I. Interpreting MFM signals
Simulation methods (like the NIST code OOMMF) for finding the magnetization in a magnetic sample do not tell you what the resulting configuration will look like when it is scanned by Magnetic Force Microscopy (MFM). That requires an additional step, computation of the magnetic field outside the sample due to the magnetization. Students in Kathy Aidala's group have used the method described in the foregoing link to interpret MFM data in light of OOMMF simulations. Below is MFM data for two magnetic rings with the same characteristic magnetic defect structure. Compare with the simulation below it (for a single ring).
II. Classical Point Contact
Kathy Aidala's lab is investigating the switching of bistable magnetization configurations using currents applied at (essentially) point contacts. It is useful to know the resulting current distribution and magnetic field, which can be calculated by standard methods of mathematical physics.
If you use IPython Notebook, you can see an implementation of the method. Below is a result showing the (dimensionless) magnetic field intensity created by the current of a small contact of radius a as a function of radial distance from the center of a disk of radius R at height z above the grounded base. It is noteworthy that the field falls off rapidly with depth into the material.
III. Ciliary Dynamics
Cilia are whiplike organelles found on the surface of cells in organisms as distant from each other as ferns and humans. Their dynamics is still mysterious. A modelling project to describe videographic data of ciliary beats was the subject of an REU in the summer of 2010. Below is a typical ciliary beat that we analyzed, the different shapes labeled by when they occurred within the periodic cycle of the beat.
IV. Multilamellar optics
Alexi Arango's lab fabricates multilamellar solar cells. For the purpose of designing optimal cells, it is useful to model the optics of reflection and transmission at the many interfaces in such devices, as well as the absorption in those layers that convert light to electric current. This is a future project.
The MHC art museum owns two Arabic astrolabes. Fikriye Idil Kaya and I are doing precise measurements and comparing these instruments with the theory that presumably underlies their construction. One of the astrolabes is shown below.
VI. Thermodynamics and Statistics
There are a number of possibilities in this area. Here is a starting point based on Markov processes, and here is a starting point for studying this topic from the quantum point of view.
VII. Low Reynolds Number Flows
Low Reynolds number flows occur frequently in biological situations, and they often present computational challenges at an interesting but manageable level of difficulty. Here is a rather formal example, done with Danti Chen and Becky Ding, a two dimensional flow through a periodic array of obstacles.