Algebraic Variety

Algebraic varieties, sets defined by polynomial equations, are central objects of study in mathematics with increasing relevance across diverse scientific fields. Current research focuses on leveraging their geometric properties for applications in machine learning (e.g., analyzing neural network expressivity and generalization), computer vision (e.g., camera resectioning and point cloud registration), and robotics (e.g., manipulator cuspidality analysis). These applications utilize advanced techniques from algebraic geometry, including Gröbner basis computations and Riemannian optimization, to solve complex problems and provide theoretical insights into model behavior and data structure. The resulting advancements offer both improved algorithms for practical tasks and a deeper understanding of fundamental mathematical structures.