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.

Using Singapore Math to teach Algebra in the Northfield Middle School

One of the teachers at NMS remediates students who are weak on math using a technique used in Singapore. The teacher likes a website called Thinking Blocks which uses the ideas of Singapore math to teach a variety of topics, but algebra is not one of them. He would like a curriculum designed to teach his students Algebra using these techniques.

This project will start by learning something about Singapore Math. The comps group will make visits to the NMS to learn about the students and what they struggle to understand. We will then create some sample lesson plans and try them out in the classroom. We may request permission to do pre- and post-tests on some students to see what style problems have the most success in helping students learn Algebra. Finally, we will devise an entire curriculum unit on Algebra, present it to the teacher, and teach the students how it works. Depending on the members of the comps group and their strengths, we may write the problems as worksheets with manipulatives or as software that the students can use on their iPads in class (perhaps with progress reports for the teacher).

Explorations of factorizations of some polynomial iterates

Description can be found here.

Directed Reading in Elliptic Functions and Modular Forms

Description can be found here.

Directed Reading in Analytic Number Theory

Description can be found here.

Mathematical and Computational Modeling of Neural Systems

Description can be found here.

Time Series Modeling for Climate Data

Data that are observed over time often have a special kind of dependence structure: recent observations are more informative about the present than are past observations. Statistical models that assume independence do not suffice in this setting, but the unique structure of temporally varying data has motivated the development of a large class of techniques that broadly fall under the umbrella of time series analysis.

This comps will be an introduction to applied time series analysis. We will learn about the relationship between “time-domain” and “frequency-domain” approaches to understanding time series. We’ll apply the techniques we learn to data and gain insights into the characteristics of variability in the underlying generating processes of the system we study.

The application will be to the Earth’s climate, with a focus on the statistical properties of the output of computer models that simulate the Earth’s atmosphere and oceans. Broad questions that might be considered include: how do ocean processes relate to variability of the Earth’s surface temperatures? And what role does natural variability play in our uncertainty about the Earth’s sensitivity to greenhouse gases?

S-GAP SHIFTS

Description can be found here.

The ABCDs and EFGs of classifying Lie algebras over C

Description can be found here.

Statistical Analysis of Networks

Description can be found here.

Data Analysis using Topology

Welcome to the age of big data.  Data analysis has a new enemy: too much data.  Collecting vast amounts of data is easy, but how do you make sense of all those variables and all those data points? Can we visualize them or quantify them in some way, beyond the known techniques of statistics?

Enter the mathematical field of topology, which studies the shape of spaces. Imagine that we can somehow represent the data as a “cloud” (think of plotting points in multivariable calculus).  What does that cloud look like?  Does it look like a solid ball or are there gaping holes?  One brand new area of research tries to answer exactly that question using the very pure technique called homology.

The goal of this comps is manifold.  First, we’ll learn about the pure math part, beautiful in its own right. Second, we’ll see how it is being applied to data analysis and hopefully get our hands dirty with computation on a real data set.  And finally, we’ll try to interpret the results in terms of the real world.