The recent developments in the research area have seen a significant shift towards the integration of advanced mathematical frameworks with biological applications, particularly in the quantification and analysis of shape variations and topological features. Researchers are increasingly leveraging differential geometry, machine learning, and computational topology to develop robust methods for shape analysis, which are not only applicable to anatomical surfaces but also extend to broader domains. Notably, there is a growing focus on the development of algorithms that can efficiently compute topological invariants and shape descriptors, enhancing the interpretability and robustness of shape quantification methods. These advancements are paving the way for more sophisticated analyses of complex biological structures, such as proteins and DNA, and are enabling the detection of novel genomic features in pangenome graphs. Additionally, the computational efficiency of these methods is being significantly improved through the use of parallel computing and optimized algorithms, making them more practical for large-scale datasets. Overall, the field is moving towards more integrated and computationally efficient approaches that bridge mathematical theory with practical applications in biology and beyond.