CAREER: Combinatorial probability, limit shapes and enumeration

研究者将研究与二维晶格上的某些结构相关的组合模型，如多米诺骨牌平铺、杨氏表和交替符号矩阵。研究的一个重点将是枚举问题，其目标是推导出满足某些属性的组合对象数量的精确公式。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，当模型的大小变大时，在渐近极限中出现的几何形状。更一般地说，在概率设置下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关于与随机矩阵理论的联系的问题。研究者希望用来解决这些问题的技术工具箱是多种多样的，包括代数方法（如生成函数、线性代数）、解析方法（如变分演算）和一般概率技术。该项目的科学教育价值是多方面的。多年来，纯粹数学家对组合对象的兴趣一直很浓厚，因为它们固有的美和优雅的结构。但令人惊讶的是，这种看似“无用”的数学却被证明与物理学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“方冰”有关，这是统计物理学家研究的冰晶的简化模型，同时出现在计算矩阵行列式的算法的研究中，这是一个几乎在所有科学和工程学科中使用的无处不在的概念。而杨格图，起源于对称的数学研究，被发现与随机矩阵理论有关，随机矩阵理论是一个重要的数学理论，它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。从科学的角度来看，理论优雅和应用实用的结合使这些研究课题具有很强的吸引力，也非常适合作为研究者计划的教育活动的重点，这些活动的目标是吸引有前途的学生学习数学和科学，并促进公众对科学研究价值的认识。

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.