Resolutions of positivity in Hopf algebras

数学的中心主题是正性：解释正数的出现，特别是当它们是整数时。这是因为它通常表明了解释现实世界或其他科学（如计算机科学或量子物理学）中现象的底层结构。相反，可以使用组合规则显式地计算底层结构的面，其中对抽象对象进行计数以给出正整数。在我的代数组合研究领域，随着新的理论方向的发展，一些长期存在的正性问题最近得到了解决。因此，现在是攻击剩余部分的理想时机。我将努力解决其中3个问题。1)舒伯特微积分中的正性：找到两个舒伯特多项式相乘的组合规则。这些多项式是由Lascoux-Schutzenberger在1982年引入的。2)色对称函数的正性：证明了Stanley-Stembridge（1993）所推测的，对于某些无抓痕图，色多项式的泛化（称为色对称函数）是初等对称函数的正线性组合。3)对角谐波空间的正性：寻找Garsia-Haiman在20世纪90年代初引入的对角谐波空间的梯度Frobenius特征的Schur函数组合规则。这将通过调查和应用未开发的工具和函数（如基本滑动多项式）以及锻造和应用新的研究领域（如我与haglund - loto - mason发现的准对称舒尔函数）的结合来实现。初级研究人员将参与所有项目的各个方面，从在SAGE和Maple中生成数据，到数据分析和形成猜想，再到证明结果，并通过文章和演讲传播它们。这将为他们提供宝贵的培训和有影响力的研究，为他们的职业生涯奠定基础。解决这三个问题中的任何一个都将对我的研究领域具有重大意义，因为它们都是活跃的研究领域。在一般的科学水平上，这些决议也将影响涉及底层结构的相关领域，如弦理论（通过量化与舒伯特多项式相关），或解决P与NP的克莱千年问题（通过几何复杂性理论与舒尔函数相关）。在全球范围内，这三个问题的解决将进一步巩固加拿大在全球组合学前沿的地位，继Robinson和Tutte等先驱之后，吸引世界各地的关注和人才。

A central theme in mathematics is positivity: to explain the occurrence of positive numbers, especially when they are integers. This is because it often indicates an underlying structure explaining a phenomenon in the real world or other sciences such as computer science or quantum physics. Conversely, facets of an underlying structure can be calculated explicitly using a combinatorial rule where abstract objects are counted to give positive integers. In my research area of algebraic combinatorics, with new directions of theory, a number of long-standing positivity problems have recently been resolved. Hence now is the ideal time to attack some of the remaining ones. I will work towards resolving 3 of these. 1) Positivity in Schubert Calculus: Find a combinatorial rule for multiplying together two Schubert polynomials. These polynomials were introduced by Lascoux-Schutzenberger in 1982. 2) Positivity in chromatic symmetric functions: Prove that for certain claw-free graphs, the generalization of the chromatic polynomial, called the chromatic symmetric function, is a positive linear combination of elementary symmetric functions, as conjectured by Stanley-Stembridge in 1993. 3) Positivity in the space of diagonal harmonics: Find a combinatorial rule in terms of Schur functions for the bigraded Frobenius characteristic of the space of diagonal harmonics, introduced by Garsia-Haiman in the early 1990s. This will be achieved through a combination of investigating and applying under-explored tools and functions, such as fundamental slide polynomials, and forging and applying new areas of research, such as my discovery of quasisymmetric Schur functions with Haglund-Luoto-Mason. Junior researchers will be involved in all aspects of all of the projects from generating data in SAGE and Maple, to data analysis and forming conjectures, to proving results, and disseminating them with articles and talks. This will give them valuable training, and impactful research on which to found their careers. Solving any of these 3 problems would have major significance in my field, as they are all active areas of research. At a general scientific level, the resolutions will also impact related fields involving the underlying structures, such as string theory (related to Schubert polynomials through quantization), or the Clay Millennium Problem of resolving P vs NP (related to Schur functions through Geometric Complexity Theory). At a global level the resolutions of these 3 problems will further reinforce Canada's position at the global forefront of combinatorics, following such pioneers as Robinson and Tutte, and attracting attention and talent from around the world.