Corpus evidence — top 10 passages

Most-relevant passages from the entire indexed corpus (67,286 paragraph chunks across YouTube transcripts, PubMed, arXiv, archive.org, Stanford Encyclopedia of Philosophy, OpenAlex, and more) ranked by semantic similarity (bge-small-en-v1.5).

1. 01 · arxiv0.749

   In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the classification of surfaces as it relates to the Poincare and Thurston Geometrization conjectures. Finally, we survey Perelman's results on Ricci flows with surgery.

   arxiv/0803.0150-perelman-poincare-and-the-ricci-flow/info.md

2. 02 · pubmed0.746

   When he first introduced the notion of a conformal boundary into the study of asymptotically empty space-times, Penrose noted that the boundary would be null, space-like or time-like according as the cosmological constant [Formula: see text] was zero, positive or negative. While most applications of the idea of a conformal boundary have been to the zero-[Formula: see text], asymptotically Minkowskian case, there also has been work on the non-zero cases. Here, we review work with a positive [Formula: see text], which is the appropriate case for cosmology of the universe in which we live. This a…

   pubmed/PMID-38219781-conformal-methods-in-mathematical-cosmology/info.md

3. 03 · _intake0.746

   > then in effect, that isn't true anymore. So the paths always have to be finite. So it's as if time stops. So what you end up with is what in a black hole is this physical path stopping, in meta-mathematics is the proof stops. So QED is like a singularity. Yes. QED in the sense of at the end of the proof. Right. And so then it gets even funkier because in general relativity, we know

   _intake/claims-allbranch/curated-low/relativity/010-then-in-effect-that-isn-t-true-anymore.md

4. 04 · yt0.745

   Oh, I was saying, okay, so in the principal   polynomial, I can imagine that there's a lot  of theoretical considerations. I can imagine   how you can get symmetry or anti-symmetry  conditions. I can imagine how you can get   signature. I can imagine how you get that it is  non-degenerate or that is degenerate. But I don't   see how you get compatibility with connection  as a condition of the principal polynomial. But there's no connection at all. So, okay, what  we call signature in the metric, if I look at it   from t…

   yt/Bnh-UNrxYZg-frederic-schuller-the-physicist-who-derived-gravity-from-ele/transcript.txt

5. 05 · yt0.740

   And you see, there are if you can have a time-like curve connecting one event with another one, then it can causally affect it. Whereas, if it's space-like connected, there's no time-like curve connecting them, then you can't causally affect it. So, there's a whole sort of body of theory which was taken over by other people and they developed in other ways of what you call causal sets. Where you may not think about continuous manifolds. They can be just sets of points with their causal connections and so on. I'm only mentioning this because I was worrying about this in connection with making i…

   yt/vC4HNcqTQXk-roger-penrose-on-mind-consciousness-closer-to-truth-chats/transcript.txt

6. 06 · yt0.739

   Actually, Schrödinger did that,   right? Schrödinger had a modified idea. Again,  Schrödinger had an idea of modified gravity.   And he said, “no, no, no, no. Einstein does  these non-symmetric metrics. But actually,   a deeper structural concept than the  metric is the connection.” And there   are connections that come from a metric, and  they're more general connections, right? So,   that was Schrödinger's idea. Actually, a wonderful  book by Schrödinger, Space-Time Structure.   A very beautiful, thin book o…

   yt/Bnh-UNrxYZg-frederic-schuller-the-physicist-who-derived-gravity-from-ele/transcript.txt

7. 07 · openalex0.739

   (1985) Pseudo holomorphic curves in symplectic manifolds — M. Gromov — cited 2077x (1987) Hyperbolic Groups — M. Gromov — cited 1512x (1982) Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds — Jeff Cheeger M. Gromov Michael E. Taylor — cited 860x (2003) Random walk in random groups — M. Gromov — cited 498x (1992) A.D. Alexandrov spaces with curvature bounded below — Yu. D. Burago M. Gromov G. Perelman — cited 428x (1983) A Topological Application of the Isoperimetric Inequality — M. Gromov Vitali Milman — cited 3…

   openalex/A5109481429/info.md

8. 08 · arxiv0.736

   Let ${\bf M}$ be a compact Riemannian manifold and the metrics $g=g(t)$ evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for $({\bf M}, g(t))$ for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the same Sobolev imbedding holds uniformly after a standard normalization. As a consequence, long time non-collapsing results are derived, which improve Perelman's local non-collapsing results. An appli…

   arxiv/0706.1594-a-uniform-sobolev-inequality-under-ricci-flow/info.md

9. 09 · yt0.734

   So that means that the the distance itself is the square root that so if I'm going to think talk about spacetime you know manowsky spacetime that distance has to be the square root of this marovian commute time that's outside of spacetime and when you take a square root there's there's not one answer there's two plus and minus you can you know like the square root of four is not just two it's also minus two because minus 2^2 is also four. So, so that means that if I start off with this marov dynamics outside of spacetime and I'm I'm forced to say that there are two different kinds of spacetime…

   yt/Hf1q-bZMEo4-what-are-traces-of-consciousness-a-new-breakthrough-unifying/transcript.txt

10. 10 · openalex-fanout0.734

    (1994) Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes — cited 1056x (1993) Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties — cited 993x (1994) Target space duality in string theory — cited 927x (1994) Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes — cited 917x (1994) A strong coupling test of s-duality — cited 885x (2007) Flux compactification — cited 815x (1991) Scale factor duality for classical and quantum strings — cited 813x (1994) A strong coupling test of S-duality — cited 771x (1994…

    openalex-fanout/W2067602374-duality-in-calabi-yau-moduli-space/info.md