bio | website | |
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location | National University of Singapore | |
age | 29 | |
visits | member for | 4 years, 11 months |
seen | Nov 12 at 3:50 | |
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I am a graduate student at the National University of Singapore studying complex geometry with To Wing Keung.
Nov 19 |
awarded | Popular Question |
Nov 11 |
awarded | Notable Question |
Nov 10 |
awarded | Nice Question |
Nov 9 |
accepted | Reverse plane geometry, anyone? |
Nov 8 |
revised |
Reverse plane geometry, anyone?
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Nov 8 |
revised |
Reverse plane geometry, anyone?
added 370 characters in body |
Nov 8 |
revised |
Reverse plane geometry, anyone?
deleted 486 characters in body |
Nov 8 |
awarded | Informed |
Nov 8 |
asked | Reverse plane geometry, anyone? |
Nov 5 |
accepted | Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism? |
Nov 4 |
awarded | Favorite Question |
Nov 4 |
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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 4 |
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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. |
Nov 4 |
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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. |
Nov 4 |
asked | Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism? |
Sep 30 |
awarded | Explainer |
Sep 18 |
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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 18 |
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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. |
Sep 18 |
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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 |
Sep 11 |
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Proof by `universal receiver'
Perhaps Chow's theorem that a complex submanifold of complex projective space is a complex variety is also required before one is able to apply the machinery of algebraic geometry. |