Computational Studies in Polyhedral Convexity: Lattice Points and Triangulations

De Loera0073815 调查研究的组合和代数性质的最佳细分，覆盖，和三角形的凸多面体。 他开发了计算这种最佳对象的算法。 探讨的最优性标准包括单形的数量最小化，单形的长度或面积的总和，以及单形的平均体积。 他还开发软件，用于计算低维多面体内的所有格点，并计算其整数外壳。 该技术还允许快速计算体积。 具体的问题被认为是评估软件的效率，例如，n个点的最佳安排，在一个球体的固定半径，最大限度地提高了网格点内的凸船体的数量。 该软件的算法是新技术的适应，由于Barvinok，这是基于覆盖多面体与单模单形。 该项目还包括凸性，组合学，整数和线性规划，交换代数，复杂性和密集的计算机实验的方法。 这项工作的结果应感兴趣的整数规划，组合数学和符号代数计算。 非正式地说，这个项目的第一部分可以被认为是试图理解如何有效地将物体（如立方体和多边形）分解或分解为基本块或块。 这也许让人想起了制作拼图游戏。 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片段。 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。 研究中的许多理论问题都是由计算机图形学和计算机可视化（通过设计用于建模的经济网格），数据安全和计算（在RSA加密的背景下，用于互联网交易）和运筹学（通过某些技术解决整数规划时，预期的不确定性水平）中的问题所激发的。 学生培训是该项目的一个重要组成部分。

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.