CAREER: Geometric and Topological Combinatorics

几何学和拓扑学是数学的分支，它们提供了测量全局现象而不是局部现象的方法。如果问题的解决取决于其信息的聚合，那么几何方法在检测全局障碍物时是有用的。从这个意义上讲，数学及其应用中的许多问题都是全球性的，从数据科学到经济学。这项资助的研究将开发新的拓扑和几何方法来解决更远的问题，主要是离散的、非连续的问题。这有望在组合学、几何学和拓扑学的融合中获得新的见解。所有阶段的学生都将参与这项资助所支持的研究工作。研究计划的总体目标如下：在等变拓扑的应用中，通常要求某个参数必须是一个素数幂，而方法在这个素数幂的情况下失败。最近对PI的研究表明，人们可以通过拓扑和组合技术的综合有效地规避这种主功率要求。这将得到进一步发展。拓扑方法的应用带来了组合问题的几何推广。为了理解拓扑技术的局限性，我们必须研究这些类型的问题，其中几何结果与它们的组合特殊情况有很大的偏差。因此，中心目标将是将刚性组合结果与它们的柔性几何对应物区分开来。除了证明某些组合问题的解的存在性之外，人们常常对量化解的空间有多丰富感兴趣。一般的拓扑方法将被开发，为给定的组合问题的解空间的拓扑结构提供下界。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Geometry and topology are mathematical branches that provide methods that measure phenomena that are global instead of local. If a problem's resolution depends on the aggregate of its information then geometric methods are useful in detecting global obstructions. Numerous problems across mathematics and its applications are global in this sense, ranging from data science to economics. The research supported by this grant will develop new topological and geometric methods to tackle problems further afield, primarily discrete, non-continuous problems. This promises new insights at the confluence of combinatorics, geometry, and topology. Students at all stages will be involved in the research effort supported by this grant.Among the general goals of the research program are the following: In applications of equivariant topology one often requires that a certain parameter has to be a prime power, and methods fail outside of this prime power case. Recent research of the PI has suggested that one may effectively circumvent this prime power requirement via a synthesis of topological and combinatorial techniques. This will be further developed. The application of topological methods brings about geometric generalizations of combinatorial problems. To understand the limitations of topological techniques one has to study those types of problems, where geometric results deviate considerably from their combinatorial special cases. A central goal will thus be to delimit rigid combinatorial results from their flexible geometric counterparts. In addition to showing the existence of a solution to some combinatorial problem, one is often interested in quantifying how rich the space of solutions is. General topological methods will be developed that provide lower bounds for the topology of the space of solutions of a given combinatorial problem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.