Adrien Sauvaget

D Zagier

VOLUMES AND SIEGEL-VEECH CONSTANTS OF H(2g -2) AND HODGE INTEGRALS

Keywords: Mathematics Subject Classification. 14H15, 14N10, 30F30, 30F60, 14C17 Moduli space of curves, translation surfaces, Masur-Veech volumes, Hodge integrals

In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller,

1. Introduction 1.1. The Hodge bundle and its stratification. Let g and n be nonnegative integers satisfying 2g -2 + n > 0. We denote by M g,n (respectively M g,n ) the moduli space of smooth (respectively stable nodal) curves of genus g with n marked points. Let π : C g,n → M g,n be the universal curve and σ i : M g,n → C g,n the sections associated to marked points for 1 ≤ i ≤ n.

The Hodge bundle p : H g,n → M g,n is the rank g vector bundle whose sheaf of sections is R 0 π * (ω Cg,n/Mg,n ). Its total space is the space of tuples (C, x 1 , . . . , x n , α): stable curves endowed with an abelian differential. We also denote by p : PH g,n → M g,n its projectivization and by p : H g,n → M g,n the restriction of the Hodge bundle to the locus of smooth curves.

The total space of the Hodge bundle is stratified according to the orders of zeros of the differential. Let µ = (k 1 , . . . , k n ) be a partition of (2g -2). We denote by H(µ) ⊂ H g the locus of curves endowed with an abelian differential with zeros of order k 1 , . . . , k n (here the zeros are not marked). The dimension of H(µ) is 2g-1+n and the Hodge bundle is the disjoint union of the H(µ) for all partitions µ of 2g -2.

The locus H(µ) is invariant under the C * -action. We denote by PH(µ) its projectivization. Besides, we denote by H(µ) (respectively PH(µ)) the Zariski closure of H(µ) in H g (respectively of PH(µ) in PH g ). The space PH(µ) is a compact (singular) DM stack that has been precisely described in [START_REF] Bainbridge | Compactification of strata of abelian differentials[END_REF].

In the present paper we are mainly interested in the strata H(2g -2) of differentials with a unique zero.

1.2. Mazur-Veech Volumes. Fix g, n and µ as above. Let (C, α) be a point in H(µ). We denote by x 1 , . . . , x n the zeros of α. Consider the relative cohomology group H = H 1 (C, {x 1 , . . . , x n }, Z).

The space H ⊗C provides a system of local coordinates of H(µ) at the neighborhood of (C, α) called the period coordinates. The transition maps between two such system of coordinates are given by matrices with integer coefficients. Therefore the space H(µ) is endowed with an affine structure and with a volume form ν on : in period coordinates, this volume form is the Lebesgue volume form normalized in such a way that the lattice H ⊗ (Z ⊕ iZ) has volume 1.

We denote by H 1 (µ) ⊂ H(µ) (respectively H ≤1 (µ)) the subspace defined by

i 2 C α ∧ α = 1 (respectively ≤ 1).
The volume form on H(µ) induces a form ν 1 on H 1 (µ), by a disintegration of ν (see [START_REF] Eskin | Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow[END_REF] for definition). The total volume of H 1 (µ) for ν 1 is finite (see [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF] and [START_REF] Veech | Gauss measures for transformations on the space of interval exchange maps[END_REF]). This is the Masur-Veech volume (or simply the volume) of H 1 (µ). We denote it by Vol(µ).

1.3. Siegel-Veech constants. The spaces H(µ) and H 1 (µ) are endowed with an action of SL(2, R). This action is defined in period coordinates: the group SL(2, R) acts simultaneously on all coordinates. The diagonal sub-group e t 0 0 e -t , t ∈ R endows H 1 (µ) with an ergodic flow with finite measure. This action lifts to the real vector bundle whose fiber at (C, α) is given by H 1 (C, R). This vector bundle is also endowed with an equivariant measure. This set-up allows to define the 2g Lyapunov exponents of the vector bundle. These 2g invariants are of the following form: [START_REF] Eskin | Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow[END_REF]) . We define the Siegel-Veech constant of H 1 (µ) as

λ 1 ≥ λ 2 . . . ≥ λ g ≥ 0 ≥ λ g+1 = -λ g ≥ . . . ≥ λ 2g = -λ 1 (see
c area (µ) = 3 π 2   λ 1 + . . . + λ g - 1 12 ki∈µ k i (k i + 2) k i + 1   .
Remark 1.1. This definition of c area (µ) is actually a theorem (see [START_REF] Eskin | Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow[END_REF]). The Siegel-Veech constant has an inner geometrical interpretation in terms of the number of families of closed geodesics of a general curve in H(µ).

