Combinatorial models in algebraic geometry and commutative algebra

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Many fundamental issues in robotics and computer vision are questions of high-dimensional geometry. For example, positions of a simple robotic arm might be specified by giving the coordinates of each of its 3 joints, for 9 coordinates in total. The coordinates of achievable positions, however, must satisfy certain relations, forced for example by the robot's arm segments being of fixed length. Hence, we might find that the space of achievable arm configurations is a complicated 7-dimensional object sitting inside a 9-coordinate space. Geometric properties of this robotic "phase space" tell us important features of our robot. For example, if the space is disconnected, we might only be able to move the arm from one position to another position by unscrewing the components and reassembling them in the new configuration -  not what we want from robots! However, it is difficult to reason visually about a 7-dimensional space (and in a more realistic robot, the dimension would be much higher still), so we rely heavily on algebraic tools such as cohomology to calculate aspects of the geometry. These algebraic calculations can then be interpreted as useful geometric information about robot design. Unfortunately, these algebraic calculations are themselves very difficult. We therefore look for discrete models of the algebraic objects involved. For example, on some particularly important spaces such as Grassmannians, we can usefully label cohomology classes by certain finite grids of boxes and compute products of these cohomology classes by counting the number of ways to fill the grids with whole numbers satisfying some simple rules. One major goal of this research is to extend these discrete models to broader classes of important spaces. While cohomology rings make it feasible to compute important information about complicated spaces, they also neglect other information that might be needed. Hence, another major goal is to develop discrete models for richer analogues of cohomology that compute more of the geometry but are correspondingly more difficult to work with. "Elliptic" cohomology, in particular, is currently extremely hard to understand for almost any space, although it is believed that a better understanding would lead to major advances in string theory and other aspects of contemporary physics. By developing new discrete models, the proposed research program will make it easier to compute geometric properties of important spaces. The impacts of this program will be felt in theoretical and practical advances in each of algebra, geometry, and discrete mathematics, as these fields cross-fertilize each other. With training in discrete mathematics and computer algebra, HQP will be well-prepared for careers in academia, data science, and computer vision.