The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

Finite fields have applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type.

In mathematics, a Lie algebra (, not ) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

In mathematics, a Lie group () is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics.

Lie groups occur in abundance throughout mathematics and physics.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled

The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics.

Despite this they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups.

The theory of Lie groups describes continuous symmetry in mathematics ; its importance there and in theoretical physics ( for example quark theory ) grew steadily in the twentieth century.

The influence on graduate education in pure mathematics is perhaps most noticeable in the treatment now current of Lie groups and Lie algebras.

A key theme from the " categorical " point of view is that mathematics requires not only certain kinds of objects ( Lie groups, Banach spaces, etc.

From this time, and certainly much helped by Weyl's expositions, Lie groups and Lie algebras became a mainstream part both of pure mathematics and theoretical physics.

Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal group ), differential geometry ( Riemannian metric ), differential topology ( intersection forms of four-manifolds ), and Lie theory ( the Killing form ).

Since Lie groups ( and some analogues such as algebraic groups ) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied.

Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory ( such as singularity theory ).

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry.

One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real reductive Lie groups.

In mathematics, G < sub > 2 </ sub > is the name of three simple Lie groups ( a complex form, a compact real form and a split real form ), their Lie algebras, as well as some algebraic groups.

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes.

In mathematics, the exponential function is the function e < sup > x </ sup >, where e is the number ( approximately 2. 718281828 ) such that the function e < sup > x </ sup > is its own derivative.

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant ( as opposed to the total derivative, in which all variables are allowed to vary ).

* Differentiation ( mathematics ), the process of finding a derivative

In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus.

In mathematics, a connector is a map which can be defined for a linear connection and used to define the covariant derivative on a vector bundle from the linear connection.

* partial derivative, in mathematics

In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative.

* In mathematics, the unit doublet is the derivative of the Dirac delta function

In calculus ( a branch of mathematics ), a differentiable function is a function whose derivative exists at each point in its domain.

One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations in mathematics, physics and engineering.

In mathematics, the Grünwald – Letnikov derivative is a basic extension of the derivative in fractional calculus, that allows one to take the derivative a non-integer number of times.

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero ( dα = 0 ), and an exact form is a differential form that is the exterior derivative of another differential form β.

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

In mathematics, the Pincherle derivative of a linear operator on the vector space of polynomials in the variable over a field is another linear operator defined as

In mathematics and physics, when using Newton's notation the dot denotes the time derivative as in.

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.

In mathematics, the Klein four-group ( or just Klein group or Vierergruppe (), often symbolized by the letter V ) is the group Z < sub > 2 </ sub > × Z < sub > 2 </ sub >, the direct product of two copies of the cyclic group of order 2.

(), better known just as Ibn Rushd (), and in European literature as Averroes (; April 14, 1126 – December 10, 1198 ), was an Andalusian Muslim polymath ; a master of Aristotelian philosophy, Islamic philosophy, Islamic theology, Maliki law and jurisprudence, logic, psychology, politics, Arabic music theory, and the sciences of medicine, astronomy, geography, mathematics, physics and celestial mechanics.

In mathematics, the sieve of Eratosthenes (), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit.

* Max (), an abbreviation for either maximal element, greatest element, or extreme value in mathematics ; see Maxima and minima

* Tian yuan shu, in Japanese tengenjutsu (), a method of algebra in Chinese and Japanese mathematics

In mathematics, the witch of Agnesi (), sometimes called the witch of Maria Agnesi is the curve defined as follows.

In mathematics, a Lissajous curve (), also known as Lissajous figure or Bowditch curve (), is the graph of a system of parametric equations

He divided mathematics into two parts Mental () and Observable (), ( or in other words, Pure and Applied.

The Instituto Nacional de Matemática Pura e Aplicada (), mostly referred to as IMPA, is widely considered to be the foremost research and educational institution of Brazil in the area of mathematics.