定量辛幾何(arXiv:math/0506191)

辛流形

( M , ω )是一個光滑流形M，其具有非退化且封閉的2形式ω。根據 Darboux 定理，這樣的流形在局部看起來像是某個R^2n=¢ n中具有標準辛形式的開集

ω_0 = dx_j ^ dy_j , ...(0–1)

因此辛流形沒有局部不變量。這與黎曼流形形成鮮明對比，黎曼流形的黎曼度量允許各種曲率不變量。然而，辛流形確實承認許多全局數值不變量，其中最突出的是所謂的辛容量。

I. Ekeland 和 H. Hofer 於 1990 年引入了辛容量 （儘管第一個容量實際上是由 M. Gromov構建的）。從那時起，許多新的容量被定義。關於辛容量的調查是[44；49; 54; 66; 95]。不同的容量以不同的方式定義，因此容量之間的關係通常會導致辛幾何和哈密頓動力學的不同方面之間出現令人驚訝的關係。這在第 2 節中進行了說明，其中我們討論了辛容量的一些示例並描述了它們存在的一些後果。在第 3 節中，我們嘗試更好地理解所有辛容量的空間，並進一步討論辛容量的一些一般性質。在第 4 節中，我們描述了橢球體和多圓盤上某些辛容量之間的幾種新關係。在整個討論過程中，我們提到了許多未解決的問題。

如下圖所示，辛幾何的許多定量方面都可以根據辛容量來表述。當然還有辛流形的其他數值不變量可以包含在定量辛幾何的討論中，例如從哈密頓微分同胚群的 Hofer 雙不變度量導出的不變量， 或 Gromov–Witten 不變量。它們與辛容量的關係還不是很清楚，我們不會在這裡討論它們。

我們首先簡要描述辛幾何與相鄰場的一些關係。

辛幾何是一門新興的、蓬勃發展的數學學科。“辛爆炸”在 [21] 中有描述。辛流形的例子是R^2n,ω_0的開子集，環面R^2n/Z^2n被賦予了導出的辛形式，表面配備了面積形式，Ka¨hler 流形如復投影空間¢P^n被賦予了他們的 Ka¨ hler 形式和余切束及其規範的辛形式。更多的例子是通過乘積和更複雜的構造獲得的，例如辛爆破操作。辛上的微分同胚φ

如果φ^∗ ω=ω ，則流形( M , ω)稱為辛或辛同胚。

辛幾何的一個迷人特徵是它處於許多其他數學學科的十字路口。在本節中，我們將提到一些此類交互的示例。

哈密​​頓動力學。

辛幾何起源於哈密頓動力學，哈密頓動力學起源於天體力學。辛流形( M , ω)上的時間相關哈密頓函數是一個光滑函數H:  R × M --> R

由於ω是非退化的，方程ω( X_H , · ) = dH ( · )

定義M上的時間相關平滑矢量場X_H 。在適當的假設下

在H上，這個向量場生成一個微分同胚族φ_H^t ，稱為

H的哈密頓流。很容易看出，每個映射φ t都是辛的。A

M上的哈密頓微分同胚 φ是φ_H^1形式的微分同胚 。

辛幾何是哈密頓系統的基礎幾何。事實證明，這種哈密頓系統的幾何方法非常有成效。下面第 2 節討論了明確的例子。

丁

體積幾何。

流形M上的體積形式是頂維無處消失的微分形式，如果φ_∗Ω=Ω ，則M的微分同胚φ是體積守恆的。遍歷理論研究體積保持映射的特性。它的發現適用於辛映射。事實上，由於辛形式ω是非退化的，ω_n是體積形式，它是保守的

在辛同胚下。在2維中，辛形式只是體積形式，因此辛映射只是體積保留映射。然而，在2n≥4維中，辛映射更為特殊。一個幾何例子是 2.2 節中陳述的 Gromov 非擠壓定理，一個動力學例子是（部分解決的）Arnol'd 猜想，該猜想指出閉合辛流形的哈密頓微分同胚至少具有與平滑函數具有臨界點一樣多的不動點。遍歷理論和辛幾何之間的另一個聯繫參見 [81]。

