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David Hilbert presents a set of self-consistent geometric axioms in Foundations of Geometry. Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert as the foundation for a modern treatment of Euclidean geometry. Hilberts Axioms
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In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Exterior Derivative
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Special relativity is the physical theory of measurement in an inertial frame of reference. It generalizes Galileo's principle of relativity—that all uniform motion is relative, and that there is no absolute and well-defined state of rest (no privileged reference frames)—from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be. Special Relativity
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Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem, Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties mapping a compact convex set into itself there is a point x0 such that f(x0) = x0. Brouwer Fixed-point Theorem
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General relativity, or the general theory of relativity, is the geometric theory of gravitation and the current description of gravitation in modern physics. General relativity generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. General Relativity
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The classical mathematical puzzle known as water, gas, and electricity, the (three) utilities problem, or sometimes the three cottage problem is proven to have no solution. Three-cottage problem
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Cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. Characteristic Class
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In mathematics, the Borsuk–Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center. Borsuk-Ulam Theorem
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. Uniform Polyhedron
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Gromov's hyberbolic theory is revolutionary to both infinite group theory and global differential geometry. Hyperbolic Group
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The classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed. Classification of finite simple groups
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Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces which are locally presented by noncommutative algebras of functions (possibly in some generalized sense). Non-commutative geometry
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The Kepler conjecture, named after the 17th-century German astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. Hales proved it by exhaustion. Kepler conjecture
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This theorem is about the characterization of the three-dimensional sphere (3-sphere), which is the hypersphere that bounds the unit ball in four-dimensional space. Poincaré conjecture
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E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. E8 mathematics
