Science
102 Research Reflection by Emil Geisler
Emil Geisler
Faculty Mentor: Sean Howe (Mathematics, University of Utah)
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.