Statistical Models of Survival
A classic problem in biostatistics is the analysis of time-to-event data. For example, when a new medical therapy is developed, we often want to know how much it extends patients’ survival. Clinical trials provide crucial information for comparing new therapies to the current standard, but for practical reasons most trials end long before all of the enrolled patients have died. The result is a set of incomplete information: we know the survival time for some, but not all, of the study population. How can we use this information to determine which therapy is superior?
This comps will be an introduction to the statistical methods used to analyze time-to-event data. These techniques allow us to identify and measure factors which impact survival time, and to develop models to predict survival. We’ll compare and contrast methods including Kaplan-Meier curves, accelerated failure time models, and Cox proportional hazards regression. If time permits, we will also explore methods to account for other causes of death.
The application can be determined by group interests; I have data from several clinical trials which could be analyzed but these methods have broad utility. In any case, we’ll use R extensively.
Advisor: Tom Madsen (Priority given to Stats majors)
Prerequisites: Math 245 and 275
Many problems in statistics such as evaluating complex integrals (for Bayesian inference) or estimating parameters of a sampling distribution can be tackled computationally by running simulations. In this comps project, we will study statistical techniques that can be handled computationally, including methods for generating random numbers, Monte Carlo integration and variance reduction including importance sampling, advanced bootstrap techniques, MCMC methods and density estimation. We will use R extensively, but also plan on learning some of the basic mathematics behind these powerful ideas.
Advisor: Laura Chihara (Priority given to Stats majors)
Prerequisite: Math 275
Meeting times: Winter, Mondays 3:10-4:10 pm & Thursdays 10:10-11:10 am
Advisor: Katie St. Clair (Priority given to Stats majors)
Prerequisites: Math 245 and 275
Modeling and optimizing production capacity at 3M
To maximize efficiency in industrial manufacturing, mathematicians, engineers and scientists need to develop strategies to plan for fluctuations in customer demand. If demand is too low, products sit in storage and machines and people are underutilized. If demand is too high, customers can’t get what they need and machines and people are overworked. This general area of study is known as capacity planning.
For this project, we will collaborate on capacity planning problems with the Minnesota based company 3M, a developer and manufacturer of a huge array of products (adhesives, dental and medical devices, office supplies, and food safety equipment, to name just a few categories). We’ll study existing methods for capacity planning and develop new models to address specific concerns encountered in manufacturing processes at 3M.
The mathematical tools we will study and use are beautiful and powerful. Intricate interdependence of manufacturing processes is modeled with graph theory. Optimizing functions which depend on many variables and must satisfy complicated constraints is done gracefully with methods and algorithms of linear programming. Incorporating uncertainty and risk into the optimization process mixes in stochastic processes and probability. All these wonderful tools will help us solve immediate and practical problems!
Advisor: Rob Thompson (Both Math and Stats majors are welcome)
Prerequisites: No extra mathematical prerequisites, but some comfort with computing is highly recommended (e.g. at the level of CS 111). We will learn what we need to know as we go!
Advisor: Alex Barrios (Priority given to Math majors)
Prerequisite: Math 342
Applied Category Theory
In linear algebra, we study vector spaces and linear transformations. In topology, we study topological spaces and continuous functions. In abstract algebra, we study groups and group homomorphisms. All these are examples of “categories,” collections of mathematical objects linked by a notion of how to get from one to another. Category theory is the study of the rules that underlie all these separate mathematical disciplines; in a sense, it is the mathematics of mathematics. This extra layer of abstraction, together with the almost tautological nature of its theorems, has earned category theory the affectionate nickname “abstract nonsense.”
On the other hand, our lives are filled with concrete problems. Do we have enough ingredients at home to make dinner? What should a self-driving car do when another car gets too close? How should we fill out a form if we don’t fit into the assumptions it’s making? Category theory can help us answer these questions, and this kind of “applied” category theory is a relatively new field in mathematics.
In this project, you will use a recently published textbook on applied category theory to learn, teach, and present the subject, to me and to each other. We will meet twice a week, spending two weeks on each of seven major concepts in category theory and their applications, and when we finish we will turn to one or more topics in more depth, depending on your interests.
Strictly speaking, this project does not require any background beyond Math Structures. However, the more 300-level classes you’ve taken, especially if you’ve had Abstract Algebra I, the more enriching you’ll find this study of the mathematics of mathematics.
Advisor: Owen Biesel (Priority given to Math majors)
Prerequisite: Math 342 (Abstract Algebra I) recommended by not required.
Advisor: Mike Cohen (Priority given to Math majors)
Prerequisites: Real Analysis I or Topology. It is helpful if you have studied groups before, but not necessary as we can learn the basics from scratch.
Advisor: Eric Egge (Priority given to Math majors)
Meeting times: Mondays 6a and Thursdays 1-2 pm
Prerequisites: Previous experience with binomial coefficients, permutations, partitions, and generating functions. Math 333 (Combinatorial Theory) will be sufficient. However, you could also gain this experience by taking certain courses in Budapest, participating in an REU, or by doing some supervised independent reading.
Advisor: Kate Hake (Priority given to Math majors)
Starting a Middle School Math Circle in Northfield
A Math Circle is based on an idea brought over roughly 25 years ago from Eastern Europe where students interested in mathematics get together with mathematics professionals for some after-school or out-of-school mathematics exploration in a friendly, relaxed environment. Each week the mathematician brings in a low-floor/high-ceiling problem for students to work on, and the students direct the conversation through their questions and suggestions.
Students see the fun of mathematics when they are conjecturing, communicating, and exploring mathematical ideas.
The goal of this project is to plan for and run a Math Circle in Northfield aimed at students in grades, approximately, 5-8. Comps students would prepare the curriculum and run the weekly meeting. This project would go all three terms at Carleton, for two credits each term.
Over the summer or at the beginning of the fall, students should read Bob and Ellen Kaplan’s Out of the Labyrinth. They should also read the Challenge Math Curriculum designed by an earlier Carleton comps group. Early in the fall students will visit math classes in grades 5-8 to get a sense of what students that age know.
Our comps group will meet once each week to prepare for the lesson, then meet once each week to run the Math Circle. The comps group will prepare a set of annotated lesson plans for running a math circle and will create a webpage with such information and notes.
Advisor: Deanna Haunsperger (Priority given to Math majors)
Terms: Fall/Winter/Spring (2 credits per term)
Advisor: Rafe Jones (Priority given to Math majors)
Prerequisites: Math 321 or equivalent. You need some experience working with ideas of open and closed sets, and uniform and pointwise convergence. You might have gotten this experience by taking certain courses in Budapest, in an REU, or having done some reading on your own. The project will involve complex analysis, but it’s fine if you haven’t had experience with that.
Advisor: Mark Krusemeyer (Priority given to Math majors)
Terms: Fall Winter or Winter/Spring
Homological Project Prerequisites: Math 342 (Abstract Algebra I); some knowledge of topology may be helpful, but is not required.
Elliptic Project Prerequisites: Math 261 or 361 (Functions of a Complex Variable/Compex Analysis; required); Math 342 (highly recommended). Experience with number theory may be helpful, but is not required.