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.

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. 

Vibrio parahaemolyticus is a gram-negative marine bacterium that causes acute gastroenteritis in humans generally following the consumption of raw or undercooked shellfish. Mice are highly resistant to many human gut pathogens, including Vibrio, Salmonella, and Shigella spp., which has hindered our understanding of bacterial pathogenesis, immunity, and the development of therapeutics. Inflammasomes are cytosolic innate immune complexes that assemble in response to pathogen infection or harmful stimuli. Once the inflammasome is assembled, inflammatory caspases like caspase 1 are activated, driving a lytic cell death termed pyroptosis and the maturation and release of pro-inflammatory cytokines (i.e., IL-1β, IL-18). Inflammasomes have recently emerged as a necessary mediator of mouse resistance to Shigella and Salmonella, suggesting that inflammasomes may also be the cause of mouse resistance to V. parahaemolyticus. Consistent with that possibility, our preliminary data suggest that inflammasomes prevent intestinal inflammation in mice infected with V. parahaemolyticus, although the mechanism of protection is unknown. To identify the inflammasome(s) responsible for mouse resistance, I reconstitute specific murine inflammasomes in HEK293T cells, which lack most components of the inflammasome pathway. Then, I assess their activation in response to V. parahaemolyticus infection. Our previous findings demonstrated that V. parahaemolyticus robustly activates the mouse NAIP-NLRC4 inflammasome. However, we unexpectedly observed that V. parahaemolyticus infection also induces inflammasome activation in HEK293T cells even in the absence of NAIP-NLRC4 inflammasome reconstitution. This suggests the presence of an inflammasome-sensor in 293T cells that is responsive to V. parahaemolyticus infection. I am currently using inflammasome inhibitors and gene knockouts to identify this unknown inflammasome, which will ultimately aid in our understanding of host factors that mediate host defense against V. parahaemolyticus.