## Senior project ideas

### Joe Champion

#### 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

#### Combinatorics and graph theory

Learn something new in one of these areas that wasn’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

#### Topics in graph theory

#### Topics concerning the topology of surfaces

#### Topics in linear algebra over rings

#### Contact Info

### Uwe Kaiser

#### Quantum Computing Algorithms

This project asks you to have some programming experience. You work on a specific quantum algorithm. You study programming in Microsoftâ€™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 algorithms are implemented using circuits. The minimal expectation is a poster to be presented in the senior showcase.

#### Tangles and Electrical Networks

In 1993 Goldman and Kauffman interpreted the continued fraction of a tangle as a conductance of a corresponding network. This projects asks to search for further relations between electrical networks and invariants of links and tangles. The starting point is a paper of mine on band-operations on links leading to a formula similar to the one by Goldman/Kauffman. Basic linear algebra and some familiarity with discrete mathematics like graph theory are helpful, also knowledge of basic notions concerning electrical networks. The minimal expectation is a poster to be presented in the senior showcase.

#### Robotics and Topology

Topology and robotics are related through notion of configuration space. For example the configuration space of a robot arm in 3-space is a product of 2-spheres. Restrictions on motions lead to more interesting topology of configuration spaces. The goal is to study the complexity of basic examples through well-known theory developed by Farber. Knowledge of basic topology (for example MATH 411) is necessary for an understanding of the theory. The minimal expectation is a poster to be presented in the senior showcase.

#### Analysis Situs

In the years 1899 to 1904, Poincare published a paper with the title Analysis Situs and five supplements introducing basic ideas of topology. This project aims to study the way he introduced a particular concept, like e.g. the orientation of a manifold, and to research pre-Poincare origins of this concept (what was a building on?) and how the concept developed into modern times. The prerequisite for this project is the maturity of a senior, the willingness to read old mathematics literature (English translations are available). The expectation is a poster to be presented in the senior showcase.

#### Contact Info

### Michal Kopera

#### 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 265 and/or 365 courses. Knowledge of basic linear algebra (matrices) is a plus. The minimum expectation is a poster presentation at Senior Showcase.

#### Ice/ocean interactions

The modeling of the interface between ice and the ocean is of utmost importance for climate science. You will experiment with models of ice/ocean boundary developed in my group and evaluate whether they produce physical results.Â No ocean science background is required, but you should be comfortable with writing simple code. The minimum expectation is a poster presentation at Senior Showcase.

#### 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. Â Knowing something about ODEs and/or PDEs is welcome. I am also open to problems that yield themselves to Machine Learning. The minimum expectation is a poster presentation at Senior Showcase.

#### Game of Life, Fractals and self-similarity

You will explore some of the concepts outlined above and write code to implement them. The minimum expectation is a poster presentation at Senior Showcase.

#### Mathematical Art

You will work with genetic (or other nature-inspired) algorithms that try to generate art. You can either focus on optimization algorithms that try to reproduce existing images or aim to generate original art and try to measure its esthetics. The minimum expectation is a poster presentation at Senior Showcase.

#### Contact Info

### Zach Teitler

#### Topics

My specialty is algebra. I can work with you on projects in algebra, graph theory, combinatorics, number theory, any other area of pure math, or any subject that you’re interested in within pure math, applied math, statistics, or math education.

#### Contact Info

I am available to work with students on undergraduate senior thesis projects. You can email me if youâ€™re looking for a senior thesis advisor, but first,

read about what you can expect if we work together and what project ideas we can work on together.

### Barbara Zubik-Kowal

#### 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 solutionâ€™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.