### Measures of Biodiversity

We speak of increases and (more often) decreases in biodiversity, but if we say that biodiversity decreases by 5%, what does that mean? There are many ways to measure the biodiversity of a population: the total number of species present (species richness), the fraction of the population occupied by the most common species (Berger-Parker dominance), the probability that two randomly chosen members belong to the same species (Simpson index), among many others. Different measures pay more or less attention to rare species, and to how evenly the population is spread among the different species, but some are also better or worse for making meaningful statements of the form “biodiversity has changed by X%”.

This project will begin with readings on the relationships between different biodiversity measures, including recent work developing a theory of biodiversity taking into account similarity between different species. From there, depending on the goals of the participants, we could explore in several different directions, such as drawing connections between biodiversity measures and the “species-area relationship,” working with research on the Arb to see how different biodiversity measures compare for real populations, or exploring applications of biodiversity research beyond the traditional population/species setting.

Terms: Winter/Spring
Prerequisites: Math 321 (Real Analysis I) recommended but not required.

### Spatial Statistics

Spatial statistics is concerned with analyzing data that has a spatial component: Is the high number of cancer deaths in this part of a state statistically significant, or could it be due to chance? Based on radon measurements taken at 100 fixed locations, can we estimate the amount of radon at unsampled locations? In this comps, we will learn some basic approaches for analyzing spatial data. We will consider spatial point processes, geostatistical data and lattice (areal) data, making heavy use of R.

Terms: Fall/Winter
Prerequisites: Math 245 and 275

### Variants on Kuratowski’s Closure-Complement Problem

Terms: Winter/Spring
Prerequisites: None

### Combinatorics of Symmetric Functions

Terms: Winter/Spring
Prerequisites: Previous experience with permutations, partitions, and generating functions. Math 333 (Combinatorial Theory), which is offered in the fall, will be sufficient. However, you could also gain this experience by taking certain courses in Budapest, in an REU, or by doing some supervised independent reading. This project will also involve some occasional computer work, so comfort learning how to use a program like Mathematica or SAGE will be important. However, you do not need to have any previous programming experience.

### Knotting Probability of Random Polygons

Terms: Winter/Spring
Prerequisites: Math 354 is recommended by not required.

### Directed Reading in Homological Algebra or Another Algebraic Subject: You Can Help Pick the Topic!

Terms: Winter/Spring
Prerequisites: Math 342; further experience in abstract algebra and related subjects might be a plus, but it’s fine if you haven’t taken any version of Math 352 (that is, you don’t need to have seen either Galois theory or representation theory).

### Statistical Analysis in Sports

Since the publication of Moneyball, the use of statistical analysis in sports has gone from a rarity to common practice. No longer are oddsmakers in Las Vegas the only parties interested in examining win probabilities, expected points, or the “value” of an athlete. Further, professional sports teams now hire Ph.D. statisticians and mathematicians to direct their analytic efforts. What models are these folks using? What types of insight can they provide?

In Math 245, the primary tools were linear and logistic regression. In this comps, we will explore models that can be used to predict the outcome in team sports, considering both how to predict the outcome before the competition and how to update the win probability throughout the competition. Specifically, we will compare and contrast the performance of statistical models, including logistic regression, random forests, and support vector machines (to name a few). If time permits, we will explore approaches to assess an individual athlete’s performance.

I have seventeen years of play-by-play data for Division I college football to explore, but the exact application will be decided by the interests of the group.

Terms: Winter/Spring
Prerequisites: Math 245 and Math 275

### Koch-Like Curves and Related Fractals

Terms: Fall/Winter
Prerequisites: None