Brandon Alexander

Major: Mathematics, Computer Science

“R-Separation of Laplace’s Equation in Rotationally-Invariant Cyclidic Coordinates”

There exist 17 conformally unique coordinate systems which admit harmonic solutions to the three-variable Laplace equation via separation of variables. Of these, four are determined by coordinate surfaces given by rotationally-invariant cyclides: bi-cyclide coordinates, flat-ring cyclide coordinates, flat-disk cyclide coordinates, and toroidal coordinates. With the exception of toroidal coordinates, these systems produce harmonics given by products of complex exponentials , where is the azimuthal coordinate, and m is the azimuthal quantum number, and simply-periodic Lamé functions or Lamé-Wangerin functions. We separate Laplace’s equation for three versions of bi-cyclide coordinates: algebraic and Jacobian elliptic forms by Miller (1977) and a Jacobian elliptic form by Moon and Spencer (1961). In the process, we rederive and correct past results, and quantify the connection between these bi-cyclide coordinates. We then present a generalized form for bi-cyclide, flat-ring cyclide, and flat-disk cyclide coordinates, that is able to separate Laplace’s equation simultaneously. The long-term goal of this project is to produce expansions of a fundamental solution for Laplace’s equation in terms of our derived harmonics.

How did you find out that you could do research in your field in the summer?
Freshman year I attended one of the “How to Get Started in Research” workshops hosted by the Office of Undergraduate Research. There I learned about all of the available resources for finding summer research. One of the most useful tools I found was the National Science Foundation’s (NSF) Research Experiences for Undergraduates (REU) program’s website, where you can search and find a list of NSF- sponsored summer research programs in your field.

How did you know that research at National Institute of Standards and Technology (NIST) was what you wanted to do?
Last year I participated in a summer research program at James Madison University. I was interested in seeing how research in a government laboratory would compare, which would help me decide on my future.

Did you apply to other places?
Last year, as a first time applicant, I applied to eight places and was only accepted into two. A few of the “rejections” were notices that they no longer had funding to support the program for that summer. Some other locations just never gave a response. This year, I applied to six locations, four of which were rejections, one (NIST) was an acceptance, and the last I informed I was no longer eligible for after accepting NIST. I was rather risky both years and probably should have applied to more places, but it worked out in the end.

Was the application difficult to do? Did you have help with this?
Almost all summer research applications are fairly straightforward and seem very similar to a college application. The hardest part is probably the personal statement, since most people aren’t used to writing about themselves. Janet McGlynn and Devon Fick from the Office of Undergraduate Education were a big help in the NIST Summer Undergraduate Research Fellowship (SURF) application process; they can even help you with general application questions like how to write a resume or personal statement. It can also be helpful to ask your friends or professors to read your personal statement and give suggestions.

What was your summer research project this year?
My project focused on finding an expansion for a fundamental solution to Laplace’s equation in rotationally-invariant cyclidic coordinates. Basically, Laplace’s equation is a fairly important partial differential equation in physics and you usually want to use a coordinate system that best matches your problem. Similar work has been done on the more well-known coordinate systems, like spherical or cylindrical coordinates, but very little has been done in these coordinates.

Who was your mentor for this project?
My mentor was Dr. Howard Cohl from the Applied and Computational Mathematics Division in the Information Technology Laboratory.

How much time do you put into this work?
We were full-time employees for eleven weeks, so we had to show up for eight hours a day, five days a week. I did take advantage of the various weekly seminars and special events, which helped break up the week into more manageable chunks.

Were you paid? Where did you live?
We were paid a stipend of $5500 ($500 per week). The program also provided free housing at a nearby apartment-style hotel with transportation to and from the NIST campus each day.

What academic background did you have before you started?
Before starting, I had taken three semesters of advanced math, physics, and computer science courses. The most relevant courses that helped with my research were Partial Differential Equations and Differential Geometry.

How did you learn what you needed to know for this project?
Anything I didn’t come in knowing, I had to learn on my own. The best sources I found were the internet and textbooks in the NIST Math Library. It’s crucial to double-check any of your sources, though. Over the course of my project, I found, and corrected, at least five errors in the literature, in both on-line and print sources.

What was the hardest part about your research?
The hardest part about my research, and probably research in general, is finding something new and interesting in so little time. Eleven weeks may seem like a lot of time, but you quickly find that a lot of time is lost going down the wrong path, usually due to a typo somewhere. The good news is that you eventually become an expert at what you’re doing: a derivation that originally took me a week to go through I could eventually do for other problems within a few hours.

What was the most unexpected thing?
I was originally assigned to a project that I quickly found was not suited to my interests. I spoke up to my advisor and we figured out a new project that was much more geared towards my skills and interests. My advisor was very accommodating and for that I am extremely thankful.

How does this research relate to your course work at UMBC?
Through research I was finally able to use the techniques and concepts I learned in the classroom. Key concepts for my research were Laplace’s equation and separation of variables from Partial Differential Equations and coordinate transformations and the metric tensor from Differential Geometry. That said, being involved in research also means learning new skills and concepts that you would likely never see in the classroom. Research is the process of continually learning and applying your knowledge.

What is your advice to other students about getting involved in research?
My advice is to start early and get to know your professors. Not only will they be writing your letters of reference for internship and graduate school applications, but some of them are just interesting people to talk to. Also, the more you interact with them, the more they can say about you. Professors who do research themselves may even be able to offer you a position in their own lab.

What are your career goals?
I hope to attend graduate school after graduating from UMBC to pursue a Ph.D. in Applied Mathematics. My long-term goal is to work at a university so I can be involved in both research and teaching.

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8/25/2021