Arithmetic topology: Difference between revisions

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Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology.

History

In the 1960s topological interpretations of class field theory were given by John Tate^[1] based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier^[2] based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots^[3] which was further explored by Barry Mazur.^[4]^[5] In the 1990s Reznikov^[6] and Kapranov^[7] began studying these analogies, coining the term arithmetic topology for this area of study.

Notes

^ J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).
^ M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole, 1964.
^ Who dreamed up the primes=knots analogy?, neverendingbooks, lieven le bruyn's blog, may 16, 2011,
^ Remarks on the Alexander Polynomial, Barry Mazur, c.1964
^ B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.
^ A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.
^ M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.

External links

Mazur’s knotty dictionary

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[1] J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).

[2] M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole, 1964.

[3] Who dreamed up the primes=knots analogy?, neverendingbooks, lieven le bruyn's blog, may 16, 2011,

[4] Remarks on the Alexander Polynomial, Barry Mazur, c.1964

[5] B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.

[6] A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.

[7] M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.

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+'''Arithmetic topology''' is an area of [[mathematics]] that is a combination of [[algebraic number theory]] and [[topology]].
-'''Arithmetic topology''' is an area of [[mathematics]] that is a combination of [[algebraic number theory]] and [[topology]]. In the 1960s topological interpretations of [[class field theory]] were given by [[John Tate]]<ref>J. Tate,  Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).</ref> based on [[Galois cohomology]], and also by [[Michael Artin]] and [[Jean-Louis Verdier]]<ref>M. Artin and J.-L. Verdier,  [http://www.jmilne.org/math/Documents/WoodsHole3.pdf Seminar on étale cohomology of number fields, Woods Hole], 1964.</ref> based on [[Étale cohomology]]. Then [[David Mumford]] (and independently [[Yuri Manin]]) came up with an analogy between [[prime ideals]] and [[Knot (mathematics)|knots]]<ref>[http://www.neverendingbooks.org/index.php/who-dreamed-up-the-primesknots-analogy.html Who dreamed up the primes=knots analogy?], neverendingbooks, lieven le bruyn's blog, may 16, 2011,</ref> which was further explored by [[Barry Mazur]].<ref>[http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf Remarks on the Alexander Polynomial], Barry Mazur, c.1964</ref><ref>B. Mazur, [http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1973_4_6_4/ASENS_1973_4_6_4_521_0/ASENS_1973_4_6_4_521_0.pdf Notes on ´etale cohomology of number fields], Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.</ref> In the 1990s Reznikov<ref>A. Reznikov, [http://www.springerlink.com/content/v9jc215brrhl4mxf/ Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold)], Sel. math. New ser. 3, (1997), 361&ndash;399.</ref> and Kapranov<ref>M. Kapranov, [http://books.google.co.uk/books?hl=en&lr=&id=TOPa9irmsGsC&oi=fnd&pg=PA119 Analogies between the Langlands correspondence and topological quantum field theory], Progress in Math., 131, Birkhäuser, (1995), 119–151.</ref> began studying these analogies, coining the term '''arithmetic topology''' for this area of study.
+==History==
+In the 1960s topological interpretations of [[class field theory]] were given by [[John Tate]]<ref>J. Tate,  Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).</ref> based on [[Galois cohomology]], and also by [[Michael Artin]] and [[Jean-Louis Verdier]]<ref>M. Artin and J.-L. Verdier,  [http://www.jmilne.org/math/Documents/WoodsHole3.pdf Seminar on étale cohomology of number fields, Woods Hole], 1964.</ref> based on [[Étale cohomology]]. Then [[David Mumford]] (and independently [[Yuri Manin]]) came up with an analogy between [[prime ideals]] and [[Knot (mathematics)|knots]]<ref>[http://www.neverendingbooks.org/index.php/who-dreamed-up-the-primesknots-analogy.html Who dreamed up the primes=knots analogy?], neverendingbooks, lieven le bruyn's blog, may 16, 2011,</ref> which was further explored by [[Barry Mazur]].<ref>[http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf Remarks on the Alexander Polynomial], Barry Mazur, c.1964</ref><ref>B. Mazur, [http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1973_4_6_4/ASENS_1973_4_6_4_521_0/ASENS_1973_4_6_4_521_0.pdf Notes on ´etale cohomology of number fields], Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.</ref> In the 1990s Reznikov<ref>A. Reznikov, [http://www.springerlink.com/content/v9jc215brrhl4mxf/ Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold)], Sel. math. New ser. 3, (1997), 361&ndash;399.</ref> and Kapranov<ref>M. Kapranov, [http://books.google.co.uk/books?hl=en&lr=&id=TOPa9irmsGsC&oi=fnd&pg=PA119 Analogies between the Langlands correspondence and topological quantum field theory], Progress in Math., 131, Birkhäuser, (1995), 119–151.</ref> began studying these analogies, coining the term '''arithmetic topology''' for this area of study.
 ==See also==

v t e Number theory
Fields	Algebraic number theory (class field theory, non-abelian class field theory, Iwasawa theory, Iwasawa–Tate theory, Kummer theory) Analytic number theory (analytic theory of L-functions, probabilistic number theory, sieve theory) Geometric number theory Computational number theory Transcendental number theory Diophantine geometry (Arakelov theory, Hodge–Arakelov theory) Arithmetic combinatorics (additive number theory) Arithmetic geometry (anabelian geometry, p-adic Hodge theory) Arithmetic topology Arithmetic dynamics
Key concepts	Numbers 0 Natural numbers Unity Prime numbers Composite numbers Rational numbers Irrational numbers Algebraic numbers Transcendental numbers p-adic numbers (p-adic analysis) Arithmetic Modular arithmetic Chinese remainder theorem Arithmetic functions
Advanced concepts	Quadratic forms Modular forms L-functions Diophantine equations Diophantine approximation Irrationality measure Simple continued fractions
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Revision as of 14:36, 26 November 2014

History

See also

Notes

Further reading

External links