Mathematical Sciences: Algebraic Methods in Optimization

该奖项支持教授亚历山大Barvinok对组合数学和代数计算问题的算法建设的研究。主要目标是通过探索底层的代数结构，找到这些问题的计算复杂性的非平凡上界。目前正在研究三个具体问题。第一个主题包括指数和的应用范围广泛的计算问题。第二部分讨论对称群的表示理论在组合优化中的应用。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未解决问题。开发的算法有潜在的实际应用真实的世界的问题，减少到硬枚举。 这项研究是在组合数学的一般领域。组合数学试图找到有效的方法来研究如何安排离散的对象集合。离散系统的行为对现代通信极为重要。例如，大型网络的设计，如电话系统中的网络设计，以及计算机科学中的算法设计，都要处理离散的对象集，这就需要使用组合研究。

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.