## Summer 2014 REU

Lafayette College’s Summer 2014 REU will tentatively run from Sunday, June 1, 2014 until Saturday, July 26, 2014. Participants are expected to be continuing their undergraduate education in the Fall of 2014, 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 you will need to

- upload a personal statement about your mathematical interests and goals.
- upload a copy of a current transcript (unofficial transcripts are acceptable).
- arrange for two letters of recommendation to be submitted to the website.

The deadline to apply for this REU is Friday, February 28, 2014. You can apply for the program HERE.

For other questions about the REU, please send an email to mathreu@lafayette.edu

**Project 1**

General Area: statistics, Bayesian econometrics

Topic: The Natural Rate of Unemployment

Advisor: Jeffrey Liebner (this is joint work with Prof. Julie Smith and Prof. Ed Gamber in the Department of Economics)

The natural rate of unemployment is the rate that prevails when the economy is in the steady state. Timely forecasts of the natural rate of unemployment could potentially aid policymakers in anticipating inflationary pressures. There are numerous methods for estimating the natural rate of unemployment, and although these are moderately successful at explaining the past, they are less successful at forecasting the future. The challenge still remains to develop a statistical model of the natural rate of unemployment that provides timely forecasts that are useful for policymakers.

Our group will develop various statistical models of the natural rate of unemployment and conduct a variety of forecast evaluations of those models. Since there are many facets to this issue, the group will determine the direction and scope of the project as it develops.

Applicants should have completed a sequence of calculus-based probability and statistics courses. Knowledge of statistical software (preferably R) as well as Bayesian methods will be helpful. In addition, familiarity with economic theory will be viewed positively.

**Project 2**

General Area: Financial Mathematics

Topic: Option Pricing

Advisor: Qin Lu

Historical stock market prices contain information about the past. In contrast, option market prices contain information that reflect people’s expectation about future risk. This is harder to quantify, and there is significant literature addressing an analysis of option market prices.

This project will investigate different methods of extracting implied risk neutral density from the market option prices. Among the existing methods, none is clearly superior to the others. This project will apply existing methods to recent option market data, including the 2008 market crash data. We will try to understand how expectation and risk preferences are incorporated into prices in the U.S. financial derivative market. This will be an empirical study.

Applicants are expected to have completed a sequence of calculus-based probability and statistics courses, and should have some background in programming. There is no requirement for specific knowledge in economics or finance, but applicants need to be interested in the questions we will address.

**Project 3**

General Area: Algebra and Number Theory

Topic: Integer Problems

Advisor: Derek Smith

This group will focus on one or both of the following areas, related by the word “integer.”

(1) There are several important “integer distance” problems like the following: Do there exist 6 points in the plane, no 3 of them on a line and no 4 of them on a circle, so that the distance between every pair of points is an integer? If so, do there exist 7 points like this? 8? More? At least one of these problems is unsolved.

(2) The 1-dimensional ring Z = {…, -2, -1, 0, 1, 2, …} of ordinary integers is very well-known, but there are other less-familiar rings of integers that exist in higher-dimensional spaces. For example, the Gaussian integers form a subring of the 2-dimensional complex numbers, and there are related rings of integers in the 4-dimensional quaternions and 8-dimensional octonions. Some important elementary questions about quaternion and octonion integers remain unanswered.

Prospective applicants: The goal for the summer is to come up to speed on current knowledge in one or both of these areas and try to make contributions to current research. You should have already completed at least one theoretical course in algebra, linear algebra, or number theory and really enjoyed the material. (Don’t worry if you’ve never heard of the quaternions or octonions before). An interest in making mathematical investigations with computer software (e.g. Mathematica, Matlab, C) is a plus.