In addition to the many off-campus possibilities for summer research, students occasionally work on projects with our faculty funded by grants. Here are some recent projects:
Mathematics has much to say about how to allocate resources and analyze performance in humanitaritan crisis situations, using techniques from the growing field of humanitarian logistics, which seeks to improve the effectiveness of aid operations. Mathematics professor Mike Veatch has worked with students on several related projects. One project helped a new aid organization called GiveDirect to decide how dispersed its beneficiaries should be in Kenya. A second project analyzed data on the response to the Haiti earthquake. Working with the MIT Humanitarian Response Lab, a Pike senior created a case study on food supplements for graduate students.
Mathematics of Voting and Statistics
The mathematics of voting and choice yields insight into why elections often don't seem to represent the "will of the people" by analyzing paradoxes and different procedures (such as plurality, primaries, voting for all you approve of). This has unexpected connections to many areas, such as statistics; in essence, elections and statistics both aggregate lots of information and try to give you just one answer. In recent summer and school year research with Dr. Crisman, a recent graduate proved several theorems about the structure of non-parametric data sets; this became both a departmental honors paper and a published paper in the field.
Designing Systems with Operation Flexibility
From workers in a call center to highly automated machines in a factory, the efficient use of resources often requires that they be flexible enough to work on more than one kind of call or job. Cross-trained workers can help out where needed. Two Gordon student math majors spent their summer with Dr. Veatch studying cross-training patterns and operating rules for which resource works on what. Funded by a National Science Foundation grant, they built mathematical models using probability, learned about optimization and approximation techniques, and performed large numerical studies using state-of-the-art optimization software and a powerful computer.
Isoperimetric Sequences of Integers
There is a famous problem asking what the smallest possible perimeter is for a figure of a given volume; it turns out that this is the circle (though this isn't trivial to prove). For a set of nonnegative integers, one can define analogous concepts of "perimeter" and "volume", and prove various bounds and formulas connecting them. Gordon students have worked on extending the initial results in various ways.
Numerical Methods for Controlling Queues
Waiting in one line is bad enough. In systems where customers, jobs, or messages circulate through several queues and the machines (or people) serving the queues split their time between more than one queue, there is literally no limit to how long they may wait, even if the system has enough capacity, if the servers make bad choices about what to work on next. Two Gordon students worked with Dr. Veatch on a National Science Foundation summer research project on approximations to this very difficult problem. Using 3-D graphics to guide the approximation and algebra to collapse infinite sets of inequalities into manageable sets, they demonstrated that accurate approximations could be found for two test problems. One student continued his work during an independent study, presenting at a regional conference and writing a departmental honors paper.
If a species of fish was separated by the formation of the Isthmus of Panama 4 to 5 million years ago, the genetic variation between the Pacific and Caribbean populations would be greater than the natural variation within an interbreeding population. As an independent study with Dr. Veatch and a marine biologist, a Gordon student adapted an algorithm that computed genetic distance and analyzed DNA data that the biologist had collected and sequenced. Several statistical methods were tested against published rates of mutation and the age of the isthmus.