# Colin's Homepage

## Welcome to my site!

### I am a Ph.D. candidate at Colorado State University. My areas of interest are in Riemannian geometry, Clifford analysis, mathematical physics, and partial differential equations.

Here is a copy of my CV and my resume.

# About Me

# Blog

# Work Experience and Involvement

# NASA Internship

# AIM COVID-19 Modeling

# AIM/MCRN Data Assimilation

# Research

As a mathematician, I am interested in physical problems and geometry. In my mind, the linkage between these two components is through partial differential equations (PDEs). PDEs are typically studied in an analytical sense with boundary value problems posed on subregions in n-dimensional Eucliean space. However, the coordinate free perspective granted by Riemannian geometry may provide new insights or techniques to make progress in both old and new problems.

### Publications

Aishah Albarakati, Marko Budišić, Rose Crocker, Juniper Glass-Klaiber, Sarah Iams, John Maclean, Noah Marshall, Colin Roberts, Erik S. Van Vleck, * Model and data reduction for data assimilation: Particle filters employing projected forecasts and data with application to a shallow water model *,
Computers & Mathematics with Applications,
2021, ISSN 0898-1221.

Brooks Adams, Henry Adams, and Colin Roberts, * Sweeping Costs of Planar Domains *, In Erin W Chambers, Brittany T Fasy, and Lori S Ziegelmeier, eds., Research in Computational Topology, pages 71-92, AWM Springer series, volume 13, 2018.

Jonathan Gilbert, Colin Roberts, and Jacob Roberts, * Near-Resonant Light Propagation in an Absorptive Spatially Anisotropic Ultracold Gas *, Journal of the Optical Society of America B, pages 718723, volume 25, number 4, 2018.

# Thesis Work

# Data Assimilation

# Teaching

Colorado State University (CSU) provides their graduate students with the unique ability to teach classes. I have been fortunate enough to have experience teaching a variety of courses at CSU which has not only helped me strengthen my mathematical knowledge but my ability to speak share my knowledge in a public forum. Moreso, I believe that teaching is a fundamental skill for mathematicians and physicists alike.

I put a large effort into providing a great learning experience for my students. Starting in the spring of 2019, I was given courses in which there is only a single section and was left to coordinating the courses myself with some oversight from the department. This allowed me to develop personalized content for these classes (Math 255, 271, and 272) which can hopefully be used for years to come.

Math 272 Spring 2022, Vector fields, partial differentiation, cylindrical and spherical coordinates, multiple integrals, line integrals, the Wave and the Schrödinger equations, separation of variables method. Inner Product Spaces. Fourier Series.

Math 271 Fall 2021, Series and limits, Taylor series, complex variables, first- and second- order ordinary differential equations, matrices, linear transformations, determinants, and eigenvalues.

Math 272 Spring 2021, Vector fields, partial differentiation, cylindrical and spherical coordinates, multiple integrals, line integrals, the Wave and the Schrödinger equations, separation of variables method. Inner Product Spaces. Fourier Series.

Math 271 Fall 2020, Series and limits, Taylor series, complex variables, first- and second- order ordinary differential equations, matrices, linear transformations, determinants, and eigenvalues.

Math 272 Spring 2020, Vector fields, partial differentiation, cylindrical and spherical coordinates, multiple integrals, line integrals, the Wave and the Schrödinger equations, separation of variables method. Inner Product Spaces. Fourier Series.

Math 271 Fall 2019, Series and limits, Taylor series, complex variables, first- and second- order ordinary differential equations, matrices, linear transformations, determinants, and eigenvalues.

Math 255 Spring 2019, Derivatives and integrals of functions of several variables, differential and difference equations, matrices, applications in the biosciences.

Math 155 Fall 2018, Limits, continuity, differentiation, and integration of elementary functions with applications in the biosciences.

Math 340 Summer 2018, First and second order equations, series, Laplace transforms, linear algebra, eigenvalues, first order systems of equations, numerical techniques.

Math 160 Spring 2018, Limits, continuity, differention, and integration of elementary functions with applications; conic sections.

Math 161 Fall 2017, Transcendental functions, integration techniques, polar coordinates, sequences and series, with mathematical software.

# Applied Mathematics for Chemists

In the Fall of 2019 I began teaching the year long sequence course Math 271/272 at CSU titled **Applied Mathematics for Chemists I/II**.
This class is rather new. It had only been taught by three others before me and had only ran for four years.
I challenged myself to build this class in my own way by bringing my own insight to the content.

Having taught Math 255, Calculus for Biological Sciences II, just prior to the chemistry version, I had some TeXed up notes prepared. Students are already paying thousands of dollars to enroll at a university. I figured that the least I could do was provide them with a free textbook of my own notes. So, my notes became the foundation for the course and I created homeworks, quizzes, exams, and projects to make the course complete. The advantage here is that the content is streamlined -- it is specifically catered to this class sequence and has no unnecessary material. Homeworks, for example, build off one another and problems are rehashed in new ways throughout.

As of late 2020 I decided to release this textbook publicly on Github. Here is a link to the text: Applied Mathematics for Physicists and Chemists. There is still much to do. I have started placing the problems I've written in the corresponding sections and I have outlined a few changes that should be made. I believe this text can be used widely and updated by any interested party. I do not need to claim complete ownership. I prefer everyone has their chance to use the material if they so wish.

Along the sidebar you can visit the various semesters for the course sequence. You can see the class progression throughout. I have learned a ton from my students and I deeply appreciate their input. From countless typos they've found, to suggestions about the content, and to returning during later semesters to be TAs for the class, I owe each one a debt of gratitude.

