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

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

This new perspective led to revolutionary advances across many areas of pure mathematics.

By the late 1960s he had started to become interested in scientific areas outside of mathematics.

In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1.

* it reveals deep connections between different areas of mathematics

Other examples are readily found in different areas of mathematics, for example, vector addition, matrix multiplication and conjugation in groups.

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.

Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces.

In 1879, Peirce was appointed Lecturer in logic at the new Johns Hopkins University, which was strong in a number of areas that interested him, such as philosophy ( Royce and Dewey did their PhDs at Hopkins ), psychology ( taught by G. Stanley Hall and studied by Joseph Jastrow, who coauthored a landmark empirical study with Peirce ), and mathematics ( taught by J. J. Sylvester, who came to admire Peirce's work on mathematics and logic ).

The mathematical community as a whole could enlist in problems, which he had identified as crucial aspects of the areas of mathematics he took to be key.

The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business accounts.

The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field.

Theoretical computer science includes areas of discrete mathematics relevant to computing.

The orthogonality of the DFT is now expressed as an orthonormality condition ( which arises in many areas of mathematics as described in root of unity ):

Examples of broad areas of academic disciplines include the natural sciences, mathematics, computer science, social sciences, humanities and applied sciences.

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.

An example of such a finite field is the ring Z / pZ, which is essentially the set of integers from 0 to p − 1 with integer addition and multiplication modulo p. It is also sometimes denoted Z < sub > p </ sub >, but within some areas of mathematics, particularly number theory, this may cause confusion because the same notation Z < sub > p </ sub > is used for the ring of p-adic integers.

( See areas of mathematics and algebraic geometry.

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 areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure ( as above ) but also the extra structure.

Artists and sculptors tried to find this ideal order in relation with mathematics, but they believed that this ideal order revealed itself not so much to the dispassionate intellect, as to the whole sentient self.

* Atom ( order theory ) in mathematics

It is useful to study sets naively at an early stage of mathematics in order to develop facility for working with them.

In mathematics a combination is a way of selecting several things out of a larger group, where ( unlike permutations ) order does not matter.

In mathematics, especially in order theory, the cofinality cf ( A ) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.

Much of mathematics is grounded in the study of equivalences, and order relations.

Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders.

Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property that is preserved by an isomorphism and that is true of one of the objects, is also true of the other.

The German philosopher Arthur Schopenhauer designates this " inner nature " with the term Will, and suggests that science and mathematics are unable to penetrate beyond the relationship between one thing and another in order to explain the " inner nature " of the thing itself, independent of any external causal relationships with other " things ".

In mathematics, especially in order theory, a preorder or quasi-order is a binary relation that is reflexive and transitive.

This change from a quasi-intensional stance to a fully extensional stance also restricts predicate logic to the second order, i. e. functions of functions: " We can decide that mathematics is to confine itself to functions of functions which obey the above assumption " ( PM 2nd Edition p. 401, Appendix C ).

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order.

In both mathematics and the relational database model, a set is an unordered collection of unique, non-duplicated items, although some DBMSs impose an order to their data.

In mathematics, a tuple has an order, and allows for duplication.

The players of the game may not need to use mathematics in order to play mathematical games.

For example, Mancala is a mathematical game, because mathematicians can study it using combinatorial game theory, even though no mathematics is necessary in order to play it.

Mathematical puzzles require mathematics in order to solve them.

In discrete mathematics, tree rotation is an operation on a binary tree that changes the structure without interfering with the order of the elements.

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order ( number of elements ) of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange.

In mathematics, a well-order relation ( or well-ordering ) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

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.

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set ( P, ≤) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.