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seen Dec 9 at 11:04

I am a graduate student at the National University of Singapore studying complex geometry with To Wing Keung.


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accepted Reverse plane geometry, anyone?
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accepted Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism?
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comment Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism?
Of course, for simplicity, we can assume that both $X$ and $Y$ has homology which is finitely generated in each degree.
Nov
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comment Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism?
Thank you Eric for your answer. You answer together with Tyler's answer leaves just one case, where we know only that one of $X$ or $Y$ is the suspension of a connected space.
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comment Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism?
Ah, in the example I have in mind, in fact $Y= \Sigma Y'$ is the suspension of some other space, and hence is simply connected. Noting Tyler's answer, I now realize that this added condition of $Y$ being a suspension space is crucial.
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asked Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism?
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Sep
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comment Stability of real polynomials with positive coefficients
In particular, as you noted, the condition that $|f(z)|<f(|z|)$ is equivalent to the associated Hermitian metric of $f$ satisfying the Sharp Global Cauchy Schwar inequality. The other condition that the first two and the last two coefficients of $f$ are strictly positive is equivalent to its associated globalizable Hermitian metric being negatively curved.
Sep
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comment Stability of real polynomials with positive coefficients
can be isometrically embedded into complex projective space. Theorem A can be deduced from this result of Catlin and D'Angelo in the case where the complex compact manifold is the Riemann sphere and using the fact that every negative holomorphic line bundle over the Riemann surface is antiample.
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comment Stability of real polynomials with positive coefficients
After reading the answers by you and Prof Handleman and having a discussion with my PhD supervisor, we realized that the affirmative answer to my question and in fact what is now Theorem A in your revised preprint is a corollary of an isometric embedding theorem of Catlin and D'Angelo, MRL 1999. There, Catlin and D'Angelo proved, in particular, that for a globalizable Hermitian metric on a holomorphic line bundle over a compact complex manifold, if this metric is negatively curved and satisfies the so called Sharp Global Cauchy Schwarz inequality, then sufficiently large powers of this metric