# Math 272 Spring 2022

### Vector fields, partial differentiation, cylindrical and spherical coordinates, multiple integrals, line integrals, the Wave and the Schrödinger equations, separation of variables method. Inner Product Spaces. Fourier Series.

I am currently teaching the Math 271/272 sequence at Colorado State University. The materials below were created by myself for Math 271. Each semester I make changes to the course in an attempt to improve the class structure. This semester we will forgo quizzes entirely and favor in class participation activities. Similarly to the end of fall 2021, we will also concentrate more on homework and collaboration. These changes were suggested by students who took 271 in the fall.

## Homework

# Math 271 Fall 2021

### Series and limits, Taylor series, complex variables, first- and second- order ordinary differential equations, matrices, linear transformations, determinants, and eigenvalues.

I am currently teaching the Math 271/272 sequence at Colorado State University. The materials below were created by myself for Math 271. Each semester I make changes to the course in an attempt to improve the class structure. This semester we started with homeworks, quizzes, exams, as well as discussion posts. I decided to forgo the 50 minute quizzes after Quiz 3 in favor for more challenging homework assignments.

## Project

The project for this class was designed to be an introduction to group theory. It ended up being a bit longer than I intended and I found after the fact that some notions were not quite clear enough.

## Homework Assignments

This semester I decided to concentrate more on homework assignments rather than quizzes and exams. I felt it gave the students more room to learn and collaborate.

### Solutions

Here are the solutions for the above homework assignments. Please let me know if you find any errors!

## Exams

Exams were done as take home exams with an oral component. I met with each student individually to ask for clarification on their solutions. I also asked questions about the content that they did not know before hand.

## Quizzes

Quizzes were administered in class and students were given 50 minutes to complete each quiz.

# Math 272 Spring 2021

### Vector fields, partial differentiation, cylindrical and spherical coordinates, multiple integrals, line integrals, the Wave and the Schrödinger equations, separation of variables method. Inner Product Spaces. Fourier Series.

I am currently teaching the Math 271/272 sequence at Colorado State University. This semester was taught completely online just like the fall 2020 semester for 271. The class had oral exams done virtually and I met with each student one-on-one.

## Project

The project for this class was to find analytical solutions to the Schrödinger equation for a central Coloumb potential. In other words, we found explicit states for electrons in a hydrogen atom.

## Homework Assignments

Homework assignments were due roughly every week.

### Solutions

Here are the solutions for the above homework assignments. Please let me know if you find any errors!

## Exams

Exams were done as take home exams with an oral component. I met with each student individually to ask for clarification on their solutions. I also asked questions about the content that they did not know before hand.

## Quizzes

Quizzes were administered in class and students were given 50 minutes to complete each quiz.

# Math 271 Fall 2020

### Series and limits, Taylor series, complex variables, first- and second- order ordinary differential equations, matrices, linear transformations, determinants, and eigenvalues.

I am currently teaching the Math 271/272 sequence at Colorado State University. This semester was heavily affected by the COVID-19 pandemic. In order to prevent further spread of the virus, I chose to run this course in an online format. After all was said and done, student learning did not seem to be diminished.

## Project

The project for this class was designed to be an exploration of quantum entanglement. The students had covered a wealth of linear algebra and this was a means of including some multilinear algebra as well. Fortunately, it tied back into the quantum mechanics they had already learned from this course!

## Homework Assignments

Homeworks were assigned roughly every week.

### Solutions

Here are the solutions for the above homework assignments. Please let me know if you find any errors!

## Exams

Exams were done partially as take home and partially in class during a 50 minute session with webcams on. This did not seem to be the best way to conduct exams in an online format and this prompted changes for the following semesters.

# Math 272 Spring 2020

### Vector fields, partial differentiation, cylindrical and spherical coordinates, multiple integrals, line integrals, the Wave and the Schrödinger equations, separation of variables method. Inner Product Spaces. Fourier Series.

I am currently teaching the Math 271/272 sequence at Colorado State University. This semester marked the beginning of the COVID-19 pandemic. Classes were greatly affected by the switch to an online format. Changes were made to the syllabus in order to accomodate students.

## Project

The project for this class was to find analytical solutions to the Schrödinger equation for a central Coloumb potential. In other words, we found explicit states for electrons in a hydrogen atom.

## Homework Assignments

Homework assignments were due roughly every week.

### Solutions

## Exams

Exams were done as take home exams with an in class component. After the switch to online, only take home versions were given.

# Math 271 Fall 2019

### Series and limits, Taylor series, complex variables, first- and second- order ordinary differential equations, matrices, linear transformations, determinants, and eigenvalues.

I am currently teaching the Math 271/272 sequence at Colorado State University. This was the first semester teaching this sequence course.

## Project

The project for this class was designed to be an exploration of the Fourier series as it arises via solutions to physical problems. It was meant to motivate the content of the sequel, Math 272.

## Homework Assignments

Homeworks were assigned roughly every week.

### Solutions

## Exams

Exams were done partially as take home and partially in class during a 50 minute session.

# Calculus for Biological Scientists II, Math 255

### Derivatives and integrals of functions of several variables, differential and difference equations, matrices, applications in the biosciences.

This course gave me the ability to design the content to my liking. I had a rough template to follow, but decided to go with my own path. I developed notes for this course on the fly. I did this alongside creating homework assignments and exams. Below you can find links to the various files used throughout.

## Homework Assignments

Homeworks were assigned roughly every week.

### Solutions

## Exams

Exams were done in class during a 50 minute session.