CAREER: Discrete Structures in Continuous Contexts

研究者的研究背后的统一思想是隔离控制或产生于连续数据的组合结构。 基于这个想法的项目范围从多面体的计算几何等变代数几何。 第一个例子包括各种开放的问题，关于凸多面体球体上的最短路径的计算复杂性，以及更深层次的几何控制这种复杂性。 多面体方法的基本度量组合，在这种情况下出现有潜在的应用图形着色问题，建设周期代表特征类分段线性流形，并在计算生物学的算法问题。 中心问题揭示了凸性和多面体在组合学和拓扑学的本质。 在从连续对象中提取离散现象的保护伞下的其他项目中，研究人员和他的同事研究代数簇退化的组合学，通常存在李群作用。 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被加数，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的技术定义的。 自然界和数学中的许多系统都涉及连续变化的量，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等同于将有限或离散的信息片段从底层的连续框架中分离出来--类似于识别材料的固相、液相和气相。 在研究者的研究中，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，特别是计算机精确算法计算的形式，往往是通过离散数据表征连续系统的副产品。 调查员使用他的研究作为背景，在其中培养一个垂直整合的指导环境，模仿在实验室科学中更常见的互动咨询计划，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.