CAREER: Discrete Structures in Continuous Contexts

研究人员研究背后的统一想法是分离出支配或产生于连续数据的组合结构。基于这一想法的项目范围从多面体的计算几何到等变代数几何。第一个例子包括关于凸多面体球面上最短路径的计算复杂性的各种公开问题，以及控制这种复杂性的更深层次的几何。在此背景下产生的度量组合学背后的多面体方法在图着色问题、表示分段线性流形上的特征类的圈的构造以及计算生物学中的算法问题上具有潜在的应用。这些中心问题从根本上阐明了组合学和拓扑学中凸性和多面体的本质。在从连续物体中提取离散现象的其他项目中，研究人员和他的同事研究了代数簇的简并的组合学，通常是在存在李群作用的情况下。这种退化中的不可约分量可以索引组合学和表示论公式中的求和，从而给出公式正性的几何解释。退化本身是使用符号计算代数中的技术来定义的。自然界和整个数学中的许多系统都涉及到连续变化的量，例如体积或温度。令人惊讶的是，在这样的系统中寻找秩序有时相当于将有限或离散的信息片段-类似于识别一种材料的固体、液体和气相-从底层的连续框架中分离出来。在研究人员的研究中，揭示的离散结构产生了具有深刻美学和理论意义的抽象数学奖励。然而，具体的应用，特别是以计算机精确算法计算的形式，经常作为用离散数据表征连续系统的副产品出现。研究人员以他的研究为背景，以实验室科学中更常见的互动建议计划为蓝本，建立一个垂直整合的指导环境，旨在让本科生、研究生和博士后学生参与他们作为教师和研究人员的发展。

The unifying idea behind the investigator's research is to isolate combinatorial structures that govern or arise from continuous data. Projects based on this idea range from computational geometry of polyhedra to equivariant algebraic geometry. First examples include a variety of open problems concerning the computational complexity of shortest paths on convex polyhedral spheres, as well as the deeper geometry controlling this complexity. The polyhedral methods underlying the metric combinatorics that arises in this context have potential applications to graph-coloring questions, to the construction of cycles representing characteristic classes on piecewise linear manifolds, and to algorithmic questions in computational biology. The central issues shed fundamental light on the nature of convexity and polyhedrality in combinatorics and topology. In other projects under the umbrella of extracting discrete phenomena from continuous objects, the investigator and his colleagues study the combinatorics of degenerations of algebraic varieties, often in the presence of Lie group actions. The irreducible components in such degenerations can index summands in formulas from combinatorics and representation theory, thereby giving geometric explanations for the formulas' positivity. The degenerations themselves are defined using techniques from symbolic computational algebra. Many systems in nature and throughout mathematics involve continuously varying quantities, such as volume or temperature. Surprisingly, finding order in such systems is sometimes equivalent to isolating finite or discrete pieces of information---analogous to identifying the solid, liquid, and gas phases of a material---from the underlying continuous framework. In the investigator's research, the revealed discrete structures produce abstract mathematical rewards of deep aesthetic and theoretical significance. However, concrete applications, especially in the form of exact algorithmic calculations by computer, often arise as byproducts of characterizing continuous systems by way of discrete data. The investigator uses his research as a context in which to foster a vertically integrated mentoring environment, modeled on interactive advising programs more common in laboratory sciences, aimed at involving undergraduate, graduate, and postdoctoral students in their development as teachers as well as researchers.