Summer Opportunities – On campus – Mathematics and Statistics

Overview

Faculty in the department have active research programs that involve students during the academic year and summer. Summer research positions are open to Carleton students and are paid positions.

The faculty research projects for the Summer of 2025 that have positions open for students are described below. Direct any questions you have about a project to the faculty running the project. The student stipend rate for the Summer of 2025 is $570/week. A list of all science research opportunities and fellowships at Carleton College is also available.

If you would like to apply, please fill out out this application form by 11:59 pm on February 16, 2025. You will be asked to list the project(s) of interest to you and provide an unofficial transcript. The number of students hired for each project is dependent on funding which is likely, but not certain, so the number of students ultimately hired for each project could change.

Summer 2025 Project Descriptions

Topology of algebraic formulas, I (Claudio Gómez-Gonzáles)

2-3 students
40 hours per week, start date June 9 (8-10 weeks)
Mode: Hybrid (students must be located in the United States). Students can be located in Northfield if desired, but meetings will often be virtual.

Background: We are motivated by one of the oldest problems in mathematics: “How can we solve algebraic equations in the simplest manner possible?” It is well-known (amongst mathematicians) that there is no solution for the roots of the general quintic in radicals (Ruffini and Abel, ~1799-1824). Less known is the earlier result of Bring (~1786) that any quintic can be reduced via radicals to the form x^5+ax+1 = 0; if we can solve this much easier problem, with only one parameter, then we can solve any quintic. Hamilton (~1836) extended Bring’s work to higher degree equations: for example, any sextic can be reduced to x^6 + ax^2 + bx+ 1, making it at most a two parameter (a and b) problem. The resolvent degree RD(n) asks how many parameters we really need to solve a degree n polynomial. Despite the fundamental nature of this invariant and the influences of associated phenomena on early developments in algebraic topology, we can still say surprisingly little about many of these questions.

Project: Students will explore mathematical phenomena in and around resolvent problems. This work will include: developing visualization tools, extracting and formalizing perspectives from the literature, translating (sometimes literally) classical results into modern language, and/or using data-scientific approaches to inform ongoing research in resolvent degree. Students may also be involved in developing classroom materials (appropriate for intermediate and advanced courses at Carleton) based on their findings.

Prerequisites: Math Structures (236). Completion of at least one of the following is strongly preferred: Introduction to Computational Algebraic Geometry (295), Abstract Algebra (342), Topology (Math 354), or Computer Graphics (CS 311).

Statistical Models for decision-making and reporting styles of forensic scientists (Amanda Luby)

2-3 students
40 hours per week (8-10 weeks, flexible start date, end August 15)
Mode: Primarily in-person

Description: I am looking for 2-3 students to work on ongoing research investigating the decision-making and reporting styles of forensic scientists.

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. 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 are outliers. 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.

Your research this summer could focus on making substantive recommendations to practitioners, developing open-source software for fitting these models, or investigating the performance of the models through simulation. You will develop a written report of your findings, design a poster presentation, and might submit to a scientific conference or journal.

Prerequisites: at least two 200-level stat courses. Preferable: Stat 230, Stat 250, Stat 330.

Modeling biological systems using flow-kick dynamics (Kate Meyer)

2 students (priority given to Class of ‘27 or ‘28)
40 hours per week, 8-10 weeks, starting June 16
Mode: mostly in-person with some Zoom meetings

Background and Motivation: Differential equations are commonly used as models of biological systems to help us understand their mechanisms and make predictions. As a simple example, if x represents the size of a population, then the equation x’(t)=ax(t) asserts that the population’s growth rate is proportional to its size. The solution x(t)=ce^(at) predicts exponential growth or decay in population size, depending on the sign of a.

Whereas differential equations model continuous changes over time, many biological systems experience repeated, discrete events that interrupt this continuity: for example, hurricanes batter coral reefs, drug doses cause a spike in blood concentrations, and fires abruptly reduce plant biomass. Flow-kick models are tailored to describe such systems. A flow-kick model intersperses periods of continuous change (“flow,” modeled by a differential equation) with discrete “kicks” that deliver an instantaneous disturbance. Some natural mathematical questions to ask about a flow-kick model are

Is it possible for the disturbance (kick) and recovery (flow) to balance one another, and if so, where?
As the frequency and magnitude of the kicks changes, how does the system behavior change? Are there critical thresholds at which a small change in the disturbance pattern leads to a big change in our predictions?

Project: You will use computer simulations to explore and interpret flow-kick dynamics in a biological model of your choice. Possibilities applications include:

hurricane impacts on coral reefs
seeding fragmented prairies to maintain biodiversity
drug addiction

and more! The patterns you uncover in a specific flow-kick model may lead to conjectures about the possible behaviors of these models in general, informing Kate’s ongoing research program. You will present your work in an undergraduate poster session at the Mathematical Association of America’s MathFest Conference in Sacramento, August 6-9, 2025.

Prerequisites: A course in single-variable calculus and at least ONE of the following:

Multivariable calculus (e.g. MATH 120, MATH 211, or equivalent)
Coding experience (e.g. CS 111 or equivalent)

This research experience is aimed for students who have just completed their first or sophomore years.

Linear Algebra of Tournaments (MurphyKate Montee)

2-4 students
40 hours per week, dates flexible, 8-10 weeks
Mode: In-person

Topic: In a round-robin tournament with n teams, each team plays every other team once. Suppose that we don’t allow ties. Then we can record the results of these games by drawing a node for each team, and an arrow from team i to team j whenever team i beat team j. The result will be a collection of n nodes, with edges between each pair of nodes and an orientation on each edge. Such an object is called a tournament (equivalently, tournaments are directed complete graphs).

Tournaments are a classical area of study in graph theory. Fundamental results include classifying which tournaments have a (directed) path that touches every node, counting the number of 3-cycles in a given tournament, computing expected values related to randomly chosen tournaments, and more. For the interested reader, Topics on Tournaments by John Moon is an excellent place to see the style of some of these arguments.

This research project will focus on results about tournaments that can be studied via linear algebraic techniques. For example, what polynomials can be the characteristic polynomial of a tournament? Given such a polynomial, can we find an explicit construction of its tournament? Can we see that two tournaments will have the same characteristic polynomial via graph theoretic techniques? Exact questions to be studied will depend on the interests of the students involved. There may be an opportunity to collaborate with other researchers in this area over the summer, both virtually and in person. Any travel expenses will be covered.

Prerequisites: Linear Algebra and Mathematical Structures. Experience with Abstract Algebra and/or Graph Theory would be helpful but is not required.