Algebraic Combinatorics and its Applications

提出的研究项目涉及三类代数组合数学问题。它们涉及舒伯特微积分的几个方面，以及它与表示论、逆边界问题、代数几何和物理学的联系。第一部分是关于某类网络的逆边界问题。这个问题是在解释、推广和简化与一般线性群和标准基的表示理论有关的代数和组合结构时出现的。它与Grassmann流形上的全正性的研究直接相关。第二部分集中讨论了量子Schubert演算中出现的组合和代数问题，即复标志流形和相应的Gromov-Witten不变量的量子上同调。第三部分研究了Schubert簇的光滑性的新方法。这位研究人员和他的同事建议如何在一般的根系环境中扩展模式避免的概念。拟议的研究项目的主要目标是调查几个最初来自不同领域的问题，如几何和物理，这些问题都具有离散的性质。这位研究人员提出了一种通过边界测量识别网络的方法。这些网络为计算机微芯片提供了一个简单的模型。这个问题可以用这样的话来表述：“如何通过外部检查来识别微芯片？”另一个问题与某些在数学物理和代数几何中起作用的几何不变量有关。这些不变量通常很难计算。这位研究人员和他的同事们提出了一种计算这类不变量的新的高效技术。最后一部分是关于光滑性这一经典问题的新方法。

The proposed research project is concerned with three classes of problems of algebraic combinatorics. They are related to several aspects of Schubert calculus and its links with representation theory, inverse boundary problems, algebraic geometry, and physics. The first part is devoted to the inverseboundary problem for certain class of networks. This problem emerged in an attempt to explain, generalize, and simplify algebraic and combinatorial constructions related to representation theory of general linear groups and canonical bases. It is directly linked to the study of total positivity onGrassmann manifolds. The second part focuses on combinatorial and algebraic problems came up in quantum Schubert calculus, which deals with quantum cohomology of complex flag manifolds and corresponding Gromov-Witten invariants. The third part is devoted to a new approach to smoothness of Schubert varieties. The investigator and his colleague suggest how to extend the notion of pattern avoidance in a general context of root systems. The main goal of the proposed research project is to investigate several problems that originally came from various areas, such as geometry and physics, and all share a discrete nature. The investigator suggests an approach to the problem of identification of networks by boundary measurements. These networks present a simple model for computer microchips. The problem can be rephrasedas follows: "How to identify a microchip by external examination?" Another problem is related to certain geometric invariants that play a role in mathematical physics and algebraic geometry. These invariants are usually extremely hard to calculate. The investigator and his colleagues suggest a new efficient technique for computing invariants of this kind. The last part concerns with a new approach to the classical problem of smoothness.