Combinatorial models in algebraic geometry and commutative algebra

机器人和计算机视觉中的许多基本问题都是高维几何问题。例如，简单机械臂的位置可以通过给出其3个关节的坐标来指定，总共有9个坐标。然而，可实现位置的坐标必须满足一定的关系，例如，机器人臂段的长度是固定的。因此，我们可能会发现，可实现的手臂构型空间是一个位于9坐标空间内的复杂7维物体。这个机器人“相空间”的几何特性告诉我们机器人的重要特征。例如，如果空间是断开的，我们可能只能通过拧开组件并在新的配置中重新组装它们来将手臂从一个位置移动到另一个位置-这不是我们想要的机器人！然而，很难从视觉上推断出7维空间（在更现实的机器人中，维度会更高），因此我们严重依赖于代数工具，如上同态来计算几何的各个方面。这些代数计算可以被解释为机器人设计的有用几何信息。不幸的是，这些代数计算本身就很困难。因此，我们寻找所涉及的代数对象的离散模型。例如，在一些特别重要的空间上，如Grassmannians，我们可以有效地用某些有限的盒子网格标记上同类，并通过计算用满足一些简单规则的整数填充网格的方法的数量来计算这些上同类的乘积。本研究的一个主要目标是将这些离散模型扩展到更广泛的重要空间类别。虽然上同环使得计算复杂空间的重要信息变得可行，但它们也忽略了可能需要的其他信息。因此，另一个主要目标是为上同调的更丰富的类似物开发离散模型，计算更多的几何，但相应的更难以处理。特别是“椭圆”上同调，目前几乎在任何空间都很难理解，尽管人们相信更好的理解将导致弦理论和当代物理学其他方面的重大进展。通过开发新的离散模型，提出的研究计划将使计算重要空间的几何性质变得更加容易。该计划的影响将在代数、几何和离散数学的理论和实践进步中感受到，因为这些领域相互交叉。通过离散数学和计算机代数方面的培训，HQP将为学术界、数据科学和计算机视觉领域的职业生涯做好充分的准备。

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.