Intersection numbers on strata of differentials.

In the present text, unless otherwise mentioned, we consider cohomology classes with rational coefficients. We use the following cohomology classes in M g,n :

• for all 1 ≤ i ≤ n, let L i = σ * i ω Cg,n/Mg,n → M g,n be the line cotangent line bundle at the i-th marked point. We denote by

ψ i = c 1 (L i ) ∈ H 2 (M g,n ); • for all 1 ≤ i ≤ g, we denote by λ i = c i (H g,n ) ∈ H 2i (M g,n
) (we will no longer mention Lyapunov exponents, therefore the notation λ i will only stand for these Chern classes);

• δ 0 ∈ H 2 (M g,n )
is the Poincaré-dual class of the divisor whose generic point is a curve with a self-intersecting node.

If no confusion arises, we use the same notation for classes in H * (M g,n ) and their pull-back to H * (PH g,n ) under p. We denote by L = O(1) → PH g,n the dual of the canonical line bundle and by ξ = c 1 (L) the canonical class (beware that here canonical class does not refer to the determinant of the cotangent bundle). We recall that the splitting principle implies

H * (PH g,n ) H * (M g,n )[ξ]/(ξ g + λ 1 ξ g-1 + . . . + λ g ).
The top cohomology group H 2(4g-4+n) (PH g,n ) is canonically identified with Q by Poincaré-duality. We consider the following intersection numbers

PH(µ) ξ 2g-2+n ∈ Q.
We will see that the intersection number PH(µ) ξ 2g-2+n vanishes if µ = (2g -2). We will denote by

a g = (-1) g PH(2g-2) ξ 2g-1 .
The line bundle O(1) is endowed with a natural singular hermitian metric (see Section 2). In all the paper we will make the same assumption as in [START_REF] Kontsevich | Lyapunov exponents and hodge theory[END_REF].

Assumption 1.2.

There exists a desingularization φ : X → PH(µ) such that the curvature form associated to the hermitian metric on φ * O(1) is good in the sense of [START_REF] Mumford | Hirzebruch's proportionality theorem in the noncompact case[END_REF] Under this assumption, we have the following relation between a g and Vol(2g-2). Proposition 1.3. For all g ≥ 1, we have

(1) Vol(2g -2) = 2(2π) 2g (2g -1)! a g . Remark 1.4.
The assumption 1.2 should be proved soon. The authors of [START_REF] Bainbridge | Compactification of strata of abelian differentials[END_REF] have announced the existence of the desingularization of PH(µ). This desingularization is essentialy obtained by a refinement of the data defining the stratification of PH(µ).

Besides, for a general µ, there exists a cohomology class

β ∈ H 2(2g-3+n) (PH(µ), R) such that c area (µ) = - 1 4π 2 • PH(µ) δ 0 ∧ β PH(µ) ξ ∧ β .
In the specific case of µ = (2g -2), under the Assumption 1.2 the cocycle β is a multiple of ξ 2g-2 (see [START_REF] Kontsevich | Lyapunov exponents and hodge theory[END_REF]). Therefore, if we denote by

d g = (-1) g-1 PH(2g-2) δ 0 ξ 2g-2 ,
then we have the relation

(2) c area (2g -2) = d g 4π 2 a g .
Remark 1.5. Volumes and Siegel-Veech constants are positive, therefore a g and d g are positive (this explains our sign convention).

1.5. Statement of the results. We define the following formal series

F(t) = 1 + g>0 (2g -1)a g t 2g , ∆(t) = g>0 (2g -1)d g t 2g , S(t) = t/2 sin(t/2)
.

The main theorem of the paper is the following.

Theorem 1.6. For all g > 0, we have

(3) [t 2g ]S(t) = 1 (2g)! [t 2g ]F(t) 2g . and (4) [t 2g-2 ]S(t) = 2 (2g -1)! [t 2g ] ∆ • F(t) 2g-1 .
(where the notation [t n ] stands for the n-th coefficient of the formal series).

