Computational Studies in Polyhedral Convexity: Lattice Points and Triangulations

De Loera0073815 研究人员研究凸多面体的最佳细分、覆盖和三角剖分的组合和代数性质。 他开发了计算此类最佳对象的算法。 所探索的最优性标准包括单纯形数量的最小化、单纯形的长度或面积的总和以及单纯形的平均体积的最小化。 他还开发了用于计算低维多胞形内所有格点并计算其整数外壳的软件。 该技术还可以快速计算体积。 考虑特定问题来评估软件的效率，例如在固定半径的球体中n个点的最佳排列，从而最大化其凸包内的格点数量。 该软件的算法是对 Barvinok 提出的新技术的改编，该技术基于用单模单纯形覆盖多面体。 该项目还包括凸性、组合学、整数和线性规划、交换代数、复杂性和密集计算机实验的方法。 这项工作的结果应该对整数规划、组合学和符号代数计算感兴趣。 非正式地说，该项目的第一部分可以被认为是尝试了解如何有效地将对象（例如立方体和多边形）分解或分解为基本块或碎片。 这也许让人想起拼图游戏。 分解中使用的块例如是四面体、三角形或更小的立方体。 有效分解的一个例子是使用最少数量的碎片。 该项目的第二部分涉及建立实用的计算机软件，用于计算规则边界内规则分布的点。 规则分布点的例子是原子或晶体的排列。 正在研究的许多理论问题都是由计算机图形学和计算机可视化（通过用于建模的经济网格设计）、数据安全和计算（在互联网交易中使用的 RSA 加密的背景下）和运筹学（通过在预期存在不确定性水平时解决整数规划的某些技术）中的问题引发的。 学生的培养是该项目的重要组成部分。

De Loera0073815 The investigator studies the combinatorial and algebraic properties of optimal subdivisions, coverings, and triangulations of convex polytopes. He develops algorithms for the computation of such optimal objects. Criteria of optimality that are explored include minimization of the number of simplices, of the total sum of lengths or areas of simplices, and of the average volume of the simplices. He also develops software for counting all lattice points inside a low-dimensional polytope and for computing their integer hulls. The technique also allows the fast computation of volumes. Specific problems are considered to assess efficiency of the software, for example the optimal arrangements of n points, in a sphere of fixed radius, that maximize the number of lattice points inside their convex hull. Algorithms for the software are adaptations of new techniques, due to Barvinok, that are based on covering polyhedra with unimodular simplices. This project also includes methods from convexity, combinatorics, integer and linear programming, commutative algebra, complexity, and intensive computer experimentation. The results of this work should be of interest in integer programming, combinatorics, and symbolic-algebraic computing. Informally speaking, the first part of this project can be thought of as an attempt to understand how to break or decompose objects, such as cubes and polygons, into elementary blocks or pieces efficiently. This is perhaps reminiscent of creating jigsaw puzzles. The blocks used in the decomposition are, for instance, tetrahedra, triangles, or smaller cubes. An example of efficient decomposition is to use the smallest number of pieces. The second part of the project involves establishing practical computer software for counting regularly distributed points within regular boundaries. Examples of regularly distributed points are arrangements of atoms or crystals. Many of the theoretical questions under study are motivated by problems in computer graphics and computer visualization (via the design of economic meshes for modeling figures), data security and computation (in the context of RSA encryption, which is used in internet transactions), and operations research (via certain techniques for solving integer programs when levels of uncertainty are expected). The training of students is an important component of the project.