Projects in Stochastic Processes and Probability
Description can be found here.

Statistical Learning
Suppose for a recently released prisoner, you know his age age, marital status, education level, the number of prior convictions, whether or not he is a felon, and whether or not he has used alcohol or drugs. Can you predict whether or not he will end up back in prison? Suppose for  given house, you have information on the number of bedrooms and bathrooms, square footage, neighborhood crime rate and distance to schools. Can you predict its current market value?

In Math 245, the primary tool was linear or logistic regression. In this comps, we will consider other approaches (including modifications to the linear model) for predicting outcomes given a set of inputs. We will consider both supervised learning—we are given the inputs and an output for a set of data and we want to find the best model for prediction, and unsupervised learning—we are given only the inputs, so we must try to determine relationships and structure. We will explore topics such as linear discriminant analysis, tree-based methods (including bagging, random forests and boosting), support vector machines and principal components. We’ll also learn tools such as cross-validation and bootstrapping to measure how well our models predict so we can compare performance between different approaches.

We will use R extensively and work with real data sets.

Empirical Bayes
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Counting Pattern-Avoiding Linear Extensions of Posets
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The Mathematics of Origami
We all delight in the beauty of the folded creations known as origami:  the intricacy and detail and wonder at how the objects were folded, but there is a mathematical beauty as well.  Interest in the mathematics of paper folding dates back to the 19th century, but much more has been published in the past twenty years.  In this directed reading, students will read, in depth, some of the mathematical articles that have been written about this ancient art, and present what they have learned in bi-weekly meetings.  Many of these articles use ideas from geometry, algebra, and computation.  This comps will culminate not only in a paper and presentation, but also an art exhibit demonstrating some of the exciting mathematical ideas hidden in the folds.

Cantor Sets
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Directed Reading in Algebraic Geometry, Homological Algebra, or Another Algebraic Subject:  You Can Help Pick the Topic!
This is an opportunity for two people to delve deeply into material beyond the standard undergraduate curriculum, and to get substantial (weekly) experience in presenting that material at the blackboard.  The only prerequisite is Math 342; although further experience in abstract algebra and related subjects might be helpful, I don’t expect that anything will come up that has been taught in Math 352 recently (in particular, we’ll have occasion to use neither Galois theory nor representation theory).

In algebraic geometry, techniques from abstract algebra are used to describe and investigate “varieties”: sets that are defined by polynomial equations (in several variables).  The interplay between algebraic and geometric points of view makes this a very rich topic, that has expanded enormously in the last hundred years or so; while it is traditionally considered “pure” mathematics, the more computational parts have applications in such areas as robotics and computer-aided design. The first major goal of the reading would almost certainly be Hilbert’s “Nullstellensatz” (theorem of the zeros), which establishes a correspondence between varieties and certain ideals in polynomial rings. After that there is a variety (pun intended) of possibilities, depending on the background, interests, and tolerance for abstraction of the participants.

Homological algebra grew out of investigations in algebraic topology and has important applications there, but it has become a separate, highly abstract area of mathematics.  The mathematical “language” of category theory, which it uses, is pervasive in much of modern mathematics.  (Some results which require only this general framework and thus pop up in all sorts of different contexts are affectionately known as “(general) abstract nonsense” – you can find a Wikipedia article with this title!)  The reading would probably cover substantial sections of MacLane’s classic book Homology.

An Age Old Question with an Algebraic Explanation:  “What Voting Procedure Best Reflects the Overall Choice of the Voters?”
Description can be found here.

Elastic Curves
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Industrial Strength Mathematics
In this project we will work on a real life problem in applied mathematics – specifically computer vision – in partnership with a Minnesota company.  Using geometric data from 3d scans and optical sensors, we will design a strategy for detecting manufacturing flaws in circuit boards.  This promises to be a fascinating blend of differential geometry, data analysis and numerical computation. These mathematical tools can be learned as needed, but proficiency with linear algebra, differential equations, and a programming language is recommended.