homotopy|homotopies in English

noun

function used to uniformly distort one mapping so as to make it coincide with a second mapping (Mathematics)

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1. In the 1960s he wrote fundamental papers on higher homotopy theory and homotopy algebras.

2. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

3. It follows that adjoint functors induce homotopy equivalences.

4. It is weakly Contractible if this map is a weak homotopy equivalence, hence if all homotopy groups of X X are trivial

5. Strongly homotopy Lie algebras are associated with manifolds equipped with a closed form.

6. Moreover, a natural transformation between two such functors induces a homotopy between the induced maps.

7. Pretalk Adjunctions and monads Weighted limits Algebras and descent data Monadicity and descent Homotopy coherent Adjunctions A homotopy coherent adjunction in an (1;2)-category K is a simplicial functor Adj !K

8. An equivalent form of the Conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.

9. The deformation of various kinds of mathematical structures can be encapsulated in the language of strongly homotopy Lie algebras.

10. Research involves Multi-disciplinary theoretical and computational research in homotopy theory, the algebra of fitting spaces together.

11. Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory.

12. Infinity categories (in the form of Joyal's quasi-categories) are a convenient framework to do homotopy theory in abstract settings.

13. Along with progress in proving the formality of the strongly homotopy Lie algebras, HQSMCF established close collaborations between leading mathematicians.

14. Lastly, mathematicians used the theory developed to construct various strongly homotopy Lie algebras as higher central extensions of Lie algebras geometrically.

15. This new tool, formulated in terms of homotopy theory, proved to be successful in formulating and solving problems in algebra, particularly in ring theory and module theory.

16. He presented his solution, that amounts in modern terms to a combination of the Durand–Kerner method with the homotopy continuation principle, in 1891.

17. In return, the abstraction allowed by lax algebras will provide a reference point for the development of generalisations of homotopy theories, ranging from metric to uniform interpretations.

18. He studies, among other things, étale homotopy theory, elliptic surfaces, computer-based algebraic geometry (such as Gröbner basis), Torelli sets and toric varieties, and history of mathematics.

19. He played a role in the beginnings of modern homotopy theory perhaps analogous to that of Saunders Mac Lane in homological algebra, namely the adroit and persistent application of categorical methods.