CAREER: Combinatorial probability, limit shapes and enumeration

职业：组合概率、极限形状和枚举

• 批准号: 0955584
• 负责人: Dan Romik
• 金额: $ 40万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0955584&HistoricalAwards=false
• 关键词: CAREER; Combinatorial; probability; limit; shapes

The investigator will study combinatorial models that arise in connection with certain structures on two-dimensional lattices, such as domino tilings, Young tableaux and alternating-sign matrices. One focus of the research will be enumeration questions, where the goal is to derive exact formulas for the number of combinatorial objects satisfying certain properties. Another emphasis will be on a probabilistic analysis of random combinatorial objects, and in particular their limit shapes, which are geometric shapes that arise in the asymptotic limit when the size of the model becomes large. More generally, the study of combinatorial models in a probabilistic setting raises many fascinating questions, for example questions about limiting fluctuations and about connections to random matrix theory. The technical toolbox of techniques that the investigator hopes to employ to attack such problems is very varied and involves both algebraic methods (e.g. generating functions, linear algebra), analytic methods (for example the calculus of variations) and general probabilistic techniques.The scientific and educational value of the project is manifold. The combinatorial objects of interest have delighted pure mathematicians for many years for their inherent beauty and elegant structure. But also, amazingly, such seemingly "useless" mathematics has turned out to have deep connections to branches of physics and to some very applied mathematical disciplines like probability theory and random matrix theory. Thus, for example, alternating-sign matrices are related to "square ice", a simplified model for an ice crystal studied by statistical physicists, while at the same time appearing in the study of an algorithm for computing matrix determinants, which are a ubiquituous concept used in practically all the sciences and engineering disciplines. And Young tableaux, which originated with the mathematical study of symmetry, have been found to be related to random matrix theory, an important mathematical theory which has its origins in attempts by nuclear physicists to model the complex interactions in the nuclei of heavy elements. The combination of theoretical elegance and applied utility makes these research topics highly attractive from a scientific standpoint, and also very suitable as the focus for educational activities planned by the investigator, which would have as their goals to attract promising students to mathematics and the sciences and to promote public appreciation of the value of scientific research.

研究人员将研究与二维晶格上的某些结构有关的组合模型，例如多米诺骨牌，Young tableaux和交替符号矩阵。研究的一个重点将是枚举问题，其目标是推导出满足某些性质的组合对象的数量的精确公式。另一个重点将是随机组合对象的概率分析，特别是它们的极限形状，这是几何形状，出现在渐近极限时，模型的大小变得很大。更一般地说，在概率环境中对组合模型的研究提出了许多有趣的问题，例如关于限制波动和与随机矩阵理论的联系的问题。研究人员希望用来解决这些问题的技术工具箱是多种多样的，涉及代数方法（例如生成函数，线性代数），分析方法（例如变分法）和一般概率技术。多年来，感兴趣的组合对象因其内在的美和优雅的结构而使纯数学家们感到高兴。但令人惊讶的是，这种看似“无用”的数学与物理学的分支以及一些非常实用的数学学科（如概率论和随机矩阵理论）有着深刻的联系。因此，例如，交替符号矩阵与“方形冰”有关，这是统计物理学家研究的一种简化的冰晶模型，同时出现在计算矩阵行列式的算法研究中，矩阵行列式是一个无处不在的矩阵行列式。使用在几乎所有科学和工程学科中的概念。杨格图起源于对对称性的数学研究，现已发现与随机矩阵理论有关。随机矩阵理论是一种重要的数学理论，起源于核物理学家试图模拟重元素原子核中复杂的相互作用。理论的优雅和应用的实用性相结合，使这些研究课题从科学的角度来看非常有吸引力，也非常适合作为重点的教育活动计划的调查员，这将作为他们的目标，以吸引有前途的学生数学和科学，并促进公众欣赏的价值科学研究。