The quantum many-body problem lies at the heart of modern physics and chemistry, yet its complexity continues to challenge both theory and computation. In this talk, I will provide a brief introduction to the quantum many-body problem and outline several mathematical questions that may help advance the field. Particular emphasis will be placed on coupled cluster–based approaches, embedding methods, and emerging quantum computational strategies. Throughout the presentation, I will highlight how mathematical analysis and algorithmic development can contribute to improving accuracy, scalability, and conceptual understanding in computational many-body theory.