Resolutions of positivity in Hopf algebras

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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.