In 1950, Iwasawa was invited to Cambridge, Massachusetts to give a lecture at the International Congress of Mathematicians on his method to study Dedekind zeta functions using integration over ideles and duality of adeles ; this method was also independently obtained by John Tate and it is sometimes called Tate's thesis or the Iwasawa-Tate theory.

In the last ten years he has focused on the study of various aspects of non-commutative Iwasawa theory, for instance, the study of the arithmetic of elliptic curves in nonabelian infinite extensions.

Tate's thesis ( 1950 ) on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring, its self-duality and harmonic analysis on it ; independently and a little earlier, Kenkichi Iwasawa obtained a similar theory.

More recently ( early 90s ), Ralph Greenberg has proposed an Iwasawa theory for motives.

Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

Combining this with the p-parity theorem by and the announced proof of the main conjecture of Iwasawa theory for GL ( 2 ) by, they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by, satisfy the Birch and Swinnerton-Dyer conjecture.

These elements — named Beilinson elements after Alexander Beilinson who introduced them in — were used by Kazuya Kato in to prove one divisibility in Barry Mazur's main conjecture of Iwasawa theory for elliptic curves.

Kenkichi Iwasawa ( Iwasawa Kenkichi, September 11, 1917 – October 26, 1998 ) was a Japanese mathematician who is known for his influence on algebraic number theory.

Iwasawa spent the next two years at Institute for Advanced Study in Princeton, and in Spring of 1952 was offered a job at the Massachusetts Institute of Technology, where he worked until 1967.

Iwasawa is perhaps best known for introducing what is now called Iwasawa theory, which developed from researches on cyclotomic fields from the later parts of the 1950s.

* In Japan, according to M. Iwasawa at the National Institute of Population and Social Security Research, less than 3 % of females between 25-29 are currently cohabiting, but more than 1 in 5 have had some experience of an unmarried partnership, including cohabitation.

* Iwasawa, Kenkichi: On some types of topological groups.

Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra ( the Iwasawa decomposition ), and this provides a canonical bicrossproduct quantum group associated to g. For su ( 2 ) one obtains a quantum group deformation of the Euclidean group E ( 3 ) of motions in 3 dimensions.

* The Iwasawa decomposition G = KAN of a semisimple group G as the product of compact, abelian, and nilpotent subgroups generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix ( a consequence of Gram – Schmidt orthogonalization ).

In mathematics, the Iwasawa decomposition KAN of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix ( a consequence of Gram-Schmidt orthogonalization ).

For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of G as G = KAN in which K occurs in a product with a contractible subgroup AN.

* Iwasawa decomposition ( KAN ) a mathematical process dealing with Lie groups

It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields.

Before that he worked on Lie groups and Lie algebras, introducing the general Iwasawa decomposition.

Ribet's methods were pushed further by Barry Mazur and Andrew Wiles in order to prove the Main Conjecture of Iwasawa theory ,< ref > a corollary of which is a strengthening of the Herbrand-Ribet theorem: the power of p dividing B < sub > p − n </ sub > is exactly the power of p dividing the order of G < sub > n </ sub >.

and the Iwasawa decomposition of G is

* Borel, A .; Chowla, S .; Herz, C. S .; Iwasawa, K .; Serre, J .- P. Seminar on complex multiplication.

Via the theory of zeta integrals initiated by Kenkichi Iwasawa and by John Tate in Tate's thesis it is related to the study of the zeta function of global fields.

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields.

In fact, is a module over the Iwasawa algebra ( i. e. the completed group ring of over ).

This idea is much used in Iwasawa theory.

It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.

Then the Iwasawa decomposition of is

A more recent Iwasawa study has shown that there has been a recent emergence of non-marital cohabitation.

The latter was explicitly introduced in papers of Kenkichi Iwasawa and John Tate.

* Iwasawa Yoshihiko 岩沢愿彥.

In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties.

Iwasawa studied classical Galois modules over by asking questions about the structure of modules over.

More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.

In order to get an interesting Galois module here, Iwasawa took the ideal class group of, and let be its p-torsion part.

The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions ( by module theory, by interpolation ) should coincide, as far as that was well-defined.

* Greenberg, Ralph, Iwasawa Theory-Past & Present, Advanced Studies in Pure Math.

In machine learning, tree decompositions are also called junction trees, clique trees, or join trees ; they

But then necessarily the reductions modulo of and also make all non-leading terms vanish ( and cannot make their leading terms vanish ), since no other decompositions of are possible in, which is a unique factorization domain.

Assuming smoothness the existence of a Heegard splitting also follows from the work of Smale about handle decompositions from Morse theory.

The ( complex ) tensor, symmetric, and exterior algebras over V < sup > C </ sup > also admit decompositions.