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.

Felix Klein argued in his Erlangen program that one can consider various " geometries " by specifying an appropriate transformation group that leaves certain geometric properties invariant.

On a " global " level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure.

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.

Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.

Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E ( n ) ( the isometry group of R < sup > n </ sup >), where two subgroups H < sub > 1 </ sub >, H < sub > 2 </ sub > of a group G are conjugate, if there exists g ∈ G such that H < sub > 1 </ sub >= g < sup >− 1 </ sup > H < sub > 2 </ sub > g.

Dunwoody works on geometric group theory and low-dimensional topology.

In cosmological models ( geometric 3-manifolds ), a compact space is either a spherical geometry, or has infinite fundamental group ( and thus is called " multiply connected ", or more strictly non-simply connected ), by general results on geometric 3-manifolds.

) More precisely, if M is a manifold with a finite volume geometric structure, then the type of geometric structure is almost determined as follows, in terms of the fundamental group π < sub > 1 </ sub >( M ):

If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples ( such as the complement of a trefoil knot ) where a manifold may have a finite volume geometric structure of either type.

Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models ; for example, the geometry of almost any non-unimodular 3-dimensional Lie group.

With the American Abstract Artists group, Johnson painted geometric abstractions that, in part, reflected the influence of Albers.

Changing the group changes the appropriate geometric language.

We identify as affine theorems any geometric result that is invariant under the affine group ( in Felix Klein's Erlangen programme this is its underlying group of symmetry transformations for affine geometry ).

Poincaré duality relates H ( E < sup > k </ sup >) to H ( E < sup > k </ sup >), which is in turn the space of covariants of the monodromy group, which is the geometric fundamental group of U acting on the fiber of E < sup > k </ sup > at a point.

For the function field of a smooth curve over a finite field the quotient of the multiplicative group ( i. e. GL ( 1 )) of its adele ring by the multiplicative group of the function field of the curve and units of integral adeles, i. e. those with integral local components, is isomorphic to the group of isomorphisms of linear bundles on the curve, and thus carries a geometric information.

Algebraic geometry now finds application in statistics, control theory, robotics, error-correcting codes, phylogenetics and geometric modelling.

The area has further connections to coding theory and geometric combinatorics.

* Cone ( category theory ), a family of morphisms resembling a geometric cone

In 1916, Albert Einstein published his theory of general relativity, which provided a unified description of gravity as a geometric property of space and time.

Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.

Their work was an important part of the transition from intuitive and geometric homology to axiomatic homology theory.

It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Differential geometry is closely related to differential topology, and to the geometric aspects of the theory of differential equations.

Although best known for its geometric results, the Elements also includes number theory.

General relativity, or the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1916 and the current description of gravitation in modern physics.

Furthermore, the theory does not contain any invariant geometric background structures, i. e. it is background independent.

An incomplete and somewhat arbitrary subdivision of model theory is into classical model theory, model theory applied to groups and fields, and geometric model theory.

The result of this synthesis is called geometric model theory in this article ( which is taken to include o-minimality, for example, as well as classical geometric stability theory ).

* Wen-Tsun Wu Work in geometric theorem proving: Wu's method, Herbrand Award 1997

This can be seen easily from the fact that the sequences do converge to a common limit ( which can be shown by Bolzano – Weierstrass theorem ) and the fact that geometric mean is preserved:

An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell – Lang conjecture for function fields.

The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic.

The geometric law behind the measurement is the Pythagorean theorem (" The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides ").

His only original contribution to geometry was the proof that any geometric construction which can be done with compass and straightedge can also be done with compasses alone, a result now known as the Mohr – Mascheroni theorem.

* Jordan – Schönflies theorem in geometric topology

For finite projective spaces of geometric dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, GF ( q ), whose order ( that is, number of elements ) is q ( a prime power ).

The Veblen-Young theorem states in the finite case that every projective space of geometric dimension n ≥ 3 is isomorphic with a PG ( n, q ), the n-dimensional projective space over some finite field GF ( q ).

The Gauss-Bonnet theorem links the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties.

Desargues ' theorem is therefore one of the most basic of simple and intuitive geometric theorems whose natural home is in projective rather than affine space.

In the mathematical fields of topology and K-theory, the Serre – Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: " projective modules over commutative rings are like vector bundles on compact spaces ".

The JSJ decomposition and Thurston's hyperbolization theorem reduces the study of knots in the 3-sphere to the study of various geometric manifolds via splicing or satellite operations.

* Georg Mohr publishes the Mohr – Mascheroni theorem, that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.

The proof of the theorem uses geometric features of Hilbert spaces ; the corresponding statement for Banach spaces is not true in general, not even for finite-dimensional Banach spaces.

The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by its geometric centroid.

The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid.

By using the Pythagorean theorem, they reduced geometric problems to algebra systematically.

In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields ( satisfying an integrability condition ) in much the same way as an integral curve may be assigned to a single vector field.

The length of the resulting segment is the geometric mean, which can be proved using the Pythagorean theorem.

Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric ; geometrically, it has one of 3 possible geometries: positive curvature / spherical, zero curvature / flat, negative curvature / hyperbolic – and the geometrization conjecture ( now theorem ) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.

In integral geometry ( otherwise called geometric probability theory ), Hadwiger's theorem characterises the valuations on convex bodies in R < sup > n </ sup >.

Haboush's theorem can be used to generalize results of geometric invariant theory from characteristic 0, where they were already known, to characteristic p > 0.