic geometry: understand and implement eigenmodes or scattering
  on the pseudosphere (constant negative curvature).

* Model an acoustic `Helmholtz resonator' (Neumann boundary conditions,
  exterior wave scattering for a nearly-closed cavity shape) relevant
  to music or architecture, study corrections from the predicted frequency.