Lafayette College’s Summer 2012 REU will run from Sunday, June 3, 2012 until Saturday, July 28, 2012. Participants are expected to be continuing their undergraduate education in the Fall of 2011, and NSF regulations require that only U.S. citizens and permanent residents are eligible to participate. Each participant will receive free housing and a $3,200 stipend.
There will be three projects, listed below. In order to apply to the program
The deadline for applications is March 2, 2012. For other questions about the REU, please contact Professor Qin Lu at luq@lafayette.edu
Project 1
General Area: Geometry
Topic: Folding and unfolding polyhedra
Advisor: Kevin Hartshorn
This project will focus on questions about the (un)foldability of polyhedra. When can polyhedra be unfolded into a polygonal net? Given a polygon, what polyhedra can be folded from that polygon? For example, it has been shown that the Latin cross – a common unfolding of the cube – can be recreased and glued along its edges to form 27 different polytopes. There is no general technique for determining what polyhedra you can get from folding a given polygon. Given a non-convex polyhedron, it is unknown if it can be cut along a finite graph so that it folds flat.
Because there are few general theories on this subject, much of our work will rely on experimentation with polyhedra, both physical and virtual. Much of our work will be developing answers to the unfoldability question for specific classes of polyhedra, or foldability questions for specific polygons.
Our work will likely touch on ideas in linear algebra and differential equations. A basic background in both subjects is recommended.
Project 2
General Area: Financial Mathematics
Topic: Risk-adjusted Returns in Stochastic Volatility Models
Advisor: Qin Lu
Risk-adjusted return plays an important role at investment firms in assessing the performance of investments and in selecting portfolio managers. As the old saying goes, “No risk, no return.” That is, we cannot pursue higher returns on investments without assuming higher risks. In this project, we will use Monte Carlo simulation to analyze the statistical properties of several commonly used risk adjusted returns/ratios, namely, the Sharpe ratio, the Treynor ratio and the Return on VaR. These ratios have been well-researched in the literature.
However, in industry and most literature, conclusions are based on the comparative size of ratios for two different portfolios/managers (i.e., a bigger ratio implies “better” performance). Much less is known about the statistical properties (uncertainty) of the ratio itself. We are interested in answering the following questions: What are the distributions of risk adjusted returns/ratios for different portfolios involving financial derivatives? We assume that the returns of the underlying assets follow a stochastic volatility model with or without jumps.
Project 3
General Area: Number Theory
Topic: Perfect Parallelepipeds
Advisor: Cliff Reiter
A famous open question in number theory is whether there is a perfect cuboid: namely, a rectangular box such that its 3 edges, 3 face diagonals and 1 body diagonal are all integers. It is unknown whether a perfect cuboid exists but there infinitely many solutions if any one of the 7 requirements is removed. A less famous problem was whether there exists a perfect parallelepiped. Perfect parallelepipeds have 3 integer length edges, 6 integer length face diagonals, and 4 integer length body diagonals. Recently it has been shown that perfect parallelepipeds exist. This group project will investigate the distribution of perfect parallelepipeds both empirically and analytically. Infinite families of perfect parallelepipeds will be sought. Participants should have some experience with upper level mathematics. A course in Number Theory or experience with computer investigations is advantageous.