Hi, I'm a current senior at Cornell University studying mathematics and computer science. My interests lie primarily in the area of combinatorial optimization, especially as it pertains to scheduling problems. However, I have also been working with Professor Bob Strichartz at Cornell to study the eigenvalue spectra of Laplacian Operator on 2-spaces of constant curvature. If you are here for that, you can click here or on the heading below.
In Progress (36 pages complete)
We have been looking into the eigenvalue spectra of the Laplacian in R^2, Hyperbolic 2-space (curvature = -1) and Spherical Space (curvature = 1). We are attempting to estimate the rate of decay to zero of the average error of Weyl's Asymptotic Law and the subsequent improvements of it.In 2-spaces, Weyl's Law gives an asymptotic estimate N'(t) = a*t + b*sqrt(t) + c of the eigenvalue counting function N(t) of a spectrum. N(t) is the number of eigenvalues of the spectrum less than t (some other sources defined it to be less than t^2). a and b are constant coefficients that are easily calculated to be the area and perimeter, respectively, of the region r on which the spectrum is defined, divided by a factor of 4pi. The constant c is more difficult to calculate, and depends on the curvature of the space as well as the corners and curvature of the region r. Click here to visit the main page for this project.
The Multiple Traveling Salesmen problem is a special version of the Vehicle Routing Problem in which all weight, time, and fuel constraints are set to positive infinity for all vehicles. I am specifically addressing the problem where the number of supply depots is zero, i.e. there are no common cities between the sales routes within a routing plan. I am addressing this problem through the use of Genetic Algorithms using Co-operative co-evolution to evolve the routes within a plan. This study began as a term project for CS 5724: Evolutionary Computation at Cornell University addressing only the efficacy of internal crossover genetic algorithms, i.e. crossover between routes within a plan rather than between plans. This limits some of the techniques that would normally be applicable to genetic algorithms but has the advantage of being easy to implement and intuitively easy to understand. While I am currently examining the efficacy of algorithms that implement external crossover between plans, because this began as a term project a preliminary paper addressing the use of internal crossover is completed and can be viewed here. A more comprehensive paper will be written after implementing external crossover genetic algorithms.