MSRI-UP 2020/2023

The titles of the projects I led were "Dessin d'Enfants from Cartographic Groups" and "To and From 2-Generated Groups and Origamis." Below are the abstracts of the students' written reports: 

• Dessin d'Enfants from Cartographic Groups:  Our main task given any triple from $S_n$ with specified conditions to draw a unique bipartite graph which can be drawn on a compact, connected Riemann surface in such a way that its cartographic group is generated by the triple.  In particular, we focused on drawing such Dessin d'Enfants when the Riemann surface has genus 1 or greater by focusing on examples which appear in the L-Series and Modular Forms Database (LMFDB).

• To and From 2-Generated Groups and Origamis: Starting from Square One:  In this research, we focus on the geometric construction of origami in detail. Initially, we construct various origami, by considering different examples of 2-generated groups. Conversely, we begin with an arbitrary collection of squares, glued together to form an origami and determine the corresponding transitive subgroup of $S_n$.

If you are interested in watching these students final presentations you can find them here.

Fun Facts:

• William Sabalan is now a graduate student at UC Riverside (my undergraduate institution) and an NSF GRFP recipient.

• Nicholas Arosamena is now a graduate student at Brown University.

• Elisa Rodriguez was a participant in the EDGE 22 program (the same year I was a mentor for the program) and is now a graduate student.

The two main areas of the projects I led were Topological Time Series Analysis and Topological Dimensionality Reduction. Below are the abstracts of the student's written reports: 

• Using Persistent Cup Products for Dissonance Detection: Quasiperiodic time series are those whose sliding window point clouds are dense in high-dimensional tori. Their topological structure can be captured via persistence diagrams, allowing for analysis of quasiperiodicity. There are, however, time series whose sliding window point clouds are not dense in tori -- and hence not quasiperiodic -- but have the persistence diagrams of one. In this study, we propose an algorithm that incorporates both persistent cohomology and cup products, introducing a persistent cup product that distinguishes between quasiperiodic and non-quasiperiodic time series. By utilizing this algorithm, we achieve a more precise detection of quasiperiodicity. Additionally, we apply this algorithm to the tritone interval to display its use for dissonance detection, as well as add levels of noise to a tritone audio file to demonstrate the robustness of the algorithm.

• (Quasi)Periodicity Unraveled: Topological Decoupling of Quasiperiodic Videos: This research presents a method for analyzing quasiperiodic videos by leveraging the principles of topological decoupling. By unraveling the intricate spatial patterns inherent in these videos, we appeal to persistent homology to identify and track persistent topological features over time. Our findings demonstrate the efficacy of topological decoupling in extracting meaningful spatiotemporal features, which can be harnessed for various applications including de-noising, anomaly detection, and video synthesis. This work bridges the fields of topology and video processing, providing valuable insights into the nature of quasiperiodic phenomena and unlocking new avenues for interpreting and manipulating complex visual patterns. This research explores the behavior of oscillators within the video context. Specifically, the focus is on a video consisting of two oscillators that continuously move back and forth. The objective is to investigate the manipulation of these oscillators, aiming to stop one while allowing the other to continue its motion.

• Manifold Modeling of Pentagon Spaces Using Laplacian Eigenfunctions: The algorithmic sampling of conformation spaces is a problem with applications across computational chemistry and biology. Often these spaces represent the configurations of a molecule at different energies. We present an algorithm for sampling the moduli space of pentagons with mathematical justification. Furthermore, we use persistent cohomology and Eilenberg-MacLane spaces to identify the underlying manifold of said moduli spaces. We use said identification as well as eigenfunctions of the Laplacian to create best fit manifolds. In the process we recover a 2D visualization of our space. The developed pipeline allows for the visualization of complicated high dimensional spaces in lower dimension.

If you are interested in watching these students final presentations you can find them here.

Fun Facts:

• Alpha Recio  is now a graduate student at Boston University. 

• Martin Martin is now a graduate student at the University of Washington. 

• Austin Mbaye is now a graduate student at Northeastern University.