Mathematical Sciences: Algebraic Methods in Optimization

该奖项支持Alexandre Barvinok教授在组合学和代数计算问题的算法构建方面的研究。主要目标是通过探索基本的代数结构来找到这些问题的计算复杂性的非平凡上界。目前正在调查三个具体问题。第一个主题包括指数和在广泛的计算问题中的应用。第二部分讨论对称群表示理论在组合优化中困难问题中的应用。第三个主题涉及整数规划中的量词消除，这是一个关于多面体中整点的特定悬而未决的问题。所开发的算法对于简化为硬枚举的现实世界问题具有潜在的实际应用价值。这项研究属于组合学的一般领域。组合学试图找到有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。

This award supports the research of Professor Alexandre Barvinok on the construction of algorithms for computational problems in combinatorics and algebra. The main goal is to find non-trivial upper bounds for the computational complexity of these problems by exploring the underlying algebraic structure. Three specific topics are being investigated. The first topic includes the applications of exponential sums to a wide range of computational problems. The second deals with applications of the representation theory of the symmetric group to hard problems in combinatorial optimization. The third topic involves quantifier elimination in integer programming, a specific unresolved question concerning integral points in polyhedra. The algorithms developed have potential practical applications to real world problems that reduce to hard enumeration. This research is in the general area of Combinatorics. Combinatorics attempts to find efficient methods to study how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.