接觸幾何。

接觸幾何起源於幾何光學。接觸流形( P , α)是具有1形式a的( 2n-1 )維流形P，使得 α^ ( d  α)^( n - 1)是P上的體積形式。矢量場X上

由d α( X , )= 0和 α( X ) =1定義的P生成所謂的 Reeb 流。這

有時可以將與時間無關的哈密頓系統限制為能量面，將其實現為接觸流形上的 Reeb 流。接觸流形也作為辛流形的邊界自然出現。人們可以通過觀察其辛化來通過辛方法研究接觸流形( P , a)

(P × R , d ( e ^t  α) ，見例 [46; 22]。

代數幾何。

一類特殊的辛流形是 Ka¨hler 流形。可以通過查看其中的全純曲線來研究此類流形（更一般地，复流形）。M. Gromov [39] 觀察到在 Ka¨hler 上下文中使用的一些工具可以適用於辛流形的研究。他的開創性工作的一部分已經發展成為現在所謂的 Gromov-Witten 理論，參見例如 [70] 的介紹。

複雜幾何的許多其他技術和構造在辛幾何中很有用。例如，有一個辛版本的爆破，它與辛包裝問題密切相關，參見 [64; 68] 和 4.1.2 下面。另一個例子是 Donaldson 構造辛子流形 [17]。相反，辛技術被證明對研究代數幾何中的問題很有用，例如 Nagata 的猜想 [5；6; 68] 和代數簇的退化 [7]。

黎曼和譜幾何。

回想一下，光滑流形M的可微分結構在其餘切叢T^∗ M上產生了規範辛形式。

在M上給出一個黎曼度量g等價於規定它的單位餘球叢S_g^ ∗ M ⊂ T ∗ M ，並且來自T ^∗ M的規範 1-形式的限制給S ^∗ M一個接觸流形的結構。S^ ∗ M上的 Reeb 流是測地線流（自由粒子運動）。

在稍微不同的方向上，某些流形M上的每個辛形式ω區分黎曼度量的類別，這些形式對於某些幾乎複雜的結構J具有ω( J , )的形式。

辛幾何和黎曼幾何之間的這些（和其他）聯繫還沒有完全探索，我們相信這裡還有很多東西有待發現。這裡有一些已知結果的例子，這些結果與幾何的黎曼和辛方面有關。

拉格朗日子流形。( M , ω)的中維子流形L如果ω在TL上消失，則稱為拉格朗日量。

體積。賦予复射影空間¢P n以通常的 Ka¨hler 度量和通常的 Ka¨hler 形式。子流形的體積是根據這個黎曼度量來計算的。根據 Givental–Kleiner–Oh 的結果， ￠P^n中的標準RP^n在其所有哈密頓變形中體積最小 [74]。￠P n中克利福德圓環的部分結果可以在 [38] 中找到。由赤道形成的環面S ^1 × S^ 1 ⊂ S ^2 × S^ 2在其哈密頓變形中也是體積最小化，[50]。如果L是封閉的拉格朗日量

(R^2n , ω 0)的子流形 ，根據存在一個取決於L 的常數C使得vol (φ _H ( L )) ≥ C 對於L的所有哈密頓變形。 (1–1)

平均曲率。Ka¨hler-Einstein 流形中拉格朗日子流形L的平均曲率形式可以通過L的辛不變量表示，參見 [15]。

拉普拉斯算子的第一個特徵值。辛方法可用於估計某些黎曼流形的拉普拉斯算子函數的第一個特徵值 [80]。

短台球軌跡。考慮具有平滑邊界的有界域U ⊂ R _n。在U上存在一個長度為l的周期性台球軌跡

l n ≤ C n vol ( U )... (1–2)

其中C_n 是一個顯式常量，僅取決於n ，參見 [98；30]。

 

Examples of symplectic capacities

In this section we give the formal definition of symplectic capacities, and discuss a number of examples along with sample applications.

