Research

Timeline

Masters Thesis

I am currently working with Dr. Stephen Wright to find efficient ways to solve the nonlinear resource allocation problem (\mathcal{P}):

\begin{array}{cll}<br /> \min &{\displaystyle \sum_{i=1}^n f_i(x_{i})} &\\[0.5em]<br /> {\rm s.t.} &{\displaystyle \sum_{i=1}^n g_i(x_i) \leq c}, & c \in \mathbb{R} \\[0.5em]<br /> & \ell_{i} \leq x_{i} \leq u_{i}, & i = 1,2 ,\ldots, n<br /> \end{array}

We will be applying both interior point methods and pegging algorithms while comparing past work in the field.

Senior Independent Study (Read More – Paper)

At the College of Wooster I completed an Independent Study entitled “The SIAM 100.0000000-Digit Challenge: A Study In High Accuracy Numerical Computing Using Interval Analysis and Mathematica.” I considered three of the ten problems.

  1. A photon moving at speed 1 in the x-y plane starts at time t = 0 at (x, y) = (1/2, 1/10) heading due east. Around every integer lattice point (i, j) in the plane, a circular mirror of radius 1/3 has been rected. How far from (0, 0) is the photon at t = 10?
  2. Find the global minimum of the function
    \begin{split}f(x,y) = e^{\sin(50x)} &+\sin(60e^y) + \sin(70 \sin x) \\ &+\sin\bigl(\sin(80y)\bigr) - \sin(\bigl(10(x+y)\bigr) + \frac{x^2+y^2}{4}\end{split}
  3. Let A be the 20, 000 x 20, 000 matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, . . . , 224737 along the main diagonal and the number 1 in all the positions a_{ij} with |i − j| = 1, 2, 4, 8, . . . , 16384. What is the (1, 1) entry of A^{-1}?

Research Experience for Undergraduates (Link – Paper)

During the summer in 2006 I participated in a REU at the University of Akron where we wrote a paper entitled, “Self-Similar Tilings of Nilpotent Lie Groups.” We described a set of conditions for self-similar tilings (which typically had fractal boundaries) to exist over nilpotent Lie groups. The major result we proved was that given an expansive automorphism that preserves some lattice, we can construct a self-similar tile. Once we have this result for all nilpotent Lie groups, we examine the Heisenberg group, a specific example. In the Heisenberg group, we can describe precisely the form of all automorphisms, and in particular we can describe all automorphisms with the property of being an expansive map. We also classify all lattices in the three-dimensional Heisenberg group, which allows us to construct specific examples of self-similar tilings on this group, which can be represented as plots in \mathbb{R}^3.