In 1904 the French mathematician Henri Poincaré requested if the three dimensional sphere is characterised as the distinctive merely connected three manifold. You could also research statistical mechanics, which covers matters equivalent to inference, multivariate advanced techniques, state variables, fluctuations, equilibrium methods, transport fashions, dynamical ordering and phase transitions, and emergent conduct in non-equilibrium programs.
During the Golden Age of Islam , particularly through the ninth and 10th centuries, mathematics noticed many important innovations building on Greek mathematics: most of them embrace the contributions from Persian mathematicians akin to Al-Khwarismi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.
Superior research of mechanics involves quantum mechanics and relativity, covering matters such as electromagnetism, the Schrödinger equation, the Dirac equation and its transformation properties, the Klein-Gordon equation, pair production, Gamma matrix algebra, equivalence transformations and negative energy states.
The trendy examine of space generalizes these ideas to include greater-dimensional geometry, non-Euclidean geometries (which play a central role typically relativity ) and topology Amount and space each play a role in analytic geometry , differential geometry , and algebraic geometry Convex and discrete geometry have been developed to unravel issues in quantity theory and purposeful analysis but now are pursued with an eye on purposes in optimization and computer science Inside differential geometry are the concepts of fiber bundles and calculus on manifolds , in particular, vector and tensor calculus Inside algebraic geometry is the outline of geometric objects as solution sets of polynomial equations, combining the concepts of amount and space, and also the study of topological teams , which mix construction and space.