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- Summer 2013 Projects
**Mathematical Modeling in Ecology: What Killed the Mammoth?**

Dr. Alex CapaldiDuring the paleolithic, mammoths, as well as other mega mammals, went extinct. The exact reasons for this have been debated for some time, and current hypotheses point to causes such as disease, climate change, over hunting by humans, or some combination thereof. However, recent developments in mathematical ecology may be able to elucidate the matter. A recent study has given strong evidence in support of the hypothesis that the Neanderthals' extinction was due to strong competition from modern humans. The goal of this project is to follow in the footsteps of the Neanderthal study and consider the similar question as to whether the mammoth's extinction was anthropogenic.

Prerequisites: At least one of the following: differential equations, statistics, programming experience.

**Estimating the Volatility in the Black-Scholes Formula by Multiple Approaches**

Dr. Hugh GongBlack-Scholes formula has been a huge success since it's introduced. However, the formula has a component of the volatility, which is not deterministic and available immediately for application of this formula. We can estimate this volatility via multiple models or methods, such as parametric approaches with different probabilistic models, nonparametric approaches and time series models approaches, such as smoothing techniques. We will derive these estimates and incorporate into the Black-Scholes formula.

Prerequisites: an introduction of statistics, understanding the basic probability distributions (uniform, normal, Chi-square and etc.), the concepts of mean, variance, and correlation.

**Graph Labelings**

Dr. Zsuzsanna SzaniszloIn this project we will look at edge-vertex graphs and investigate a certain assignment of numbers to the edges and vertices. These so-called labeling problems were introduced in the 1960s as possible tools for solving graph decomposition problems. Since then different labelings were studied for their own sake and for other applications as well. In the summer project we will investigate when we can distribute labels (numbers) equally in a graph. Answering this question for trees would settle the famous graceful tree conjecture.

Prerequisite: a proof based course