Mathematics Comps (priority given to Mathematics majors)
Communicating Mathematics
Businesswoman Adena Friedman said that “Ideas are only as good as your ability to communicate them.” National Academy of Science mathematician and my advisor Donald Saari said that if you can’t explain your idea to any guy sitting next to you in a bar, then you don’t understand it well enough. We may have the best of ideas, but without the ability to articulate them to colleagues, friends, or family, do they have value?
In this comps, we will practice writing mathematics in a variety of different genres. Possible genres could include a book review, a blog post, a NYT op ed, a research paper summary, a mathematically-based short story, a news article on recent advances or issues in mathematics, an article for a STEM magazine for teenagers, an interview, or an article for Math Horizons. The comps students will be divided into two groups. Each week, one of the groups will be writing and the other group will be editing someone else’s work. Those responsibilities change week by week. At the end of the comps experience, each student will have a portfolio of edited writing which could be used by the student to demonstrate their writing ability to a potential employer.
The final written product for this comps will be a portfolio for each individual, with a cover sheet reflecting on the writings. Additionally, the group will give a group presentation.
Meeting time: Tuesdays 10:10-11:55am
Advisor: Deanna Haunsperger
Terms: Fall/Winter (Fall Math 400; 3 credits & Winter Math 400; 3 credits)
Prerequisites: None
Target number of students: 8
Commuting to Infinity: Infinite Abelian Groups
Advisor: Rafe Jones
Terms: Fall/Winter (Fall Math 399: 6 credits & Winter Math 400; 3 credits)
Prerequisites: Math 342 or equivalent
Target number of students: 4
Methods of Modeling Complex Systems
Across two terms we will focus on: (1) learning various modeling techniques, (2) analyzing how models are formed and validated and (3, if there is time) applying or tweaking models. Specifically, the goal of each person in the group will be to study two out of the five subjects:
- Discrete Maps
- Ordinary Differential Equations
- Partial Differential Equations
- Networks
- Agent Based Modeling
You must select at least one subject that you have not taken a class in to study. Members of the group can choose to study vastly different subjects, but it might be beneficial to choose some overlaps as to have someone to work with throughout the week.
We will be using the book Introduction to the Modeling and Analysis of Complex Systems by Hiroki Sayama (you can acquire the PDF version for free by following the link) and sample papers illustrating models using the five techniques. Sample papers will include models from various backgrounds (opinion dynamics, disease spread, chemotaxis, population spread, ecology, etc.).
Coding will be done in Python and Mathematica (although not quite the main focus). The book by Sayama has plenty of scaffolded code in Python, so people with little experience should still be able to start implementing these in Python.
A successful completion of comps will be a group paper and presentation that outlines the methods learned throughout the course, supplemented by figures that the group members will generate. Additionally, there must be a section that outlines how these methods are implemented and analyzed in published models or well-written pre-prints.
Advisor: Joseph Johnson
Terms: Winter/Spring (Winter Math 400; 3 credits & Spring Math 400; 3 credits)
Prerequisites: Math 232
Target number of students: 4
Flow-Kick Models of Environmental Disturbances
Advisor: Kate Meyer
Terms: Fall/Winter (Fall Math 400; 3 credits & Winter Math 400; 3 credits)
Prerequisites: Math 241 (ODEs). You should also be open to computing in MATLAB, but previous experience with this program is not expected.
Target number of students: 4
Tournaments
Advisor: MurphyKate Montee
Terms: Fall/Winter (Fall Math 399; 6 credits & Winter Math 400; 3 credits)
Prerequisites: Math 232 and 236. Any experience with abstract algebra, graph theory, or combinatorics will be helpful, but not required.
Target number of students: 12
Statistics Comps (priority given to Statistics majors)
Statistical Computing
Many problems in statistics can’t be tackled analytically, or are just very difficult to tackle with pencil and paper. In this group comps, we will explore numerous algorithms that are used to provide numerical solutions to these analytically difficult situations. Specifically, we’ll explore optimization of the likelihood function, numerical integration, Monte Carlo methods, resampling methods (e.g., more advanced bootstraps), and Markov chain Monte Carlo methods (MCMC).
