Editing Differential geometry (section)

=== Modern differential geometry (1900–2000) ===

The subject of modern differential geometry emerged from the early 1900s in response to the foundational contributions of many mathematicians, including importantly [[Analysis Situs (paper)|the work]] of [[Henri Poincaré]] on the foundations of [[topology]].<ref name="dieudonne">Dieudonné, J., 2009. A history of algebraic and differential topology, 1900-1960. Springer Science & Business Media.</ref> At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as [[Hilbert's program]]. As part of this broader movement, the notion of a [[topological space]] was distilled in by [[Felix Hausdorff]] in 1914, and by 1942 there were many different notions of manifold of a combinatorial and differential-geometric nature.<ref name="dieudonne" />

Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation <math>g</math> for a Riemannian metric, and <math>\Gamma</math> for the Christoffel symbols, both coming from ''G'' in ''Gravitation''. [[Élie Cartan]] helped reformulate the foundations of the differential geometry of smooth manifolds in terms of [[exterior calculus]] and the theory of [[moving frames]], leading in the world of physics to [[Einstein–Cartan theory]].<ref name="fre">Fré, P.G., 2018. A Conceptual History of Space and Symmetry. Springer, Cham.</ref><ref name="spivak2" />

Following this early development, many mathematicians contributed to the development of the modern theory, including [[Jean-Louis Koszul]] who introduced [[connection (vector bundle)|connections on vector bundles]], [[Shiing-Shen Chern]] who introduced [[characteristic class]]es to the subject and began the study of [[complex manifold]]s, [[W. V. D. Hodge|Sir William Vallance Douglas Hodge]] and [[Georges de Rham]] who expanded understanding of [[differential forms]], [[Charles Ehresmann]] who introduced the theory of fibre bundles and [[Ehresmann connection]]s, and others.<ref name="fre" /><ref name="spivak2" /> Of particular importance was [[Hermann Weyl]] who made important contributions to the foundations of general relativity, introduced the [[Weyl tensor]] providing insight into [[conformal geometry]], and first defined the notion of a [[gauge (mathematics)|gauge]] leading to the development of [[gauge theory]] in physics and [[gauge theory (mathematics)|mathematics]].

In the middle and late 20th century differential geometry as a subject expanded in scope and developed links to other areas of mathematics and physics. The development of [[gauge theory]] and [[Yang–Mills theory]] in physics brought bundles and connections into focus, leading to developments in [[gauge theory (mathematics)|gauge theory]]. Many analytical results were investigated including the proof of the [[Atiyah–Singer index theorem]]. The development of [[complex geometry]] was spurred on by parallel results in [[algebraic geometry]], and results in the geometry and global analysis of complex manifolds were proven by [[Shing-Tung Yau]] and others. In the latter half of the 20th century new analytic techniques were developed in regards to curvature flows such as the [[Ricci flow]], which culminated in [[Grigori Perelman]]'s proof of the [[Poincaré conjecture]]. During this same period primarily due to the influence of [[Michael Atiyah]], new links between [[theoretical physics]] and differential geometry were formed. Techniques from the study of the [[Yang–Mills equations]] and [[gauge theory]] were used by mathematicians to develop new invariants of smooth manifolds. Physicists such as [[Edward Witten]], the only physicist to be awarded a [[Fields medal]], made new impacts in mathematics by using [[topological quantum field theory]] and [[string theory]] to make predictions and provide frameworks for new rigorous mathematics, which has resulted for example in the conjectural [[mirror symmetry (string theory)|mirror symmetry]] and the [[Seiberg–Witten invariant]]s.