- Please provide a brief description of your STEP Signature Project. Write two or three sentences describing the main activities your STEP Signature Project entailed.
For my STEP Signature Project, I participated in the Knots and Graphs math research group, under the supervision of Professor Sergei Chmutov. The main components of the research were (1) individually reading math papers and exploring related concepts, (2) meeting with a small group of 3-4 peers to discuss our findings, and (3) presenting our findings to a larger group of 12-15 fellow undergraduate students.
- What about your understanding of yourself, your assumptions, or your view of the world changed/transformed while completing your STEP Signature Project? Write one or two paragraphs to describe the change or transformation that took place.
The most important thing that my project made me realize is that math is inherently a community activity. When starting the project, I though that the best way for me to do math was to sit alone in a room and read and doodle and ask myself questions and practice. However, by the end of this project, I realized that many more ideas and perspectives can be discovered when working with a group.
Another valuable lesson that I learned was the vast difference between reading a paper or textbook and actually absorbing the information to a level that permits discussion, presentation, and synthesis of new related information. It is one thing to be able to defend each step of a logical proof as valid, but quite another to be able to understand why the proof is laid out in the way that it is, and why the author may have thought to explore this. The responsibility of a presenter is to tell a meaningful story with their presentation, and this process of summarizing information and seeing high-level motivating structure can be invaluable in allowing existing proofs to inspire elements of new proofs.
- What events, interactions, relationships, or activities during your STEP Signature Project led to the change/transformation that you discussed in #2, and how did those affect you? Write three or four paragraphs describing the key aspects of your experiences completing your STEP Signature Project that led to this change/transformation.
Before this project, most of my mathematical experience consisted of going to a lecture, followed by doing homework independently. There were occasions of my peers and I helping each other with homework, but my experience had predominantly been independent: almost all interactions were either giving hints on specific pre-defined homework problems, or surface-level musings about what could be explored.
This all changed this summer. During the project, I was forced to look not at specific problems with a clearly defined box of mathematical tools, but at a hard, general problems where it was unknown if a solution existed, let alone what it was. Having multiple people to provide a constant stream of new ideas made the brainstorming work much easier.
Teamwork also provided motivation and accountability to the work: I constantly wanted to come up with a cool new proof so that I could share with my teammates, and I didn’t want to let them down by not being prepared. I remember one Saturday in particular, my teammate John and I were sitting in an otherwise empty classroom in Smith Lab, and I had come up with a way to uniquely represent an immersed plane curve (the main focus of our studies), by a particular method of labeling faces and vertices, followed by listing the labels in a particular order. I wasn’t sure about the details of the proof, but I was confident in the result. I was explaining my reasoning to John, and we ended up listing out all of the cases we needed to check, and we arrived at that result. Following that though, John asked if labeling and listing the vertices was even necessary; could we use only the faces? We then checked the relevant cases and made some arguments about that, and arrived at a new, more elegant way of representing a plane curve by a simple list of numbers. Now, not only could we uniquely describe different curves, but we could also pinpoint exactly what the differences were between the curves in this representation. This was a result that neither of us could have come up with individually, but that required both of our inputs bouncing off one another.
Another valuable experience for me was preparing for presentations to the larger research group. I got into a good habit of annotating the papers that I read, asking questions, double-checking things for myself, and writing comments about the overall structure. One of the papers that I read, I was able to condense a densely and formally written 12-page math paper into an “executive summary” that fit on the front and back of one sheet of printer paper, which I was then able to expand into an hour-long presentation. Simply preparing to transfer the information to other people with this process of “summarize then re-expand” really made me understand the information for myself.
I also gained a good amount of experience giving math presentations. I gave at least three presentations to the smaller research group, along with one presentation to a bunch of mixed undergraduate and graduate students for the “What is..?” seminar. Professors in both contexts had good positive and constructive feedback that will make me a better technical presenter in the future, and which also greatly increased my confidence in my ability to stand in front of a group of people who may be smarter than me, and present to them something they might not have heard about before.
- Why is this change/transformation significant or valuable for your life? Write one or two paragraphs discussing why this change or development matters and/or relates to your academic, personal, and/or professional goals and future plans.
My presentation skills and outlook on the community of mathematics will help me in my future career as a mathematician: the criteria for professors getting tenure is often composed of (1) research, (2) teaching, and (3) service, and all three of these were expanded for me this summer. I gained research skills by practicing extracting the essence of what a paper is trying to say, as well as by practicing research as a group. I gained teaching skills with my technical presentations, and the process for preparing for those will inform my process for preparing to teach classes as a TA this year. And an important aspect of service as a professor at a university is knowing the community around you, and I got good practice this summer learning about the mathematical community that surrounded me, and how I can interact with a mathematical community in general.
My actions this summer didn’t change my career plans much, but rather cemented them: I am more certain than ever that I want to study theoretical mathematics for the rest of my life, and the encouragement and positive experiences of this summer give me something to point to that makes me secure in that decision.