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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

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three double-cones cut by planes in different ways, resulting in the four conic sections
three double-cones cut by planes in different ways, resulting in the four conic sections
The four conic sections arise when a plane cuts through a double cone in different ways. If the plane cuts through parallel to the side of the cone (case 1), a parabola results (to be specific, the parabola is the shape of the planar graph that is formed by the set of points of intersection of the plane and the cone). If the plane is perpendicular to the cone's axis of symmetry (case 2, lower plane), a circle results. If the plane cuts through at some angle between these two cases (case 2, upper plane) — that is, if the angle between the plane and the axis of symmetry is larger than that between the side of the cone and the axis, but smaller than a right angle — an ellipse results. If the plane is parallel to the axis of symmetry (case 3), or makes a smaller positive angle with the axis than the side of the cone does (not shown), a hyperbola results. In all of these cases, if the plane passes through the point at which the two cones meet (the vertex), a degenerate conic results. First studied by the ancient Greeks in the 4th century BCE, conic sections were still considered advanced mathematics by the time Euclid (fl. c. 300 BCE) created his Elements, and so do not appear in that famous work. Euclid did write a work on conics, but it was lost after Apollonius of Perga (d. c. 190 BCE) collected the same information and added many new results in his Conics. Other important results on conics were discovered by the medieval Persian mathematician Omar Khayyám (d. 1131 CE), who used conic sections to solve algebraic equations.

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A homotopy from a circle around a sphere down to a single point.
Image credit: Richard Morris

The homotopy groups of spheres describe the different ways spheres of various dimensions can be wrapped around each other. They are studied as part of algebraic topology. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere, consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar circle (1-sphere) and sphere (2-sphere).

The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century as a tool for such classification, in effect using the set of mappings from a c-sphere into a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology. (Full article...)

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General Foundations Number theory Discrete mathematics


Algebra Analysis Geometry and topology Applied mathematics
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