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Senior Project Marketplace

Senior Project (MATH 401) is designed to be a capstone experience for Math undergraduates, where the students engage in research activities guided by a faculty member. To enroll in MATH 401, you need to choose a project topic and get in touch with a faculty member. You can come up with your own project idea, or choose one from the list below. The list is by no means exhaustive but will give you an idea of what kind of research each researcher is working on.

 

Senior project ideas

Joe Champion - Math education

Statistical Modeling of Math Education Achievement

Learn about large-scale educational achievement data and techniques for predicting students’ math achievement. Involves data wrangling, intermediate coding in R or Python (mostly adapting existing code), and a focus on data visualization. Background in mathematics education and/or statistics preferred.

High School Mathematics Curriculum Development

For mathematics education students – modify and create Desmos Teacher activities to align with high school mathematics standards. Focus on data, modeling, and technology-assisted representations.

Middle School Mathematics Curriculum Development

Adapt activities from the Algebra through Visual Patterns curriculum for delivery in the Desmos Teacher Activities platform. Involves some testing with students and collaboration with math education researchers.

Other project ideas

Dig into the history of a K-12 math topic and write a paper / make a poster, create an original math video.

Contact info

Samuel Coskey - Set theory, logic, and combinatorics

Combinatorics and graph theory

Learn something new in one of these areas that weren’t covered in Math 189/287/387. Use a new book, book chapter, online notes, or published article as a resource. Present the motivation, examples, and results with a poster.

Algebra or analysis or geometry

Learn something new in one of these areas that wasn’t covered in Math 305/311/314/405/414. Use a new book, book chapter, online notes, or published article as a resource. Present the motivation, examples, and results with a poster.

Math education

Choose a topic in college-level mathematics to present at the middle or high school level. Create a detailed lesson plan.

Other project ideas

I am open to exploring anything in pure mathematics (and applied mathematics if you can take the lead). The important thing is to find resources at the right level for you.

Contact Info

Jens Harlander - Topology and Algebra (Group Theory)

Topics in graph theory

Topics concerning the topology of surfaces

Topics in linear algebra over rings

Contact Info

Uwe Kaiser - Geometric and Algebraic Topology, Quantum Computing

Problems in Knot theory

There are many problems in knot theory, which are easily stated using only elementary mathematics like basic graph theory. In https://arxiv.org/pdf/1604.03778.pdf several such problems are stated. The goal is to study one of these problems like e.g.: (1) Given two diagrams of a link, find a sequence of Reidemeister moves relating them where all diagrams in the sequence have small crossing number, or (2) Find bounds for crossing numbers for certain classes of links. The idea is to approach interesting theoretical problems through the study of explicit examples or classes of examples. The result will be a paper or poster, which can be of survey nature, but maybe contain some new examples and ideas. No programming skills are required. The project will involve understanding and writing proofs.

Quantum computing with braids

In 2016 the Nobel prize in physics was awarded to a group of physicists for their achievement in topology. Microsoft is promoting ideas based on their theories, using representations of braids (related to specific particles called anyons) to construct quantum gates. This topic allows you to study the mathematics of specific examples like so-called Fibonacci particles. Introductory literature is e.g. https://arxiv.org/pdf/1802.06176.pdf . A survey of the topic is possible but also a study of error metrics or the implementation of explicit algorithms using the gates constructed from the Fibonacci anyons. Program skills can be helpful but are not required. Good knowledge in algebra and linear algebra will be necessary, including understanding of proofs and proof-writing skills.

Quantum computing algorithms

This project asks you to have some programming experience. You work on a specific quantum algorithm like e.g. Deutsch’s or Simon’s algorithm for factoring into primes. You study programming in Microsoft’s quantum computing kit, see https://www.microsoft.com/en-us/quantum/development-kit, try out examples, and study properties of the algorithm. Basic linear algebra skills are necessary in order to understand how the algorithm is implemented using circuits.

Realizing linking numbers

In 1996 I defined generalized linking numbers in a large class of 3-dimensional manifolds (objects locally looking like 3-space). The problem of realizing those invariants is still unknown, even in simple cases like the solid doughnut. The project will study examples of links (two disjoint knotted circles) and the calculation of corresponding invariants. The understanding of proofs and proof-writing skills will be necessary

Contact Info

Michal Kopera - Computational Math, Ocean Modeling

BroncoRank - a new university ranking

In this project, you will explore the idea of using a PageRank algorithm, which Google is using to rank websites in their search engine, for creating a university ranking which does not depend on some editorial board decision but emerges from each university peer institution lists. You will get a chance to work at an intersection of mathematics, programming, data science, and contribute to creating a more fair tool to rank universities across the U.S.

To be successful in this project, you need some background in programming and ideally have enjoyed your MATH 365 course. Knowledge of basic linear algebra (matrices) is a plus.

Visualization of ocean simulations

You will work with data produced by an ocean model to produce a visualization of the results. The visualization of the model output is important in the development process to confirm whether the model is working correctly. You will get a chance to create a 3D rendering of ocean simulations, and participate in the extraction and analysis of the data.

No ocean science background required. The project will require you to learn a 3D  visualization tool Paraview. You will learn how to transfer large data sets from a supercomputer to your local computer.

Computational modeling using ODEs and PDEs

The bulk of my work is using computational methods to simulate phenomena described by ordinary or partial differential equations. I am open to your ideas on what you would like to model, and we can create a project based on your input.

You will likely need to be able to program in MATLAB, Python, Julia, or other languages.  Knowing something about ODEs and/or PDEs is welcome. I am also open to problems that yield themselves to Machine Learning.

Contact Info

Zach Teitler - Algebra and algebraic geometry

Transforming integer sequences using determinants

A sequence of integers can be transformed into a new one using determinants of Hankel matrices. What patterns can we find? Are there sequences that stay the same, or “friendly” pairs of sequences that transform into each other? What if we switch the Hankel determinants into something else? This will be an exploration. Students need to be comfortable with determinants and induction.

Schur polynomials and other symmetric polynomials

 Goals:
1. Learn what are symmetric polynomials and what is a Schur polynomial. Write an introduction to this subject.
2. What do we get when we take a derivative of a symmetric polynomial or Schur polynomial? Given a symmetric polynomial or Schur polynomial, what are the differential equations that have it as a solution?
In one sense that is much less dramatic than it sounds, because the question is just what are the linear dependencies among the derivatives. In another sense, it is the first of many open questions about this subject in an interesting area of research.

Exact and parametrized algorithms and Waring rank

Students will read a paper by Kevin Pratt, learn about the theory of exact and parameterized algorithms, and how it relates to algebra.

Derivatives of symmetric determinants

What do we get when we take a derivative of a symmetric determinant? What are all the differential equations that have a symmetric determinant as a solution?

From graphs to rings: using zero divisor graphs for ring theory

Given any ring, the associated “zero divisor ring” encodes information about the ring. Under some conditions, it actually fully encodes all the information of the ring. Can we use these graphs to prove theorems about rings?

Barbara Zubik-Kowal - Applied mathematics

Difference equations and applications

Difference equations arise naturally in real-world applications involving discrete sets or populations, or as approximations to continuum models in science and engineering. Mathematically, difference equations can be described as mathematical equalities involving the values of a function of a discrete variable. A recurrence relation such as the logistic map, relevant to population dynamics, or the sequence of Fibonacci numbers, are simple examples. Many difference equations can be solved analogously to how one solves ordinary differential equations. However, it is well-known that most difference equations depicting real-life phenomena cannot be solved in closed form and other methods are necessary to obtain qualitative or quantitative information about the desired solutions, including their stability properties. This senior project can go in a number of directions depending upon the interests of the student. The project may involve theoretical aspects, including theoretical derivations and proof-writing, or computations, including writing new codes or modifying existing ones.

Integro-differential equations and applications

Integro-differential equations are central to modelling numerous natural and industrial phenomena across physics, biology, medicine, engineering, and other fields. As an example in the field of epidemiology, integro-differential equations are frequently used in the mathematical modelling of epidemics, such as when the age-structure of the population is important in determining the dynamics of an epidemic. Integro-differential equations involve both integrals and derivatives of a function. As very few systems of integro-differential equations have a closed-form solution, a range of mathematical methods are often used to obtain qualitative information about the solutions of classes of problems involving integro-differential equations, and approximation techniques are often used to obtain quantitative information about the corresponding solutions given some initial data. In contrast to ordinary and partial differential equations, initial data for integro-differential equations is frequently provided on a whole interval, rather than a single initial point in time. This means more initial data is used to supplement systems of integro-differential equations. This senior project can go in a number of directions depending upon the interests of the student. The project may involve theoretical aspects, including theoretical derivations and proof-writing, or computations, including writing new codes or modifying existing ones.

Differential inequalities and applications

Mathematical models for a range of biological, physical or industrial phenomena may be grouped into general classes of systems of differential equations. Even if the underlying mathematical models may involve complexities that make it hard or impossible to solve by hand, it is frequently possible to extract useful qualitative information about its solutions. Such qualitative information frequently suffices to answer key questions about a solution’s behaviour. Examples are its long-term behavior, existence and uniqueness, convergence properties, and its upper and lower bounds, such as maximal and minimal solutions. These properties, in turn, help us derive information about not only one, but a whole family of mathematical models constituting a given class of differential equations. This senior project can go in a number of directions depending upon the interests of the student. The project may involve theoretical aspects, including theoretical derivations and proof-writing, or computations, including writing new codes or modifying existing ones.

Principles of approximation and applications

Smooth functions arise frequently in the mathematical modeling of numerous real-world phenomena in the sciences and engineering, including both natural and industrial processes. An example is the solution to a SIR model of susceptible, infectious, or recovered individuals in epidemiology, or solutions to mathematical models of tumor growth. It is well known, however, that solutions to most mathematical models depicting real-world phenomena cannot, in general, be expressed in closed form. It is, however, possible to make progress by making appropriate approximations to obtain an estimate of the desired solution. Such approximations involve discretizing the domain from a continuous interval to a finite subset of grid points, solving the discrete systems of equations, computing continuous extensions, or interpolations, and performing error analysis. There are many ways of doing this, but it is important to understand how to do it in a way that preserves certain desired properties, in order to ensure that the resulting approximate solutions that we are getting are indeed approximate solutions to the problem we started out with, rather than spurious output. This senior project can go in a number of directions depending upon the interests of the student. The project may involve theoretical aspects, including theoretical derivations and proof-writing, or computations, including writing new codes or modifying existing ones.