Complex singularities and contact topology

Popescu-Pampu, Patrick

doi:10.5802/wbln.14

Cited by 11 publications

(28 citation statements)

References 181 publications

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“…Short and A. Winter [4,5], and we follow them fairly closely. We will be giving quantitative arguments in order to convince the reader about the validity of this result and we will try to explain all the needed mathematical tools.…”

Section: Introductionmentioning

confidence: 79%

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Quantum thermodynamics and canonical typicality

Facchi

Garnero

2017

Int. J. Geom. Methods Mod. Phys.

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Section: Introductionmentioning

confidence: 79%

“…One of the most controversial issue is the validity of the postulate of equal a priori probability, which cannot be proved. In these notes we are going to discuss a set of ideas based on typicality put forward by several authors [2,3,4,5], who have been looking for a different approach.…”

Section: Introductionmentioning

confidence: 99%

Quantum thermodynamics and canonical typicality

Facchi

Garnero

2017

Int. J. Geom. Methods Mod. Phys.

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“…This hyperplane distribution in M may or may not be a contact structure: if the real hypersurface M in the complex manifold X is strongly pseudoconvex, then the distribution ζ defined above is a naturally oriented contact structure (see for instance [234,Proposition 5.11]). Pseudoconvex means that M can be defined locally, in a neighborhood of each of its points, as a regular level of a strictly plurisubharmonic function.…”

Section: Open-books and Contact Structuresmentioning

confidence: 99%

“…Every Milnor fillable contact manifold (M, ξ) is holomorphically fillable, since every resolution of a singularity whose contact boundary (∂(X, p), ξ(X, p)) is contactomorphic to (M, ξ) gives a holomorphic filling of it. Moreover, if there is a singularity germ (X, p) with contact boundary (∂(X, p), ξ(X, p)) which is smoothable (see Definition 7.5), then it is easy to construct Stein representatives of its Milnor fibers and these are Stein fillings of the contact boundary (∂(X, p), ξ(X, p)) (see [234,Proposition 6.8]).…”

Section: Open-books and Contact Structuresmentioning

confidence: 99%

“…We refer to [234,Section 6] for a clear account of the subject discussed in this subsection, including a fairly complete bibliography and a list of interesting open questions. (V, 0) is a normal surface singularity in some affine space C N , then its link L V := V ∩ S ε is a closed oriented 3-manifold with a rich geometry.…”

Section: Open-books and Contact Structuresmentioning

confidence: 99%

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On Milnor’s fibration theorem and its offspring after 50 years

Seade

2018

Bull. Amer. Math. Soc.

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To Jack, whose profoundness and clarity of vision seep into our appreciation of the beauty and depth of mathematics.Abstract. Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is two-fold. On the one hand, it should serve as an introduction to the topic for non-experts, and on the other hand, it gives a wide perspective of some of the work on the subject that has been and is being done. It includes a vast literature for further reading. IntroductionMilnor's fibration theorem in [189] is a milestone in singularity theory that has opened the way to a myriad of insights and new understandings. This is a beautiful piece of mathematics, where many different branches, aspects and ideas, come together. The theorem concerns the geometry and topology of analytic maps near their critical points.Consider the simplest case, a holomorphic map (C n+1 , 0) f → (C, 0) taking the origin into the origin, with an isolated critical point at 0. As an example one can have in mind the Pham-Brieskorn polynomials:(0.1) z → z a 0 0 + · · · + z an n , a i ≥ 2 for all i = 0, 1, · · · , n .Since f is analytic, there exists r > 0 sufficiently small so that 0 ∈ C is the only critical value of the restriction f | Br , where B r is the open ball of radius r and center at 0. Set V := f −1 (0) and V * := V \ {0} .

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Open books decompositions of links of minimally elliptic singularities

Bhupal

2021

Mathematische Nachrichten

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We present an explicit Milnor open book decomposition supporting the canonical contact structure on the link of each minimally elliptic singularity whose fundamental cycle satisfies −3 ≤ ⋅ ≤ −1. For the Milnor open books whose pages have genus less than three, we give a factorization of the monodromy which does not involve any left-handed Dehn twists around interior curves. Necessary results regarding the roots of reducible, and irreducible elements in mapping class groups are proved, and some new relations in the mapping class groups are presented.

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Complex singularities and contact topology

Cited by 11 publications

References 181 publications

Quantum thermodynamics and canonical typicality

Quantum thermodynamics and canonical typicality

On Milnor’s fibration theorem and its offspring after 50 years

Open books decompositions of links of minimally elliptic singularities

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