"Geometric morphometric data derived from biological shapes are widely used for species classification, but different representation frameworks may lead to different pre-dictive performance. In this study, we compare Functional Data Geometric Morphometrics (FDGM) and classical Geometric Morphometrics (GM) for classifying three shrew species (Suncus murinus, Crocidura monticola, and Crocidura malayana) from Peninsular Malaysia. We construct a workflow for extracting skull outlines from images and representing them using pseudo-landmarks across three craniodental views. Classifiers are trained using bootstrap aggregation, and predictions are combined using majority voting, out-of-bag (OOB) aggregation, and a proposed Bayesian probability aggregation approach. Model performance is evaluated using repeated Monte Carlo resampling. The results show that bootstrap aggregation consistently improves classification accuracy over single-classifier predictions for both FDGM and GM, with the dorsal view yielding the highest accuracy overall. While GM slightly outperforms FDGM at the single-classifier level in the dorsal view, FDGM achieves comparable or higher accuracy than GM under aggregation, particularly with the proposed Bayesian scheme, which produces probability estimates that are reasonably well calibrated and competitive with standard majority voting. These findings indicate that functional representations, when combined with the proposed probabilistic aggregation approach, can be as effective as classical landmark-based approaches for morphometric classification tasks."

The purpose of this study is to see whether restricting 4-moves to a single orientation is enough to reduce any knot to the trivial knot, and to find knots that cannot be trivialized by a single oriented 4-move. Exploring this question is important because it provides insight into the structure of the full 4-move conjecture, which asks whether every knot is 4-move equivalent to the unknot and said conjecture is currently one of the central open problems in knot theory. The Dabkowski-Sahi invariant R4(K) is a quotient of the knot group that is invariant under 4-moves, restricting to positive or negative crossings yields oriented versions DS+(K) and DS-(K). To gather data, we use the computational system GAP to search for finite non-abelian groups generated by a single conjugacy class satisfying the oriented DS relation abab equals baba, which serve as targets for obstructions. We then search for knots whose knot groups admit surjective homomorphisms onto these target groups with all Wirtinger generators mapping into the “correct” conjugacy class. Based on computations completed to date, the right-handed trefoil knot group surjects onto the alternating group A4, the smallest DS- target group of order 12, providing a verified obstruction showing the trefoil cannot be trivialized by negatively oriented 4-moves alone. Further findings will clarify the role of orientation in 4-move equivalence and constrain the possible structure of any future proof of the full 4-move conjecture.

It is well-known that when taken modulo a prime integer, the entries in Pascal’s triangle exhibit a fractal behavior. Taking an integer modulo another integer is an operation that finds the remainder after dividing one number by another. The purpose of this study was to investigate more general phenomenon of this sort, both by taking the entries of Pascal’s triangle modulo a composite integer and by considering the behavior of the Trinomial triangle (whose entries are the trinomial coefficients) taken modulo various integers, including but not limited to, prime integers, prime powers, and composite integers. Similar, but more complicated, patterns emerge in these other settings, as can be seen with the aid of computer-generated examples. The goal of this project was to find mathematical explanations for these experimental results.

"Interval graphs are a well-studied class of graphs used to depict how intervals on a line overlap. Each interval is represented as a vertex and the connection or edge between two vertices correspond to two intervals intersecting. Thus, interval graphs provide an elegant framework for analyzing pairwise relationships. Because of this simple but powerful structure, they have widespread applications in optimizing scheduling, mapping DNA, modeling food webs, and solving other problems where conflict or overlap matters. However, standard, low dimensional interval graphs are limited to recording pairwise overlaps. To model higher-dimensional phenomena and analyze complex interactions among multiple pieces of data simultaneously, we can extend this concept to a simplicial complex and clutters which will essentially functions as a higher-dimensional interval graph. While interval graphs have many well-known characterizations through vertex orderings, underclosed, chordality, and intersection properties, their higher-dimensional analogues are less fully understood. Building on previous work by Dr. Bennet Goeckner from the University of San Diego on vertex orderings and chordality in interval graphs, our goal is to extend that our understanding to simplicial complexes and clutters. Specifically, how properties such as chordality and underclosedness translate from their well-understood, lower-dimensional versions to higher dimensions like if how any properties are constant, changed, or lost. By exploring these relationships, we hope to better understand how interval-based structures behave in higher dimensions and contribute to the broader study of how combinatorial, topological, and algebraic ideas interact."