Population dynamics

In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of differential forms of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M.

It was developed by Hodge in the 1930s as an extension of de Rham cohomology, and has major applications on three levels:

• Riemannian manifolds
• Kähler manifolds
• algebraic geometry of complex projective varieties, and even more broadly, motives.

In the initial development, M was taken to be a closed manifold (that is, compact and without boundary). On all three levels, the theory was very influential on subsequent work, being taken up by Kunihiko Kodaira (in Japan and later, partly under the influence of Hermann Weyl, at Princeton) and many others subsequently.

Applications and examples

De Rham cohomology

The original formulation of Hodge theory, due to W. V. D. Hodge, was for the de Rham complex. If M is a compact orientable manifold equipped with a smooth metric g, and Ω^k(M) is the sheaf of smooth differential forms of degree k on M, then the de Rham complex is the sequence of differential operators

$0\rightarrow \Omega ^{0}(M){\xrightarrow {d_{0}}}\Omega ^{1}(M){\xrightarrow {d_{1}}}\cdots {\xrightarrow {d_{n-1}}}\Omega ^{n}(M){\xrightarrow {d_{n}}}0$

where d_k denotes the exterior derivative on Ω^k(M). The de Rham cohomology is then the sequence of vector spaces defined by

$H^{k}(M)={\frac {\ker d_{k}}{\mathrm {im} \,d_{k-1}}}.$

One can define the formal adjoint of the exterior derivative d, denoted δ, called codifferential, as follows. For all α ∈ Ω^k(M) and β ∈ Ω^k+1(M), we require that

$\int _{M}\langle d\alpha ,\beta \rangle _{k+1}\,dV=\int _{M}\langle \alpha ,\delta \beta \rangle _{k}\,dV$

where $\langle \ ,\ \rangle _{k}$ is the metric induced on Ω^k(M). The Laplacian form is then defined by Δ = dδ + δd. This allows one to define spaces of harmonic forms

${\mathcal {H}}_{\Delta }^{k}(M)=\{\alpha \in \Omega ^{k}(M)\mid \Delta \alpha =0\}.$

Since $d{\mathcal {H}}_{\Delta }^{k}(M)=0$, there is a canonical mapping $\varphi :{\mathcal {H}}_{\Delta }^{k}(M)\rightarrow H^{k}(M)$. The first part of Hodge's original theorem states that φ is an isomorphism of vector spaces. In other words, for each de Rham cohomology class on M, there is a unique harmonic representative.

One major consequence of this is that the de Rham cohomology groups on a compact manifold are finite-dimensional. This follows since the operators Δ are elliptic, and the kernel of an elliptic operator on a compact manifold is always a finite-dimensional vector space.

Hodge theory of elliptic complexes

In general, Hodge theory applies to any elliptic complex over a compact manifold.

Let $E_{0},E_{1},\dots ,E_{N}$ be vector bundles, equipped with metrics, on a compact manifold M with a volume form dV. Suppose that

$L_{i}:\Gamma (E_{i})\rightarrow \Gamma (E_{i+1})$

are differential operators acting on sections of these vector bundles, and that the induced sequence

$\Gamma (E_{0})\rightarrow \Gamma (E_{1})\rightarrow \cdots \rightarrow \Gamma (E_{N})$

is an elliptic complex. Introduce the direct sums:

${\mathcal {E}}^{\bullet }=\bigoplus _{i}\Gamma (E_{i})$

$L=\bigoplus L_{i}:{\mathcal {E}}^{\bullet }\rightarrow {\mathcal {E}}^{\bullet }$

and let L* be the adjoint of L. Define the elliptic operator Δ = LL* + L*L. As in the de Rham case, this yields the vector space of harmonic sections

${\mathcal {H}}=\{e\in {\mathcal {E}}^{\bullet }\mid \Delta e=0\}.$

So let $H:{\mathcal {E}}^{\bullet }\rightarrow {\mathcal {H}}$ be the orthogonal projection, and let G be the Green's operator for Δ. The Hodge theorem then asserts the following:

1. H and G are well-defined.
2. Id = H + ΔG = H + GΔ
3. LG = GL, L*G = GL*
4. The cohomology of the complex is canonically isomorphic to the space of harmonic sections, $H(E_{j})\cong {\mathcal {H}}(E_{j})$, in the sense that each cohomology class has a unique harmonic representative.

Hodge structures

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. An abstract definition of (real) Hodge structure is now given: for a real vector space W, a Hodge structure of integer weight k on W is a direct sum decomposition of W^C = W ⊗ C, the complexification of W, into graded pieces W^{p, q} where k = p + q, and the complex conjugation of W^C interchanges this subspace with W^{q, p}.

The basic statement in algebraic geometry is then that the singular cohomology groups with real coefficients of a non-singular complex projective variety V carry such a Hodge structure, with $H^{k}(V)$ having the required decomposition into complex subspaces H^{p, q}. The consequence for the Betti numbers is that, taking dimensions

$b_{k}=\dim H^{k}(V)=\sum _{p+q=k}h^{p,q},\,$

where the sum runs over all pairs p, q with p + q = k and where

$h^{p,q}=\dim H^{p,q}.\,$

The sequence of Betti numbers becomes a Hodge diamond of Hodge numbers spread out into two dimensions.

This grading comes initially from the theory of harmonic forms, that are privileged representatives in a de Rham cohomology class picked out by the Hodge Laplacian (generalising harmonic functions, which must be locally constant on compact manifolds by their maximum principle). In later work (Dolbeault) it was shown that the Hodge decomposition above can also be found by means of the sheaf cohomology groups $H^{q}(V,\Omega ^{p})$ in which Ω^p is the sheaf of holomorphic p-forms. This gives a more directly algebraic interpretation, without Laplacians, for this case.

In the case of singularities or noncompact varieties, the Hodge structure has to be modified to a mixed Hodge structure, where the double-graded direct sum decomposition is replaced by a pair of filtrations. This case is much used, for example in monodromy questions.

See also

• Hodge–Arakelov theory
• Hodge cycle
• Hodge conjecture
• Period mapping
• Torelli theorem
• Variation of Hodge structure
• Mixed Hodge structure
• Logarithmic form

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Ofer Gabber, Lorenzo Ramero (2009). Foundations for almost ring theory.