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# Geometric Areas of Mathematics

Here we consider all the fields which exercise our geometric intuition: from Euclidean and analytic geometry to tilings and tessellations, from the Klein bottle to knots, along with curvature, soap bubbles and the very idea of dimension.

One of the oldest areas of mathematical discovery, geometry has undergone several rebirths over the centuries. At one extreme, geometry includes the very precise study of rigid structures first seen in Euclid's Elements; at the other extreme, general topology focuses on the very fundamental kinships among shapes. (There is also a more subtle notion of "geometry" implied in Algebraic Geometry (14), which is frankly quite algebraic.)

Here the MathMap shows the Geometry areas in a yellow-orange and the Topology areas in a yellow-green. Other fairly geometric areas are K-theory (19), Lie Groups (22), Several Complex Variables (32), and to some extent Global Analysis (58) and the Calculus of Variations (49).

• 51: Geometry is studied from many perspectives! This large area includes classical Euclidean geometry and synthetic (non-Euclidean) geometries; analytic geometry; incidence geometries (including projective planes); metric properties (lengths and angles); and combinatorial geometries such as those arising in finite group theory. Many results in this area are basic in either the sense of simple, or useful, or both!
• 52: Convex and discrete geometry includes the study of convex subsets of Euclidean space. A wealth of famous results distinguishes this family of sets (e.g. Brouwer's fixed-point theorem, the isoperimetric problems). This classification also includes the study of polygons and polyhedra, and frequently overlaps discrete mathematics and group theory; through piece-wise linear manifolds, it intersects topology. This area also includes tilings and packings in Euclidean space.
• 53: Differential geometry is the language of modern physics as well as an area of mathematical delight. Typically, one considers sets which are manifolds (that is, locally resemble Euclidean space) and which come equipped with a measure of distances. In particular, this includes classical studies of the curvature of curves and surfaces. Local questions both apply and help study differential equations; global questions often invoke algebraic topology.

The remaining three areas are collectively known as Topology.

• 54: General topology studies spaces on which one has only a loose notion of "closeness" -- enough to decide which functions are continuous. Typically one studies spaces with some additional structure -- metric spaces, say, or compact Hausdorff spaces -- and looks to see how properties such as compactness are shared with subspaces, product spaces, and so on. Widely applicable in geometry and analysis, topology also allows for some bizarre examples and set-theoretic conundra.
• 55: Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants illustrate some of the rigidity of the spaces. This includes various (co)homology theories, homotopy groups, and groups of maps, as well as some rather more geometric tools such as fiber bundles. The algebraic machinery (mostly derived from homological algebra) is powerful if rather daunting.
• 57: Manifolds are spaces like the sphere which look locally like Euclidean space. In particular, these are the spaces in which we can discuss (locally-)linear maps, and the spaces in which to discuss smoothness. They include familiar surfaces. Cell complexes are spaces made of pieces which are part of Euclidean space, generalizing polyhedra. These types of spaces admit very precise answers to questions about existence of maps and embeddings; they are particularly amenable to calculations in algebraic topology; they allow a careful distinction of various notions of equivalence. These are the most classic spaces on which groups of transformations act. This is also the setting for knot theory.

The geometric areas share with the fields of algebra the tendency to distill their inquiry to the study of certain axioms and their consequences; during the last half-century the ties between these broad areas have increased. On the other hand, some of the geometric areas remain close to analysis, particularly General Topology (to measure theory and functional analysis) and Differential Geometry (to differential equations and complex analysis).

You might want to continue the tour with a trip through analysis.