Noah Graham
The first three topics listed here are directly related to my research program, while the others involve subjects in which I have interest and/or experience, but are likely to require the student to take more initiative in formulating and guiding the project; to quote physicist Victor Weisskopf, "I will contribute the necessary don't-know-how." I am also happy to discuss other potential topics in theoretical physics based on a student's particular interests. These topics could all be pursued in either Fall or Spring Term, and, depending on progress, interest, and logistics, all could potentially be continued as a PHYS 0705.
1. Computational Scattering Theory
We understand many physical systems by studying how they scatter light or other waves. This project will develop computational algorithms for wave scattering, which can in turn enable calculations of the effects of quantum-mechanical fluctuations in a variety of physical systems; possible applications, depending on student interest, could include Casimir forces in nanotechnology, magnetic flux vortices in superconductors, cosmic strings, or quantum effects in curved spacetime. Possible implementation platforms range from high-level Mathematica code to low-level C++ code, with both cases potentially taking advantage of large-scale parallel computation.
2. Solitons and Oscillons
With the exception of electromagnetism, the fundamental forces of Nature are all nonlinear - they do not obey the superposition principle - and even electromagnetism is nonlinear once quantum mechanics is included. One striking feature of many systems of nonlinear waves is the appearance of self-organized, localized lumps, held together by their own self-interactions. These lumps can be static (solitons) or oscillatory (oscillons). Such objects cannot occur in a linear theory, where disturbances simply disperse, like ripples in a pond or the beam of a flashlight. Solitons and oscillons have the potential to play important roles in physical systems ranging from superconductors to the early universe, but relatively little is known about them except in a few special cases. The power of modern parallel computation, however, offers significant opportunities to change this situation. This project will investigate the formation and stability of solitons or oscillons in a particular nonlinear problem chosen commensurate with background and interest.
3. Special Functions in Mathematica
Motivated by applications from my research, my students and I have created packages for computing Mathieu and spheroidal functions in Mathematica, which are generalizations to problems involving elliptical and spheroidal geometries of the trigonometric, Bessel, and Legendre functions that we study for circular and spherical geometries in PHYS 0212. These packages are already publicly available and used by other researchers around the world, but there are a number of ways in which they could be improved. This project will create new versions of one or both packages with improved capabilities and performance.
4. Signal Processing and Pattern Matching
The same techniques of Fourier analysis that we use to study waves in quantum mechanics and electromagnetism can also be used to analyze a wide range of other wave-like signals, including speech, music, and images. This project will identify and implement a particular application of these techniques, such as identification of words or language in speech, chords or tones in music, or visual features in images.
5. Create Your Own STEM Innovation
Fostering innovation in STEM (science, technology, engineering and mathematics) disciplines has been a high priority in academia and industry around the world. Toward this goal, this project will create a technological solution to a concrete problem, drawing (of course) on topics in physics, likely in combination with other fields of science and mathematics. Although in most cases a single term will not be enough time to create a fully polished product, the project should be designed to produce a concrete "deliverable" - something that answers the question "Does it work?" - which could take the form of a prototype, a proof-of-concept, an extension of an existing technology, or a key component of a bigger system. Students taking on this project will need to take significant initiative before the start of the term to formulate the project; in particular, the proposal should identify the deliverable goal (recognizing that this goal may evolve over the course of the project), and should also lay out the experimental or computational "platform" on which exploratory work toward the project's goals can be carried out, which could take the form of software, electronics, or laboratory work.