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Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. [1] The original problem was posed as the Problem of the topology of algebraic curves and surfaces (Problem der Topologie algebraischer Kurven und Flächen).
The seven selected problems span a number of mathematical fields, namely algebraic geometry, arithmetic geometry, geometric topology, mathematical physics, number theory, partial differential equations, and theoretical computer science. Unlike Hilbert's problems, the problems selected by the Clay Institute were already renowned among ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments. 14. Proof of the finiteness of certain complete systems of functions. 15. Rigorous foundation of Schubert's enumerative calculus. 16. Problem of the topology of algebraic curves and surfaces. 17. Expression of definite forms by ...
The question of whether all linear multiplicative functionals on an -convex Frechet algebra are continuous is known as Michael's Conjecture. [16] For a long time, this conjecture was perhaps the most famous open problem in the theory of topological algebras. Michael's Conjecture was solved completely and affirmatively in 2022. [17]
Using the smooth structure, one can define the Lie algebra of G, an object of linear algebra that determines a connected group G up to covering spaces. As a result, the solution to Hilbert's fifth problem reduces the classification of topological groups that are topological manifolds to an algebraic problem, albeit a complicated problem in general.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence .
Steenrod algebra; Bott periodicity theorem; K-theory. Topological K-theory; Adams operation; Algebraic K-theory; Whitehead torsion; Twisted K-theory; Cobordism; Thom space; Suspension functor; Stable homotopy theory; Spectrum (homotopy theory) Morava K-theory; Hodge conjecture; Weil conjectures; Directed algebraic topology
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