Algebraic topology in the context of "Differential (mathematics)"

Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

Many areas of computer science also rely on category theory, such as functional programming and semantics.

Jules Henri Poincaré (UK: /ˈpwæ̃kɑːreɪ/, US: /ˌpwæ̃kɑːˈreɪ/; French: [ɑ̃ʁi pwɛ̃kaʁe] ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. He has further been called "the Gauss of modern mathematics". Due to his success in science, along with his influence and philosophy, he has further been called "the philosopher par excellence of modern science".

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. Poincaré is regarded as the creator of the field of algebraic topology, and is further credited with introducing automorphic forms. He also made important contributions to algebraic geometry, number theory, complex analysis and Lie theory. He famously introduced the concept of the Poincaré recurrence theorem, which states that a state will eventually return arbitrarily close to its initial state after a sufficiently long time, which has far-reaching consequences. Early in the 20th century he formulated the Poincaré conjecture, which became, over time, one of the famous unsolved problems in mathematics. It was eventually solved in 2002–2003 by Grigori Perelman. Poincaré popularized the use of non-Euclidean geometry in mathematics as well.

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds. An example of a Riemannian manifold is a surface, on which distances are measured by the length of curves on the surface. Riemannian geometry is the study of surfaces and their higher-dimensional analogs (called manifolds), in which distances are calculated along curves belonging to the manifold. Formally, Riemannian geometry is the study of smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in R. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

In topology, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós 'same, similar' and τόπος tópos 'place') if one can be "continuously deformed" into the other, such a deformation being called a homotopy (/həˈmɒtəpiː/ hə-MOT-ə-pee; /ˈhoʊmoʊˌtoʊpiː/ HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.

In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.

The fundamental concepts in point-set topology are continuity, compactness, and connectedness:

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by ${\displaystyle \chi }$ (Greek lower-case letter chi).

The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra.

In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be a non-integer. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise.

Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory).