102 Research Reflection by Emil Geisler

In my sophomore year (Fall 2020), I began my first undergraduate research experience under Dr. Ganesh Gopalakrishnan in computer science. The general project goal was to develop efficient neural network compression techniques. Once a neural network is trained, it is typically efficient at computing an output for any given input. However, the limiting constraint in settings like mobile devices tends to be the memory cost of storing the entire neural network architecture. Due to this high memory cost, there is a need to compress the neural network without losing specificity and accuracy. Specifically, my goal was to study and optimize rank selection algorithms for tensor decomposition techniques. This was my first experience with independent college-level academic work. Due to my lack of familiarity with machine learning and software tools, along with the challenges of the COVID-19 pandemic at the time, this research project was especially challenging for me to produce results. However, I was able to gain knowledge about deep learning by watching a lecture series on modern architectures, including convolutional neural networks and graph neural networks. Furthermore, I learned about tensors in the setting of computer science and optimization and their main methods of compression. Additionally, I developed skills required to read and interpret scientific research papers. As part of my work with Dr. Gopalakrishnan, I read several current research papers and presented findings on novel techniques in model compression to the research group.

In Spring 2021, I began working with Dr. Sean Howe on a mathematics research project in the area representation stability. It has been previously shown that a certain mathematical object (the cohomology of complex configuration space when viewed as a representation of the symmetric group) stabilizes as its degree tends toward infinity, but very little is known about its stable asymptotic structure. My research has been concerned with describing this stable structure. Due to a connection between geometry and arithmetic established by Grothendieck in the 1960’s, there is a way to phrase the geometric problem of interest as the weighted average of a family of random variables. Using this framework, it is possible to explicitly solve for the simplest parts of the desired stable structure. To understand the problem, I studied Representation Theory by Fulton and Harris and Algebraic Topology by Hatcher to gain the background needed to understand the definition of the stable structure. I was concurrently enrolled in graduate-level algebra and undergraduate topology courses, which supplemented the learning required for my research. At first, I was able to write a computer program which determines the stable structure in relation to the exterior powers of the standard representation of the symmetric group. This was the simplest non-trivial computation and a natural starting point. After studying more prerequisite mathematics, I developed an algorithm in Summer 2022 to compute the desired stable structure for any representation of the symmetric group. The results are now solely limited by computational time. I computed coefficients of the first 100 simplest cases, producing novel data which can be used by the mathematical community to better understand the desired stable structure. The field of representation stability is relatively new, having been developed within the last 10 years. Complex configuration space represents the simplest non-trivial example of representation stability. Before the results of my research, it was known that certain properties of this space stabilized as its degree tended toward infinity, but relatively little was known about this stable structure. My novel computational results provide an explicit example of a stable structure which provide a starting point for future theorems. I developed conjectures arising from my algorithmic results and am currently attempting to construct proofs. While the theoretical results are still preliminary, it is exciting to be working on understanding an unsolved math problem. I am currently preparing to submit my results to the Journal of Experimental Mathematics and to undergraduate mathematics journals so other researchers can benefit from my computational results.

In Summer 2021, I attended a summer math research experience for undergraduates (REU) at UC Davis where I worked on a problem in topological data analysis on the evolution of non-segmented RNA based viruses. One of the primary challenges of treating RNA viruses is their rapid and unpredictable mutations. Non-segmented RNA viruses are subject to random pointwise mutations over generations, but also more complex mutations broadly termed recombination events where entire portions of the RNA sequence are rearranged, replaced, or a combination of both. The standard method for modelling evolution in RNA viruses is phylogenetic trees which is effective for simpler pointwise mutations. However, phylogenetic trees have proven ineffective at modelling recombination events. Persistent homology, a technique from topological data analysis, has previously been proposed as a method to detect recombination events in RNA viruses. During the 8-week program, I studied the previous applications of persistent homology and identified which homological statistics best detect recombination, especially in large datasets.

My undergraduate research experiences have been instrumental in my decision to pursue a Ph.D. In my first years of college, I had still not decided whether to pursue a PhD in computer science or mathematics and sought advice to help inform my decision. For years, I have dreamed of one day pursuing a PhD in pure math and becoming a mathematical researcher. However, I felt discouraged from pursuing this career path because of my perception that math could only be pursued by a limited number of mathematical prodigies and career opportunities were limited. Furthermore, I was concerned that math research was primarily focused on solving Hilbert’s 23 problems. As I talked with pure mathematicians and conducted research with Dr. Howe, I learned there are exciting, rich new areas of mathematics which are relatively unstudied and ample career opportunities are available for pure math PhD graduates both inside and outside academia. Furthermore, I realized that to succeed in a PhD program, it is essential to study a subject which is deeply motivating and inspiring. While pursuing research in mathematics and computer science, I have found myself to be deeply motivated to solve problems in pure mathematics. I have set my sights on a career in academia, as a researcher and instructor in the field of pure mathematics. While I decided to pursue math, I intend to leverage my strong computer science background and skills to aid in mathematical problem solving.