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Homotopy
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==Applications== Based on the concept of the homotopy, [[Numerical methods|computation methods]] for [[algebraic equations|algebraic]] and [[differential equations]] have been developed. The methods for algebraic equations include the [[homotopy continuation]] method<ref>{{Cite book |last=Allgower |first=E. L. |url=https://www.worldcat.org/oclc/52377653 |title=Introduction to numerical continuation methods |date=2003 |publisher=SIAM |others=Kurt Georg |isbn=0-89871-544-X |location=Philadelphia |oclc=52377653}}</ref> and the continuation method (see [[numerical continuation]]). The methods for differential equations include the [[homotopy analysis method]]. Homotopy theory can be used as a foundation for [[Homology (mathematics)|homology theory]]: one can [[Representable functor|represent]] a cohomology functor on a space ''X'' by mappings of ''X'' into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian group ''G'', and any based CW-complex ''X'', the set <math>[X,K(G,n)]</math> of based homotopy classes of based maps from ''X'' to the [[Eilenberg–MacLane space]] <math>K(G,n)</math> is in natural bijection with the ''n''-th [[singular cohomology]] group <math>H^n(X,G)</math> of the space ''X''. One says that the [[Spectrum (topology)|omega-spectrum]] of Eilenberg-MacLane spaces are [[representing space]]s for singular cohomology with coefficients in ''G''. Using this fact, homotopy classes between a CW complex and a multiply connected space can be calculated using cohomology as described by the [[Hopf–Whitney theorem]]. Recently, homotopy theory is used to develop deep learning based generative models like [[diffusion model]]s and [[flow-based generative model]]s. Perturbing the complex non-Gaussian states is a tough task. Using deep learning and homotopy, such complex states can be transformed to Gaussian state and mildly perturbed to get transformed back to perturbed complex states.<ref>{{Citation |last1=Rout |first1=Siddharth |title=Probabilistic Forecasting for Dynamical Systems with Missing or Imperfect Data |date=2025-03-15 |arxiv=2503.12273 |last2=Haber |first2=Eldad |last3=Gaudreault |first3=Stéphane}}</ref>
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