Note: This is a Large Group Comps which will enroll 9-12 students and has a different structure from regular Group Comps. The main difference is that you will enroll in a 6-credit seminar course in the Fall that will meet on a regular course schedule with the full group of 9-12 students. In Winter you will enroll in the usual 3 credit comps, and you will work in a small group that will meet with me individually while completing a project.
Advisor: Adam Loy
Terms: Fall/Winter (Fall Stat 399; 6 credits & Winter Stat 400; 3 credits)
Prerequisites: Stat 230, Stat 250, and Stat 220 or CS 111
Target number of students: 9-12
Statistical Models for Reporting Styles in Forensic Science
Think of the last time you filled out a survey. The questions likely contained some sort of rating scale for you to use: “strongly agree” to “strongly disagree”; “very often” to “never”; or “very good” to “poor” are a few common examples. But how do you know whether your “very good” response is similar to mine? In general, you don’t! There are several ways that individual responses can be biased. Some people avoid the extreme categories, while others avoid the middle category. Some people change their answers if the question is phrased as agreement or disagreement. Some people provide inconsistent answers if a different number of categories is used. While it’s important to understand these differences when analyzing any type of survey data, it’s especially important when high-stakes decisions depend on the results.
In many forensic science disciplines, there is no objective way to determine whether two fingerprints, bullets, or handwritten documents come from the same person or not. Instead, it is the responsibility of each individual analyst to make a subjective decision and communicate their results to a judge or jury. Just like the survey setting, differences have been observed among examiners when analyzing the same piece of evidence: some examiners appear to be well-calibrated with each other and utilize the full range of possible outcomes, while other examiners overuse the extreme categories or middle categories. Recently, there have been proposals to shift from a 3-category scale (e.g., “match”, “non-match”, and “inconclusive”) to 5 or even 7 category scales (e.g., “strong support for match”, “moderate support for match”, “inconclusive”, “moderate support for non-match”, “strong support for non-match”). Since examiner conclusions can influence investigator, judge, and jury decisions, it is important to measure and understand the range of individual differences in reporting styles before adopting a more complicated scale.
For this comps experience, we’ll investigate existing approaches for modeling individual response styles and apply them to forensic datasets. We’ll focus on two broad classes of models. The first is categorical outcome models, starting with extensions of logistic regression for more than two categories. The second is latent variable models, which assume there is a variable that cannot be observed directly but governs the observed responses.
Goals and Deliverables
The first part of this project will focus on reading:
- Gelman and Hill’s Data Analysis Using Regression and Multilevel/Hierarchical Models
- DeBoeck’s Explanatory Item Response Models
- Examples of peer-reviewed papers in quantitative psychology, forensic science, or other disciplines that use models for response styles
In the second part of the project, you will apply these model(s) to dataset(s) in forensic science. This will include:
- Becoming familiar with a “black box” study in fingerprints, firearms, handwriting, or another forensic evidence discipline
- Selecting an appropriate model(s) for response styles and studying it via simulation
- Analyzing the dataset using your chosen model(s)
- Writing up your results and submit a paper to the USRESP Competition
Depending on interests, we may also:
- Develop open-access software for forensic scientists to fit these models to their own data using R/Shiny
- Submit a conference paper to the International Meeting of the Psychometric Society and the corresponding refereed Quantitative Psychology proceedings
- Submit an article to a peer-reviewed forensics or statistics journal
Advisor: Amanda Luby
Terms: Winter/Spring (Winter Stat 399; 6 credits & Spring Stat 400; 3 credits)
Prerequisites: Stat 230; Stat 250
Target number of students: 4
Statistical Analysis of Networks
Advisor: Katie St. Clair
Terms: Winter/Spring (Winter Stat 400; 3 credits & Spring Stat 400; 3 credits)
Prerequisites: Stat 230 and Stat 250
Target number of students: 8