Perspectives on geometric analysis

Yau, Shing–Tung

doi:10.4310/sdg.2005.v10.n1.a8

Cited by 11 publications

(7 citation statements)

References 640 publications

(705 reference statements)

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“…To this end, let be a compact Kähler manifold and the corresponding Kähler class. When is the first Chern class of some ample line bundle L over X , such questions are closely related to the Yau–Tian–Donaldson (YTD) conjecture [ 27 , 49 , 54 ]: A polarised algebraic manifold ( X , L ) is K-polystable if and only if the polarisation class admits a Kähler metric of constant scalar curvature. This conjecture was recently confirmed in the Fano case , i.e.…”

Section: Introductionmentioning

confidence: 99%

K-Semistability of cscK Manifolds with Transcendental Cohomology Class

Dyrefelt

2017

J Geom Anal

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We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson–Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.

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Section: Introductionmentioning

confidence: 99%

K-Semistability of cscK Manifolds with Transcendental Cohomology Class

Dyrefelt

2017

J Geom Anal

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“…One of the main issues in Kähler geometry is the existence problem of Kähler metrics with constant scalar curvature on a given Kähler manifold. Through Yau's conjecture [20] and the works of Tian [17], Donaldson [4], this problem is formulated as follows; The existence of Kähler metrics with constant scalar curvature in a fixed integral Kähler class would be equivalent to a suitable notion of stability of manifolds in the sense of Geometric Invariant Theory. Though remarkable progress is made recently in this problem, we shall focus only on the related results to our purpose.…”

Section: Introductionmentioning

confidence: 99%

An example of an asymptotically Chow unstable manifold with constant scalar curvature

Ono¹,

Sano

Yotsutani

2012

Ann. inst. Fourier

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“…Looking at the distribution of killers' dimension, the largest simplex in H 0 is made of 27 concepts, while both in H 1 and H 2 has 38 concepts (it is actually the same article in both). Interestingly and not surprisingly, these two articles are both surveys, the largest for H 0 regards open questions in number theory [36] and the largest for H 1 is a survey on differential geometry [38]. The the most frequent killers' dimensions for H 0 , H 1 and H 2 are respectively 4,7,11.…”

Section: Resultsmentioning

confidence: 99%

Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge

et al. 2018

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In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. k-dimensional holes die when every concept in the hole appears in an article together with other k+1 concepts in the hole, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the size of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We provide further description of the conceptual space by looking for the simplicial analogs of stars and explore the likelihood of edges in a star to be also part of a homological cycle. We also show that authors’ conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.

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Perspectives on geometric analysis

Cited by 11 publications

References 640 publications

K-Semistability of cscK Manifolds with Transcendental Cohomology Class

K-Semistability of cscK Manifolds with Transcendental Cohomology Class

An example of an asymptotically Chow unstable manifold with constant scalar curvature

Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge

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