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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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animation showing what looks like a smaller inner cube with corners connected to those of a larger outer cube; the smaller cube passes through one face of the larger cube and becomes larger as the larger cube becomes smaller; eventually the smaller and larger cubes have switched positions and the animation repeats
animation showing what looks like a smaller inner cube with corners connected to those of a larger outer cube; the smaller cube passes through one face of the larger cube and becomes larger as the larger cube becomes smaller; eventually the smaller and larger cubes have switched positions and the animation repeats
A three-dimensional projection of a tesseract performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom. Also called an 8-cell or octachoron, a tesseract is the four-dimensional analog of the cube (i.e., a 4-D hypercube, or 4-cube), where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. The tesseract is to the cube as the cube is to the square. Tesseracts and other polytopes can be used as the basis for the network topology when linking multiple processors in parallel computing.

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  • ... that Ewa Ligocka cooked another mathematician's goose?
  • ... that mathematician Daniel Larsen was the youngest contributor to the New York Times crossword puzzle?
  • ... that after Florida schools banned 54 mathematics books, Chaz Stevens petitioned that they also ban the Bible?
  • ... that Fathimath Dheema Ali is the first Olympic qualifier from the Maldives?
  • ... that despite published scholarship to the contrary, Andrew Planta neither received a doctorate nor taught mathematics at Erlangen?
  • ... that the music of math rock band Jyocho has been alternatively described as akin to "madness" or "contemplative and melancholy"?
  • ... that in the aftermath of the American Civil War, the only Black-led organization providing teachers to formerly enslaved people was the African Civilization Society?
  • ... that circle packings in the form of a Doyle spiral were used to model plant growth long before their mathematical investigation by Doyle?

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  • ...that as of April 2010 only 35 even numbers have been found that are not the sum of two primes which are each in a Twin Primes pair? ref
  • ...the Piphilology record (memorizing digits of Pi) is 70000 as of Mar 2015?
  • ...that people are significantly slower to identify the parity of zero than other whole numbers, regardless of age, language spoken, or whether the symbol or word for zero is used?
  • ...that Auction theory was successfully used in 1994 to sell FCC airwave spectrum, in a financial application of game theory?
  • ...properties of Pascal's triangle have application in many fields of mathematics including combinatorics, algebra, calculus and geometry?
  • ...work in artificial intelligence makes use of swarm intelligence, which has foundations in the behavioral examples found in nature of ants, birds, bees, and fish among others?
  • ...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
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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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Topics in mathematics

General Foundations Number theory Discrete mathematics


Algebra Analysis Geometry and topology Applied mathematics
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