Mentor: Eric Sharpe (Spring 2007) & Don Tucker (Summer 2006, Fall 2006)
Majors: Mathematics & Physics
Minor: Computer Science
Spring 2007 project description:
The universe appears to have 4 macro-dimensions, one of time and three spatial dimensions. It has been proposed that the universe has additional dimensions which are extremely small and folded into elaborate shapes rendering them difficult to detect. These extra dimensions help theoretical physicists in attempts to unify the four fundamental forces. Perhaps one of the earliest forays into the possibility of a higher number of physical dimensions was made by Theodor Kaluza. He extended General Relativity to a 5 dimensional space time. This extension can be separated out into a series of equations which correspond to the equations for general relativity and the set of Maxwell equations and an additional scalar field. This amazing ability to generate the Maxwell equations from an extrapolation of General Relativity to a higher dimensional space is a perfect example of the reason that theorists postulate the existence of such extra dimensions. While the general consensus is that these dimensions must be curled into extremely small shapes so that the universe appears to have a smaller number of dimensions than it actually does. Just as a wire appears one dimensional because its diameter is insignificant compared to its length. There are other conditions under which a dimension might be masked however. For instance the inverse squares law of force is theorized to be a consequence of our three dimensional space. More generally the power law of a d dimensional space would have a r^(-d+1) dependence because of the spreading of force carriers onto a d-1 dimensional surface during propagation. If we consider the case of a charged wire the coulomb force appears to change relative to it with a 1/r dependence which would seem to suggest a 2 dimensional space. In this case the reduction of the power law is not due to the dimensionality of the space but the sameness of the wire along one dimension. I would like to investigate whether it might be possible for such a case to be extended to allow for hidden dimensions which are not microscopic but true spatial macro-dimensions. In such a world our universe would be sandwiched between a seemingly infinite number of universes with almost exactly the same conditions as ours effectively reducing the apparent number of dimensions of our own to the traditional three spatial dimensions.
Fall 2006 project description:
In the summer REU I explored the possibility of creating a program to search for odd perfect numbers efficiently. During that exploration I produced an inequality that allowed me to begin sifting through numbers of different forms looking for ones that allowed for odd perfect numbers. During the fall I would like to continue along this line of search. However, during the summer I primarily looked at the effect of the upper bound of my inequality on the numbers that could be perfect. However for sufficiently large numbers the lower bound of the inequality becomes of primary importance and the upper bound rapidly becomes unable to sift possible perfects from other numbers. I would like to use the fall semester to further analyze the properties of this inequality and its possible uses. Also, if it proves feasible, I would like to utilize what I have learned about this inequality to sufficiently limit the search space of the program I proposed for my summer REU to allow for a more effective search.
Fall 2006 final report
Previous VIGRE project