# 2001 Top Ten of Polemic for Mathematics

Symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of
Trigonometry
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios such as sine
Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element
Integer
An integer is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not
Computational complexity theory
Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm
Natural number
In mathematics, the natural numbers are those numbers used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers", and words used for ordering are "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes, that is, as what linguists call nominal numbers, forgoing many or all of the properties of being a number in a mathematical sense
Mathematics
Mathematics includes the study of such topics as numbers, formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes. There is no general consensus about its exact scope or epistemological status
Insertion sort
Insertion sort is a simple sorting algorithm that builds the final sorted array one item at a time. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort. However, insertion sort provides several advantages:Simple implementation: Jon Bentley shows a three-line C version, and a five-line optimized version Efficient for (quite) small data sets, much like other quadratic sorting algorithms More efficient in practice than most other simple quadratic
Katal
The katal is the unit of catalytic activity in the International System of Units (SI). It is a derived SI unit for quantifying the catalytic activity of enzymes and other catalysts
Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has