In particular, the above equality implies that the a g 's and d g 's can be computed inductively using the coefficients of S.

1.6. Asymptotic behavior for large genera. In the past few years, algebraic geometers started to study the large genus asymptotic behavior of numerical invariants associated to moduli spaces of curves. For example, M. Mirzakhani and P. Zograf identified the large genus asymptotics of the Weil-Petersson volumes (see [START_REF] Mirzakhani | Towards large genus asymptotics of intersection numbers on moduli spaces of curves[END_REF]). For strata of differentials, A. Eskin and A. Zorich proposed the following conjectures.

Conjecture 1.7 (see [START_REF] Eskin | Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera[END_REF]). Volumes of strata satisfy

Vol(k 1 , . . . , k n ) = 4 (k 1 + 1)(k 2 + 1) . . . (k n + 1) (1 + 1 (µ))
where lim

g→∞ max µ 2g-2 | 1 (µ)| = 0.
Conjecture 1.8. We have

c area (µ) = 1 2 + 2 (µ)
where lim

g→∞ max µ 2g-2 | 2 (µ)| = 0.
We will use the induction Formulas (3) and (4) to compute the asymptotic expansion of a g 's and d g 's. Then, using Formulas (1) and (2), we will deduce the following result. Theorem 1.9. For all R ≥ 1, we have

(2g -2) 4 Vol(2g -2) = 1 - π 2 12g + 24π 2 -π 4 288g 2 + . . . + c R g R + O 1 g R+1 , and c area (2g -2) = 1 2 - 1 4g + . . . + c R g R + O 1 g R+1 ,
where the coefficients c k and c k lie in Q[π 2 ] and can be effectively computed. In particular conjectures 1.7, and 1.8 hold for H(2g -2).

Remark 1.10. For g ≥ 4, the space H(2g -2) has three connected components: hyperelliptic, odd and even (see [START_REF] Kontsevich | Connected components of the moduli spaces of Abelian differentials with prescribed singularities[END_REF]). Our formulas do not separate the volumes and Siegel-Veech constants of these three components. The volume is the sum of volumes of the connected components and the Siegel-Veech constant is a mean of the Siegel-veech constants of the connected components that is weighted by the volumes.

The hyperelliptic component has an explicitly computable volume and Siegel-Veech constants (see [START_REF] Jayadev | Right-angled billiards and volumes of moduli spaces of quadratic differentials on CP 1[END_REF]). We will check that the volume of the hyperelliptic component is asymptotically negligible in comparison to the total volume. It remains to separate the volumes (and Siegel-Veech constants) of the odd and even components. These are conjectured to be equivalent as g goes to infinity (see [START_REF] Eskin | Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera[END_REF]). 1.7. Some comments on Theorem 1.9. In [START_REF] Chen | Quasimodularity and large genus limits of siegel-veech constantss[END_REF], D. Chen, M. Möller and D. Zagier computed the asymptotic expansion of volumes and Siegel-Veech constants of strata of differentials with simple zeros:

Vol 1, 1, . . . , 1 2g-2 ∼ g→+∞ 1 4 g-1 1 - π 2 24g + 60π 2 -π 4 1152g 2 + . . . ,
and

c area (1, . . . , 1) = 1 2 - 1 8g + . . . .
Therefore Conjectures 1.7 and 1.8 also hold for these strata.

Their proof relies on a careful analysis of the formula of Eskin and Okounkov for volumes of strata. The main ingredient of this formula is the generating function of the number of ramified coverings of the punctured torus. These formal series are quasi-modular forms and the volumes (and Siegel-Veech constants) of strata are expressed using the asymptotic of the coefficients (see [START_REF] Eskin | Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials[END_REF]).

By numerical experiments, one observes that, for a fixed value of g, | 1 (µ)| and | 2 (µ)| are maximal for µ = (2g -2) and minimal for µ = (1, . . . , 1) (see Figure 1).

We can observe that the dominating term of 1 (2g -2) is -π 2 /12g which is twice the leading term in the expansion of 1 (1, . . . , 1) (the same holds for 2 ). This leads us to the following straightening of the conjecture of Eskin and Zorich (see Figure 2).

