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Cohen [4] [7] showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof. To prove his result, Cohen developed the method of forcing, which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds, and constructs another model which contains more sets than ...
In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω 2 × ω to {0,1} and p < q if p ⊇ q. This poset satisfies the countable chain condition. Forcing with this poset adds ω 2 distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum ...
This is one of the central problems of number theory, incorporating earlier conjectures by Deligne and Beilinson. The Birch–Swinnerton-Dyer conjecture is a special case. More precisely, the conjecture predicts the leading coefficient of the L-function at an integer point in terms of regulators and a height pairing on motivic cohomology.
The Cohen–Lenstra heuristics [6] are a set of more precise conjectures about the structure of class groups of quadratic fields. For real fields they predict that about 75.45% of the fields obtained by adjoining the square root of a prime will have class number 1, a result that agrees with computations.
Then there exists an R-algebra B R that is a balanced big Cohen–Macaulay algebra for R, an S-algebra that is a balanced big Cohen-Macaulay algebra for S, and a homomorphism B R → B S such that the natural square given by these maps commutes. Serre's Conjecture on Multiplicities. (cf. Serre's multiplicity conjectures.
In algebraic topology, the doomsday conjecture was a conjecture about Ext groups over the Steenrod algebra made by Joel Cohen, named by Michael Barratt, published by Milgram (1971, conjecture 73) and disproved by Mahowald (1977). Minami (1995) stated a modified version called the new doomsday conjecture.
In deformation theory, a branch of mathematics, Deligne's conjecture is about the operadic structure on Hochschild cochain complex.Various proofs have been suggested by Dmitry Tamarkin, [1] [2] Alexander A. Voronov, [3] James E. McClure and Jeffrey H. Smith, [4] Maxim Kontsevich and Yan Soibelman, [5] and others, after an initial input of construction of homotopy algebraic structures on the ...
Chudnovsky et al. introduced some technical constraints on skew partitions, and were able to show that Chvátal's conjecture is true for the resulting "balanced skew partitions". The full conjecture is a corollary of the strong perfect graph theorem. [8] A homogeneous pair is related to a modular decomposition of a graph.