Tools for Positivity in Algebraic Combinatorics

非负整数不变量为理解复杂的代数或几何对象提供了重要的途径。例如，多项式的次数和曲面上的孔数。这个项目的目标是开发通用的方法，以获得对数学中几个不同领域中出现的非负整数不变量的详细理解。这可能为量子信息论、纽结理论、物理学和信号处理的发展奠定基础。特别是，这个项目为张量分解问题提供了潜在的新见解，张量分解问题本质上是从混合信号中恢复单个信号的问题，并在医学、计算机视觉、化学和快速矩阵乘法中应用。代数组合数学中的正性问题要求为几何和表示理论中的非负量找到正组合公式。这个项目的目标是开发工具来解决在两个活跃的研究领域--麦克唐纳理论和几何复杂性理论--中出现的正性问题。麦克唐纳多项式是一族两参数对称多项式，它与几何学、物理学和纽结理论等多个领域有着密切的联系。这一领域的重大突破来自于Macdonald正性猜想的证明，该猜想表明与Macdonald多项式有关的重要结构系数是非负的。对这些系数给出一个积极的组合解释仍然是一个基本的开放问题。几何复杂性理论是利用代数几何和表示论来研究复杂性理论中的P与NP及相关问题的方法。Kronecker问题是表示理论中的一个基本问题，被认为对这种方法很重要。Kronecker问题要求一个正的组合公式，将对称群的两个不可约表示的张量积分解为不可约。这个项目将进一步发展非对易Schur函数的理论，这是一个解决正性问题的强大工具，特别是专注于Macdonald多项式和Kronecker问题的应用。

Nonnegative integer invariants provide an important way of understanding complicated algebraic or geometric objects. Examples include the degree of a polynomial and the number of holes of a surface. The goal of this project is develop general methods to obtain a detailed understanding of nonnegative integer invariants arising in several different areas of mathematics. This may form the foundation for developments in quantum information theory, knot theory, physics, and signal processing. In particular, this project offers potential new insights into the tensor decomposition problem, which is essentially the problem of recovering individual signals from a mixture of signals and has applications in medicine, computer vision, chemistry, and fast matrix multiplication.Positivity problems in algebraic combinatorics ask to find positive combinatorial formulae for nonnegative quantities arising in geometry and representation theory. The goal of this project is to develop tools to solve positivity problems arising in two areas of active research, Macdonald theory and geometric complexity theory. Macdonald polynomials are a two-parameter family of symmetric polynomials, which have ties to many areas including geometry, physics, and knot theory. A major breakthrough in this area came with the proof of the Macdonald positivity conjecture, which showed that important structure coefficients related to Macdonald polynomials are nonnegative. It remains a fundamental open question to give a positive combinatorial interpretation of these coefficients. Geometric complexity theory is an approach to P versus NP and related problems in complexity theory using algebraic geometry and representation theory. A fundamental problem in representation theory, believed to be important for this approach, is the Kronecker problem, which asks for a positive combinatorial formula for decomposing the tensor product of two irreducible representations of the symmetric group into irreducibles. This project will further develop the theory of noncommutative Schur functions, a powerful tool for solving positivity problems, particularly focusing on applications to Macdonald polynomials and the Kronecker problem.