1. Definition. Denote by Symp2n the category of all symplectic manifolds of dimension 2n, with symplectic embeddings as morphisms. A symplectic category is a subcategory 9 of Symp2n such that (M, ω) 2 9 implies (M, αω) 2 9 for all α > 0.

   Convention. We will use the symbol Œ to denote symplectic embeddings and

   ! to denote morphisms in the category 9 (which may be more restrictive).

   D ×

   Let B2n(r 2) be the open ball of radius r in R2n and Z2n(r 2) B2(r 2) R2n—2 the open cylinder (the reason for this notation will become apparent below). Unless stated otherwise, open subsets of R2n are always equipped with the canon-

   j =1

   ical symplectic form ω0 D Pn dyj ^ dxj . We will suppress the dimension

   2n when it is clear from the context and abbreviate

   
   

   B WD B2n(1), Z WD Z2n(1).

   ⊂

   Now let 9 Symp2n be a symplectic category containing the ball B and the cylinder Z. A symplectic capacity on 9 is a covariant functor c from 9 to the category ([0, 1], ≤) (with a ≤ b as morphisms) satisfying

   ≤ !

   (MONOTONICITY): c(M, ω) c(M r, ωr) if there exists a morphism (M, ω) (M r, ωr);

   (CONFORMALITY): c(M, αω) D α c(M, ω) for α > 0; (NONTRIVIALITY): 0 < c(B) and c(Z)< 1.

   Note that the (Monotonicity) axiom just states the functoriality of c. A symplectic capacity is said to be normalized if

   (Normalization): c(B) D 1.

   ⊂

   As a frequent example we will use the set Op2n of open subsets in R2n. We make it into a symplectic category by identifying (U, α2ω0) with the symplectomorphic manifold (αU, ω0) for U R2n and α > 0. We agree that the morphisms in this category shall be symplectic embeddings induced by global symplectomorphisms of R2n. With this identification, the (Conformality) axiom above takes the form

   (CONFORMALITY) r: c(αU ) D α2c(U ) for U 2 Op2n, α > 0.

2. Examples of symplectic capacities

   In this section we give the formal definition of symplectic capacities, and discuss a number of examples along with sample applications.

   1. Definition. Denote by Symp2n the category of all symplectic manifolds of dimension 2n, with symplectic embeddings as morphisms. A symplectic category is a subcategory 9 of Symp2n such that (M, ω) 2 9 implies (M, αω) 2 9 for all α > 0.

      Convention. We will use the symbol Œ to denote symplectic embeddings and

      ! to denote morphisms in the category 9 (which may be more restrictive).

      D ×

      Let B2n(r 2) be the open ball of radius r in R2n and Z2n(r 2) B2(r 2) R2n—2 the open cylinder (the reason for this notation will become apparent below). Unless stated otherwise, open subsets of R2n are always equipped with the canon-

      j =1

      ical symplectic form ω0 D Pn dyj ^ dxj . We will suppress the dimension

      2n when it is clear from the context and abbreviate

      
      

      B WD B2n(1), Z WD Z2n(1).

      ⊂

      Now let 9 Symp2n be a symplectic category containing the ball B and the cylinder Z. A symplectic capacity on 9 is a covariant functor c from 9 to the category ([0, 1], ≤) (with a ≤ b as morphisms) satisfying

Riemannian and spectral geometry

 Riemannian and spectral geometry. Recall that the differentiable structure of a smooth manifold M gives rise to a canonical symplectic form on its cotangent

g

⊂

g

bundle T ∗M . Giving a Riemannian metric g on M is equivalent to prescribing its unit cosphere bundle S ∗M T ∗M , and the restriction of the canonical 1-form from T ∗M gives S ∗M the structure of a contact manifold. The Reeb flow on S ∗M is the geodesic flow (free particle motion).

