The Mars Rover Sojourner is an autonomous robotic vehicle that was used by NASA for space exploration on Mars. The Mars Sojourner landed in a location called Ares Vallis in 1997 where it explored and took several photos and collected data. Google Maps and network routing programs, often use the popular Dijkstra's algorithm used to find the shortest and most cost-effective routes between various nodes on a graph. Here, we used the free open Cyberbotics Webots code to simulate the Mars Sojourner which was designed to successfully navigate over the rocky terrains of Mars. Robotic simulation software like Webots offers an excellent testing platform to study the stability and routing of autonomous navigation vehicles on different terrains prior to their deployment in outer space. The robotic Mars Sojourner was equipped with a GPS sensor which kept provided measurements of position using geodetic coordinates involving latitude and longitude. We used the Haversine formula to calculate distances between various places it traversed on Mars. We used a python code to map the shortest optimized route to go from point A to point B on Mars using Dijkstra’s iterative algorithm. Such studies are important for future development of robot motion controller software that can be effectively used for cost effective autonomous navigation in outer space.

Determining the genus of a graph—the smallest surface on which it can be embedded without edge crossings—is a fundamental problem in graph theory with applications in circuit design, transportation networks, and data visualization. Existing genus computation methods often require extensive case-by-case analysis, and the problem is NP-hard in general. To address this, we developed and analyzed PAGE, a novel algorithm that efficiently determines a graph’s genus by enumerating non-backtracking closed directed trails and optimizing search iterations. We established an upper bound on its runtime as O(n(4m/n)^(n/t)) for graphs of girth t and implemented performance optimizations using hash sets for cycle tracking and adjacency list storage for rotation systems. We then conducted empirical comparisons against existing methods, including SAGEMath and MULTI_GENUS, demonstrating PAGE’s superior scalability, particularly for large 3-regular cage graphs. These results suggest PAGE’s potential to improve genus computation, contribute to open problems in topological graph theory, and enhance applications in fields requiring efficient graph embeddings.

Image processing using dithering finds important applications for data compression, data encryption, data security and cryptography. Dithering is used for the design of high-quality printers, computers and game consoles. Digitization and compression of images often involve reduction of color palette which gives rise to color bands due to quantization errors. Dithering involves the application of noise to randomize quantization errors which helps preserve key features of images when converted to black and white and other color reduced formats. We applied mathematical matrix-algebra based methods for image digitization, image recognition and image processing. We wrote Python programs to study image quality before and after application of different algorithms for dithering including thresholding, randomization, ordered methods like Bayer’s method, void and cluster methods, error diffusion methods like Floyd-Steinberg techniques, etc. This project offered excellent hands-on opportunities to integrate programming in python with matrix algebra methods for image processing and provides insights into how different dither algorithms enhance visual capabilities along with compressing image size. Dithering introduces some textural contours and color shifts but preserves most features in image data. We also find that in addition to enhancing image quality, dithering modulations offer powerful techniques for digital watermarking and image embedding indicating their key role in data security.

In this work, we present two new geometric properties of ellipses that emerged while studying problems related to mass transportation. The first property states that if we shrink the ellipse proportionally and consider any tangent to the smaller ellipse, the two points where this tangent intersects the original ellipse always sum to the point of tangency. This reveals a fundamental symmetry in how ellipses scale and interact with their tangents. The second property describes a special set of interior points that arise by tracing how tangents from a fixed boundary point behave across a family of smaller ellipses. We prove that this set forms another ellipse, exactly half the size of the original, centered at the midpoint between the origin and the fixed boundary point. These findings offer new insights into classical geometry and may have potential applications for problems in mathematics and other fields. 

In this project, we study the order polytopes of generalized snake posets, a particular family of polytopes (geometric shapes with flat sides) defined in recent work by von Bell and coauthors. We specifically investigate triangulations of these polytopes, which are subdivisions of the polytopes into simplicies (arbitrary-dimensional generalizations of the triangle). To do so we examine their secondary polytopes. The secondary polytope of a polytope is a related polytope in which each vertex corresponds to a triangulation of the original polytope and each edge corresponds to a bistellar flip (a specific type of transformation between triangulations), capturing the combinatorial relationships between different triangulations in a geometric object. Von Bell and coauthors conjecture that the secondary polytopes of order polytopes of snake posets have multiple unusually nice properties similar to those of the associahedra, a well-known family of polytopes that admit nice combinatorial descriptions and appear in many areas of mathematics. The overall goal of this project is to prove these conjectures. Collaborating with fellow undergraduates Molly Bradley (University of Pennsylvania), Mario Tomba (Dartmouth), and Katherine Tung (Harvard), I have written code to compute details of the smallest examples of these secondary polytopes, proved related theorems/results, proposed some new conjectures based on observed patterns in the examined examples, and made partial progress towards proving multiple of the von Bell conjectures. This project is ongoing, and we continue to explore multiple ways to approach the conjectures and more broadly understand the secondary polytopes.