Grant: Discovery Launch Supplement

My research does not involve Indigenous communities, societies or individuals. My research focuses on geometric combinatorics, a growing field where combinatorics, discrete geometry, and group theory come together. My area of expertise involves blending combinatorial and geometric ideas within computational frameworks to obtain pioneering results used by mathematicians and physicists. Specifically, the vision of my Discovery Research is to achieve remarkable practical applications in algebraic geometry, commutative algebra, and quantum physics through groundbreaking discoveries related to symmetric geometric objects, including permutahedra, and infinite algebraic structures such as infinite Coxeter groups. My strategy for achieving this vision entails three interconnected components, each aiming for a well-defined long-term objective related to a specific object with close relations to the other two. First, I aim to analyze lineup polytopes in order to establish a comprehensive theory of symmetric polytopes to discover new exclusion principles in quantum theories. Second, my goal is to find and study examples of high-dimensional Lorentzian Coxeter groups, and ultimately characterize the obstacles to their Lorentzian property thereby solving a long-standing problem in hyperbolic geometry. Finally, my program aims to clarify a link between the geometry and combinatorics of Coxeter groups by analyzing the oriented matroid structure of infinite Coxeter arrangements to uncover the bridge between limit roots and infinite reduced words. I highlight the importance of this research program by illustrating the novel use of symmetric polytopes in quantum chemistry, and their potential for future applications. Indeed, Density Functional Theory (DFT) is an efficient computational method for computing electronic structure properties, for which Kohn and Pople received the 1998 Nobel Prize in Chemistry. Traditional DFT concentrates on electron density instead of the complex many-body wavefunction, while Reduced Density Matrix Functional Theory (RDMFT) is a theoretical framework that examines the electron density matrix. RDMFT precisely characterizes excited states, including their energies and properties, facilitating the study of excited-state phenomena and spectroscopic properties that are challenging for conventional DFT. However, more efficient computational methods are needed before RDMFT can replace DFT as the effective electronic structure method. Recently, we introduced highly symmetric polytopes called lineup polytopes to make RDMFT a practical method. Understanding them in depth will enhance the efficiency and accuracy of RDMFT in calculating excited states, optical gap, and finite temperature behavior of large quantum systems. A long-term ambition is to broaden the capabilities of RDMFT to incorporate spin and temperature, thereby unlocking its potential for quantum information theory by analyzing the corresponding symmetric lineup polytopes.