· ·

In a somewhat different direction, each symplectic form ω on some manifold M distinguishes the class of Riemannian metrics which are of the form ω(J , ) for some almost complex structure J .

These (and other) connections between symplectic and Riemannian geometry are by no means completely explored, and we believe there is still plenty to be discovered here. Here are some examples of known results relating Riemannian and symplectic aspects of geometry.

Lagrangian submanifolds. A middle-dimensional submanifold L of (M, ω)

is called Lagrangian if ω vanishes on T L.

1. Volume. Endow complex projective space ¢Pn with the usual Ka¨hler metric and the usual Ka¨hler form. The volume of submanifolds is taken with respect to this Riemannian metric. According to a result of Givental–Kleiner–Oh, the standard RPn in ¢Pn has minimal volume among all its Hamiltonian deformations [74]. A partial result for the Clifford torus in ¢Pn can be found in [38]. The torus S1 × S1 ⊂ S2 × S2 formed by the equators is also volume minimizing among its Hamiltonian deformations, [50]. If L is a closed Lagrangian

   submanifold of R2n, ω0 , there exists according to [98] a constant C depending on L such that

   
   

   vol (φH (L)) ≥ C for all Hamiltonian deformations of L. (1–1)

2. Mean curvature. The mean curvature form of a Lagrangian submanifold L in a Ka¨hler–Einstein manifold can be expressed through symplectic invariants of L, see [15].

The first eigenvalue of the Laplacian. Symplectic methods can be used to estimate the first eigenvalue of the Laplace operator on functions for certain Riemannian manifolds [80].

Short billiard trajectories. Consider a bounded domain U ⊂ Rn with smooth boundary. There exists a periodic billiard trajectory on U of length l with

ln ≤ Cn vol(U ) (1–2)

where Cn is an explicit constant depending only on n, see [98; 30].

辛幾何的了解

在一個名為叭啦的遊戲公會裏，有一個成員名為小雪。她是一個來自北京的學生，對數學非常感興趣。一天，她發現公會的管理團隊正在建立一個成員名單的Wiki百科，她便開始著手寫下了她對辛幾何的了解。

小雪：大家好，我今天想和大家分享一個叫做辛幾何的數學分支。這門學科和微分幾何以及代數幾何是平行的三個分支。微分幾何的定義比較狹隘，對辛幾何的理解也就相對有所限制。而我們中國的很多做微分幾何的人也很少了解辛幾何。辛幾何的定義是一個non-degenerate closed 2-form，這個2-form不需要是positive-definite，也就不一定是一個metric。這對我們理解辛流形的重要結構是有一定影響的。

公會成員：啊？講得太高深了吧，我不是數學家，不太能理解。

小雪：別擔心，我會用一些通俗易懂的方法解釋。要理解辛幾何，就需要從Kahler流形入手。如果一個symplectic manifold上存在和辛結構compatible的近複結構，就可以得到Kahler manifold。這裏的一個重點是近複結構的可積性。如果不存在這樣的結構，辛幾何就很難被研究了。Gromov在這方面的貢獻是非常大的。

公會成員：哦，我懂了，你是在說辛幾何需要有一些rigidity。

小雪：對，就是這樣。另外，在辛幾何中還存在一個重要概念，就是symplectic vector space。這個概念比較簡單，可以通過一些線性代數的知識來理解。此外還有一些比較複雜的概念，比如symplectic toric manifold/orbifold、Lagrangian fibration和Lefschetz fibration等等。這些都是辛幾何中的重要結構，可以通過reduce到一些組合數學來研究。

