Arithmetic topology is an area of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 that is a combination of algebraic number theory

Algebraic number theory

Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

 and topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

. In the 1960s topological interpretations of class field theory

Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

 were given by John Tate

John Tate

John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...

 based on Galois cohomology

Galois cohomology

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...

, and also by Michael Artin

Michael Artin

Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...

 and Jean-Louis Verdier

Jean-Louis Verdier

Jean-Louis Verdier was a French mathematician who worked, under the guidance of Alexander Grothendieck, on derived categories and Verdier duality...

 based on Étale cohomology

Étale cohomology

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...

. Then David Mumford

David Mumford

David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...

 (and independently Yuri Manin) came up with an analogy between prime ideals and knots

Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...

 which was further explored by Barry Mazur

Barry Mazur

-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959,...

. In the 1990s Reznikov and Kapranov began studying these analogies, coining the term arithmetic topology for this area of study.

See also

• Arithmetic geometry
• Arithmetic dynamics
  Arithmetic dynamics
  Arithmetic dynamicsis a field that amalgamates two areas of mathematics, dynamical systems and number theory.Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line...
  
  
• Topological quantum field theory
  Topological quantum field theory
  A topological quantum field theory is a quantum field theory which computes topological invariants....
  
  
• Langlands program
  Langlands program
  The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
  
  

Further reading

• Masanori Morishita (2011), Knots and Primes, Springer, ISBN 978-1-4471-2157-2

• 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

• Mazur’s knotty dictionary