Computational Ridgidity and Motion Planning

研究者继续研究由Karoly Bezdek教授开始的Kneser-PoulsenConjecture。这个猜想说，如果欧几里得平面上的任何有限的圆盘集合被重新排列，使得任何一对中心之间的距离不增加，那么联合的面积不会增加，相交的面积也不会减少。他们已经解决了平面上的猜想，但还有许多更高维度的问题有待解决。例如，在三维空间中，有一个候选的构型，它可能是类似语句的反例，并且在磁盘的并集和磁盘的交集的情况下，可能存在一些有趣的差异。另一个主要课题是研究平面上被锁定的连杆，即它们只能移动一小部分而不交叉。框架和张拉整体理论在这里起着重要的作用，可以用来给出合理的准则来检测这种联系何时被锁定。这是研究者Erik Demaine和Gunter Rote的继续工作，他们最初解决了木匠规则问题，该问题表明任何多边形弧都可以在不产生任何自交的情况下打开。研究者研究一些关于离散物体几何的基本问题。正如物理学提出了关于物质、空间和宇宙的本质以及它们必须遵守的物理定律的基本问题一样，几何学提出了关于几何物体如何相互作用以及它们满足的隐含约束的基本问题。当圆心被移开时，圆盘并集的面积是如何变化的？研究者（与Karoly Bezdek一起）已经证明，该区域在飞机上的表现（它增加或保持不变）与预期的一样，但在太空中的情况并不那么清楚。机械臂在平面上什么时候是刚性的，什么时候可以打开？研究者（与Erik Demaine和Gunter Rote一起）已经证明了双臂在飞机上张开是预期的，但是更复杂的联系呢？作为粒状材料的圆球填料的刚性如何？调查人员是这方面的专家，但对于装有少量磁盘的容器的刚性，存在一些微妙的问题。被柔性曲面包围的体积是常数，那么其他几何不变量呢？这些都是具体的、有形的物体，但可以通过适当的几何洞察力来理解。这些问题对细胞生物学、蛋白质折叠、运动学和颗粒材料等广泛的学科都是相关的，而且可能非常有用。例如，人们普遍认为细胞的几何结构与其功能有很大关系。因此，适当的离散对象的刚性是相关的。正如物理学和数学一再证明的那样，如果问题中肯，应用就会随之而来。

The investigator continues work on the Kneser-PoulsenConjecture that was started with Professor Karoly Bezdek. Thisconjecture says that if any finite collection of disks in theEuclidean plane is rearranged so that the distance between anypair of centers does not increase, then the area of the uniondoes not increase and the area of the interesection does notdecrease. They have solved the conjecture in the plane, but manyhigher-dimensional questions remain. For example, there is acandidate for a configuration that might be a counterexample forthe analogous statement in three-space, and there are someinteresting possible differences between the case for a union ofdisks and the intersection of disks. Another major topic is thestudy of linkages in the plane that are locked in the sense thatthey can only move a small amount without crossing. The theoryof frameworks and tensegrities plays an important role here andcan be applied to give reasonable criteria to detect when such alinkage is locked. This is continuing work of the investigator,Erik Demaine, and Gunter Rote, who originally solved thecarpenter's rule problem, which states that any polygonal arc canbe opened without creating any self-intersection. The investigator works on certain basic, fundamentalquestions about the geometry of discrete objects. Just asphysics asks fundamental questions about the nature of matter,space, and the universe, and the physical laws they must obey,geometry asks fundamental questions about how geometric objectsinteract and the implicit constraints they satisfy. How does thearea of the union of round disks change as the centers are movedapart? The investigator (with Karoly Bezdek) has shown that thearea behaves (it increases or stays the same) as expected in theplane, but the situation in space is not so clear. When is arobot arm in the plane rigid, and when can it be opened? Theinvestigator (with Erik Demaine and Gunter Rote) has shown thatarms open as expected in the plane, but what about morecomplicated linkages? What can be said about the rigidity ofpackings of round balls as a granular material? The investigatoris an expert in such matters, but there are delicate questionsabout the rigidity in a container with a small number of disks.The volume enclosed by a flexible surface is constant, but whatabout other geometric invariants? These are concrete tangibleobjects, but accessible to the appropriate geometric insight.These questions are both relevant and potentially quite usefulfor subjects as wide-ranging as cell biology, protein folding,kinematics, and granular materials. For example, it is widelybelieved that the geometric structure of a cell has a great dealto do with its function. So the rigidity of appropriate discreteobjects is relevant. As has been shown over and over again inphysics and mathematics, if the questions are to the point,applications follow.