公會成員：原來辛幾何可以通過組合數學來研究啊，這太神奇了。

ENGLISH111

trepidation

2know

https://www.google.com/search?safe=active&sxsrf=ALeKk02oB6LyE4ZbHwZmFB-kxQu2UoJkTw%3A1602590215621&ei=B5aFX7q9JeHemAW0rq3IBQ&q=+Katz-Pavlovic&oq=+Katz-Pavlovic&gs_lcp=CgZwc3ktYWIQA1DawwFY2sMBYMHFAWgAcAB4AIABNIgBNJIBATGYAQCgAQGqAQdnd3Mtd2l6wAEB&sclient=psy-ab&ved=0ahUKEwj63JbkwbHsAhVhL6YKHTRXC1kQ4dUDCA0&uact=5

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*

https://en.wikipedia.org/wiki/Path_integral_formulation **M.A. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics * Borrelli A. (2009) Spin Statistics Theorem. In: Greenberger D., Hentschel K., Weinert F. (eds) Compendium of Quantum Physics. Springer, Berlin, Heidelberg

*Massimi M. (2009) Exclusion Principle (or Pauli Exclusion Principle). In: Greenberger D., Hentschel K., Weinert F. (eds) Compendium of Quantum Physics. Springer, Berlin, Heidelberg ******* definite. From this he concluded, as Fierz had done before him, that a field ψ(x) with half-integer spin had to be quantised with anticommutation relations so that, by using the ensuing exclusion principle, an infinite number of negative-energy states could be regarded as being already occupied. In this way, one would in the end recover a physical system with positive energy. To prove the second part of the theorem, Pauli implemented locality by requiring that operators derived from ψ(x) and associated to physical quantities should commute for spacelike separations, i.e. for events which, in some reference frame, occur at the same time in two different places. He showed that, when a field with integer spin was quantized according to anticommutation rules, this condition would lead to a relation implying that the field is identically zero. This result was based on a mathematical argument whose legitimacy was proved only years later. ****** https://en.wikipedia.org/wiki/Descartes%27_theorem

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ahanorv-casher effect(PRL53)/Lineweaver–Burk equation/conundrum/Golay code/supercuspidal representations/Weil

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factor/Aharonov-Bohm phase//Wigner-Eckart Theorem//Dirichlet principle/Hopf theorem/Schreier’s theory of group extensions/Fubini’s theorem/Clausen function/heterotic E8 × E8/Majorana–Weyl fermions/self-dual even lattice/Kac–Moody algebras/group embedding/model building/model building/Casimir invariants /Dynkin indices/heterotic string/K theory/index theorem/loop analysis/De Rham complex/skein relation/vassilief invariants/string coupling/quantum cohomology/non-commutative geometry/combinatorical knot invariants/gromov/witten/connes/vassiliev/kontsevich/donaldson/de rham differential forms/segal functorial quantum mechanics/atiyah-singer index theorem/chern character/moduli space/module(math)/winding number/narain lattices/topological strings/mirror symmetry/holomorphic maps/kahler manifold/sympletic manifold/instanton action/pseudo-holomorphic spheres/pseudodifferential

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surface/Killing Fields/Central Extensions/Bargmann’s Theorem/Virasoro Algebra/Wick Rotation/Highest-Weight Representations/Verma Modules /Kac Determinant/Diff+(S)/vehement/spin sums and Dirac matrices/Kontsevich–Zagier period/Eyring Equation/Cartan subalgebra/Georgi–Glashow model/SO(10) model/Peccei–Quinn symmetry/MacDowell-Mansouri formalism/Coleman-Mandula theorem/Klein geometry/Coleman-Mandula theorem/Einstein-Proca model (e.g. [9,13]), the Einstein-

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TO BUY:rubik cube/isogeny/

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2BS: bell number/stirling and euler-maclaurin/so(n),so(4) and quartnion,so(8) and octonion/laplacian in high div/spherical trig/Hamilton-Jacobi theory/WKB/Gauss-Bonnet/

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*Wirtz D, Konstantopoulos K & Searson PC (2011) The physics of cancer:

the role of physical interactions and mechanical forces in metastasis.

Nat. Rev. Cancer 11, 512–522.

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