 differential geometry of various groups of transformations broader than Euclidean motion developed rapidly, especially the differential geometry of h-dimensional spaces. Stimulated by the emergence of the general theory of relativity, geometric investigations in this direction were initiated by T. Levi-Civita, E. Cartan, and H. Weyl.
Because of the development of more general viewpoints of set theory and the theory of functions of a real variable, the theory of analytic functions at the end of the 19th century lost its previous position as the core of all mathematical analysis. However, it continued to be developed intensively both in accordance with its intrinsic requirements and because of its newly discovered connections with other branches of analysis and natural science. Especially significant in this last direction was the elucidation of the role of conformal mappings in the solution of boundary value problems for partial differential equations (for example, Dirichlet’s problems for the Laplace equation), in the study of planar flows of an ideal fluid, and in problems of the theory of elasticity.
Klein and H. Poincare created the theory of automorphic functions in which Lobachevskii’s geometry found remarkable applications. C. E. Picard, Poincare, Hadamard, and E. Borel worked out in depth the theory of entire functions, which in particular made it possible to derive the theorem of the density distribution of primes that was mentioned above. Poincare, Hilbert, and others developed the geometric theory of functions and the theory of Riemann surfaces. Conformal mappings found application in aeromechanics (N. E. Zhukovskii, S. A. Chaplygin).
As a result of systematic development of mathematical analysis, a new branch of mathematics—the theory of functions of a real variable—arose on the basis of the rigorous arithmetic theory of irrational numbers and set theory. Whereas previously only functions arising “naturally” from various special problems were studied systematically, interest in the complete elucidation of the true extent of the general concepts of analysis is typical of the theory of functions of a real variable. (At the very start of the theory’s development B. Bolzano and later Weierstrass found, for example, that a continuous function may not have a derivative at any point.) Investigations in the theory of functions of a real variable led to general definitions of the concepts of the measure of a set, measurable functions, and integral, which play an important role in modern mathematics. The foundations of the modern theory of functions of a real variable were laid by mathematicians of the French school (Jordan, Borel, H. Lebesgue, R. Baire), and later the leading role was assumed by the Russian and Soviet school.
In addition to its immediate interest, the theory of functions of a real variable greatly influenced the development of many other branches of mathematics. Its methods proved to be essential in the construction of the foundations of functional analysis. Although the methods of functional analysis developed under the influence of the theory of functions of a real variable and set theory, the content and nature of the problems of functional analysis border directly on classical analysis and mathematical physics. Functional analysis is virtually indispensable (chiefly in the form of the theory of operators) in quantum physics. The first conscious isolation of functional analysis as a special branch of mathematics was made by V. Volterra in the late 19th century. The calculus of variations and the theory of integral equations, whose systematic development was also begun by Volterra and continued by E. Fredholm, were now perceived as parts of functional analysis. The most important special case of operators in Hilbert space, whose primary role was brought out in the works of Hilbert on integral equations, was analyzed with particular intensity.
Most of the mathematical problems posed by natural science and technology reduce to the solution of differential equations, both ordinary (in the study of syst