A Unified Survey of Grand Challenges and Open Problems at the Intersection of Physics, Mathematics and Computation
Mohamed Fakhry Ahmed Salem · 2025
The current report can be considered a coherent and transdisciplinary review of the main unsolved challenges and curiosities that characterize the current boundaries of modern science and engineering. It systematically synthesizes foundational problem lists from both the twentieth and twenty-first centuries, most prominently Hilbert’s problems , the Millennium Prize problems , and Smale’s issues , with contemporary outcome-oriented Grand Challenges outlined by organizations such as the National Academy of Engineering . The scope of the surveyed foundational questions in theoretical computer science and mathematics contains the P versus NP problem and the Riemann Hypothesis; frontier questions in theoretical physics, particularly quantum gravity, dark matter, and the nature of the glass transition; and outstanding issues in applied disciplines like computational electromagnetics, solid mechanics, and biophysics. Revealing the thematic links and conceptual frameworks common to these various fields, primarily the ubiquity of nonlinearity, the emergence of phenomena at varying scales of space or time, and the limitations of computing power, this work provides a unified picture of the boundaries of human cognition. It is hoped that it will become the authoritative and basic source for researchers interested in traveling and helping to address the questions that constantly face the entire scientific world.