Connes has applied his work in areas of mathematics and theoretical physics, including number theory, differential geometry and particle physics.

The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y ( x ) of Bessel's differential equation:

The survey of Hanover fueled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces.

In mathematics, the Cauchy – Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

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.

Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of formal mathematical analysis, beginning with calculus ; and, increasing in complexity up to differential geometry and partial differential equations.

Undergraduate physics students must also take extensive mathematics courses ( calculus, differential equations, advanced calculus ), and computer science and programming.

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.

In mathematics, particularly in differential geometry and global analysis, spinors have since found broad applications to algebraic and differential topology, symplectic geometry, gauge theory, complex algebraic geometry, index theory, and special holonomy.

As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point.

The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics.

His writings present a fresh, often geometric approach to traditional mathematical topics like ordinary differential equations, and his many textbooks have proved influential in the development of new areas of mathematics.

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

Other important contributions to mathematics that he made involved discontinuous functions and divergent series, differential equations, interpolation, the gamma function and elliptic functions.

* Autonomous system ( mathematics ), a system of ordinary differential equations which does not depend on the independent variable

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.

Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.

In mathematics, a partial differential equation ( PDE ) is a differential equation that contains unknown multivariable functions and their partial derivatives.

* Atlas ( topology ), a collection of local coordinate charts in mathematics

It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

Dyson also did work in a variety of topics in mathematics, such as topology, analysis, number theory and random matrices.

Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.

In mathematics, graphs are useful in geometry and certain parts of topology, e. g. Knot Theory.

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.

In topology and related branches of mathematics, a Hausdorff space, separated space or T < sub > 2 </ sub > space is a topological space in which distinct points have disjoint neighbourhoods.

Likewise, analysis, geometry and topology, although considered pure mathematics, find applications in theoretical physics — string theory, for instance.

Another aspect of mathematics, set-theoretic topology and point-set topology, concerns objects of a different nature from objects in our universe, or in a higher dimensional analogue of our universe.

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.

The mathematician Leonhard Euler was one of the first to analyze plane mazes mathematically, and in doing so made the first significant contributions to the branch of mathematics known as topology.

Both these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.

In mathematics, more specifically in general topology and related branches, a net or Moore – Smith sequence is a generalization of the notion of a sequence.

In mathematics, pointless topology ( also called point-free or pointfree topology ) is an approach to topology that avoids mentioning points.

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.

But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.

Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.

In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.

So, too, is the mathematical competence of a college graduate who has majored in mathematics.

The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.

In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.

There is no prize awarded for mathematics, but see Abel Prize.