Using a Mathematical Lens for Better Understanding 

Independent Study/Research in Mathematics

By definition, an independent study is exactly that - a chance for a student to independently study problems or topics that are of interest. The role of the advisor changes from independent study to independent study, but in general, it is to help students when they run into difficulty and to direct the studies of the students in "promising" directions. 

During the first two weeks of the semester (or, preferably during the previous semester), the student wishing to pursue an independent study should approach a faculty member and express interest in doing an independent study. If the faculty member is willing, then both the faculty member and the student together should determine a direction for the independent study and write a short description of the planned study. The faculty member will then discuss the independent study with the rest of the department and must receive the permission of the department to advise the independent study.

 

Honors Projects in Mathematics

The designation of Honors applies to a student who has completed an outstanding independent study project and is in good standing in the department.  An outstanding independent study project is usually at the 400-level and lasts two semesters. Furthermore, in order for an independent study project to be considered outstanding, it must meet certain criteria.  Please refer to the links above for more description of the honors process and requirements.

Xiao Receives Honors for Damped Wave Equation Research 

  • Zhengyi Xiao ’21 (top right in the computer screenshot) developed a more advanced technique to solve the damped wave equation, advised by mathematician Prof. Satbir Malhi, using the general idea of C0-semigroup operator theory.  After graduation,  Zhengyi will be pursuing a masters of science in mathematical modeling and scientific computing, a field where emphasis is placed on the formulation of real-world problems to math problems, and then using analytical and numerical techniques to find the solutions. Zhengyi Xiao ’21 (top right in the computer screenshot) developed a more advanced technique to solve the damped wave equation, advised by mathematician Prof. Satbir Malhi, using the general idea of C0-semigroup operator theory. After graduation, Zhengyi will be pursuing a masters of science in mathematical modeling and scientific computing, a field where emphasis is placed on the formulation of real-world problems to math problems, and then using analytical and numerical techniques to find the solutions.

Research in mathematical modeling at F&M gave Zhengyi Xiao ’21 more than a glimpse into the life of a graduate student:  it provided the achievement of successfully completing an honors project to graduate with distinction in mathematics, and submission of a paper for publication in a professional journal.

The senior investigated, “A class of finite difference methods for solving inhomogeneous damped wave equations,” with the guidance of Prof. Satbir Malhi. In this project, a class numerical technique of finite difference method was studied for the solution of the one-dimensional inhomogeneous damped wave equation.

The damped wave equation is an important evolution model and is widely used by physicists and engineers in describing the propagation of water waves, sound waves and electromagnetic waves.  Zhengyi says, “For instance, in the case of a 3D tsunami wave, we would like to know the size and structure of the damping force to bring the amplitude of the tsunami to a safe level before it hits the shore. Our research focused on developing numerical solutions to such problems.”     

  • Figure 1. The behavior of the solution by the implicit finite difference method. Figure 1. The behavior of the solution by the implicit finite difference method.

In Fall 2019, Zhengyi completed MAT338, Computational Mathematics, with Prof. Christina Weaver. She informed the student that Prof.Malhi was working on a project focusing on using numerical analysis to solve the damped wave equation. Zhengyi found research intriguing, and approached Prof. Malhi’s to join his research.

The team made significant progress during Spring 2020, and Zhengyi decided to keep working on the project, securing approval from the Math Department to pursue the research as an honors project for Fall 2020.  The designation of Honors applies to a student is in good standing with the department and who has completed an outstanding independent study project, usually at the 400-level and lasting two semesters,.

At the end of Fall 2020, Zhengyi defended his honors thesis orally before a panel or “board” of faculty members. A typical defense consists of three parts. After an introduction by the advisor, the student gives a presentation lasting approximately 50 minutes on his/her project.  The audience is then allowed and encouraged to ask questions.  When the general question period ends, everyone leaves except for the board, the student, and the advisor. The board members then ask the student questions about the thesis for as long as necessary to determine the qualification for Honors.   Zhengyi also was required to submit a written thesis, describing the project and the results, finalizing the process to be conferred with the distinction of honors in mathematics.

(Click on the buttons above for more details on the honors project process, honors defense, and past honors theses.)

 

  • Figure 2. The absolute errors generated by different methods. The last two (yellow and purple curves) are methods developed by Zhengyi and Prof. Malhi. Figure 2. The absolute errors generated by different methods. The last two (yellow and purple curves) are methods developed by Zhengyi and Prof. Malhi.

The project and process were challenging and difficult in many ways for the senior, particularly with embarking on completing an honors thesis during the pandemic; but, both Zhengyi and Prof. Malhi found it a satisfying and rewarding experience, none-the-less.

“Due to the effect of COVID-19, our research went remote. Instead of meeting in person, we met 2-3 days per week via Zoom. There is no doubt that this made our research activities much more challenging. For example, the central part of our research was to derive the math formulae for a given problem.  If we could have worked in-person, we would have borrowed a classroom so that we could take advantage of blackboards. The larger a blackboard is, the more information and thoughts we can put on board and discuss together,” explains the student researcher.  “Also, when questions arose, I could have normally just knocked on my professor’s door to discuss my inquiries with him. However, facilities and communications were limited during COVID-19. We adjusted ourselves to these changes. We used an application on the iPad to write math and shared our work via Zoom. We also increased the frequency of our virtual meetings.”  

Zhengyi comments, “I had a chance to get to know and learn materials that are not generally taught in most of the undergraduate courses. Moreover, I had an opportunity to experience a graduate student’s life: a professor gives you a topic and a reading list, and you try your best to solve the research question and meet with your professor to discuss your progress and questions. And you will find, you cannot discover the solution to the problem on Google!  Last, but not least, if one wants to pursue graduate school, one needs to learn and exercise how to read academic papers. Believe it or not, it is not easy!”

“As professor, I embrace undergraduate research as a way to enhance student learning. I find it a useful tool to attract talented students, to improve their engagement in academics, and to prepare them for graduate school,” reflects project advisor Prof. Malhi.  “I love working with student researchers. It keeps me connected with students and engaged in high-level intellectual collaborations. For instance, working with Zhengyi Xiao has broadened my research interest to numerical analysis.”

 “My research with Professor Malhi aligned with my future graduate program, incorporating the methods to solve the partial differential equation numerically that I learned in the project. The MSc program will be a continuous study of our research” stated Zhengyi. “Doing research helped me discover my interest in computational mathematics and motivated me to pursue my career in this direction. I also submitted my first math journal paper in collaboration with Prof. Malhi.”

Zhengyi’s final advice: “If you are interested in being a student researcher, you should take advanced-level courses in your area of interest. There will be many new things you will learn and you should be ready to face these challenges. Also, you should find your research advisor early.  Do not wait until the last moment when the professor might be busy with something else or he/she is on sabbatical.  And a tip for doing research virtually is making a daily schedule. Prioritize tasks, then complete in order from urgent to important and then less important.  I use this strategy and it works for me.”

 

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