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]]. In the 1960s [[Barry Mazur]]<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> and [[Yuri Manin]] pointed out a series of analogies between [[prime ideals]] and [[Knot (mathematics)|knots]]. In the 1990s Reznikov<ref>A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.</ref> and Kapranov<ref>M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.</ref> began studying these analogies and coined the term '''arithmetic topology'''. |
'''Arithmetic topology''' is an area of [[mathematics]] that is a combination of [[algebraic number theory]] and [[topology]]. In the 1960s [[Barry Mazur]]<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> and [[Yuri Manin]] pointed out a series of analogies between [[prime ideals]] and [[Knot (mathematics)|knots]]. 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), 361–399.</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 and coined the term '''arithmetic topology'''. |
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==See also== |
==See also== |
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Revision as of 19:21, 4 August 2009
Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. In the 1960s Barry Mazur[1] and Yuri Manin pointed out a series of analogies between prime ideals and knots. In the 1990s Reznikov[2] and Kapranov[3] began studying these analogies and coined the term arithmetic topology.
See also
Notes
- ^ 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.
Further reading
- Masanori Morishita (2009), Analogies Between Knots And Primes, 3-Manifolds And Number Rings
- Christopher Deninger (2002), A note on arithmetic topology and dynamical systems
- Adam S. Sikora (2001), Analogies between group actions on 3-manifolds and number fields
- Curtis T. McMullen (2003), From dynamics on surfaces to rational points on curves
External links
 | This number theory-related article is a stub. You can help Wikipedia by expanding it. |
 | This topology-related article is a stub. You can help Wikipedia by expanding it. |