-1 (µ) = π 2 6 dim(H(µ)) (1 + 1 (µ)), -2 (µ) = 1 2 dim(H(µ)) (1 + 2 (µ)),
where 1 and and 2 tends uniformly to 0 as g goes to infinity. 1.8. Plan of the paper. We will follow linearly the general strategy of the introduction. In Section 2 we show how to express the volumes of the strata H 1 (2g -2) in terms of integrals of ξ-classes. In Section 3 we prove the induction formula for the integrals of ξ-classes (Theorem 1.6). The main ingredient in this proof is the computation of the cohomlogy classes Poincaré-dual to [PH(µ)] ∈ H * (PH g ) as in [START_REF] Sauvaget | Cohomology classes of strata of abelian differentials[END_REF] (we will recall a simplified version of this computation here). Finally, in Section 4 we analyze the asymptotic behavior of the integrals of ξ-classes to deduce Theorem 1.9.

Acknowledgement. I would like to thank Dimitri Zvonkine, Martin Moëller, Dawei Chen, Xavier Blot, Siarhei Finski and Felix Janda for very useful conversations on intersection of tautological classes over spaces of differentials. I am also very thankful to Anton Zorich and Charles Fougeron for having introduced me to the topic of large-genus invariants (and for having provided tables of numerical computations that allowed me to understand part of the results of the paper). Finally, I am very grateful to Elie de Panafieu for his precious help to handle the asymptotic analysis.

Volumes and integrals of canonical classes

In this section we prove that Vol(2g -2) = 2(2π) 2g (2g-1)! a g for all g ≥ 1 (i.e. Proposition 1.3). This identity may be known to experts. However, we could not find a reference to cite so we give the details of the proof.

The symplectic affine structure on

H(2g -2). Let g ≥ 1. Let (C, α) ∈ H(2g-2). Let x ∈ C
be the unique zero of α. We have seen that a local parametrization of H(2g -2) is given by the relative cohomology group H 1 (C, {x}, C). This space is isomorphic to H 1 (C, C). Moreover, this space contains the lattice H 1 (C, Z⊕ iZ).

We choose a symplectic basis of H 1 (C, Z) made of closed curves (A i , B i ) 1≤i≤g on C. With this basis, the coordinates on H 1 (C, C) are given by

z Ai = Ai α and z Bi = Bi α.
Therefore the space H(2g -2) is endowed with an affine structure whose transition map are matrices in Sp(2g, Z). Besides, the reciprocity law defines a hermitian metric on H 1 (C, C) given by

(5) α, α = i 2 C α ∧ α = i 2 g i=1 (z Ai z Bi -z Ai z Bi ).
The lattice

H 1 (C, Z ⊕ iZ) ⊂ H 1 (C, C) defines a volume form ν in H 4g (H(2g - 2), R) (normalized
in such a way that the volume of a unit cube is 1). Both the volume form and the hermitian metric are independent of the choice of the basis (A i , B i ) 1≤i≤g . Now, for all R ∈ R + , we denote by H ≤R (2g -2) the space of (C, α) with ||α|| ≤ R.

Two volume forms on PH(2g -2).

Using the period coordinates we define the following volume forms.

• Let us consider the complex projectivization of the space of period coordinates PH 1 (C, C). The line bundle O(-1) → PH 1 (C, C) is endowed with the hermitian metric h induced by the above hermitian metric •, • on H 1 (C, C). This hermitian metric extends to a singular hermitian metric on O(-1) → PH(2g -2). We define ω to be the curvature form associated to h in PH(2g -2), i.e.

ω = 1 2iπ ∂∂ log (h(σ))
for any local holomorphic non-vanishing section σ of O(-1). With this 2-form we define ν h = ω 2g-1 . The form ω is a closed 2-form on PH(2g -2). However, this form cannot be extended to a regular form on the whole boundary of PH(2g-2). Indeed, the function h is ill-defined along the divisor of pairs (C, α) with infinite area; i.e. when C has at least one non-seperating node and the form α has a pole of order exactly 1 at the two branches of this node. This obstruction is the motivation for Assumption 1.2.

• The second volume form on PH(2g -2), ν, is obtained by disintegration of ν. We have a projection map p : Lemma 2.1. We have

H(2g -2) → PH(2g -
(6) ν h = - (2g -1)! 2(2iπ) 2g dim R (H(2g -2)) ν. Proof. Let us fix a point (C, α) ∈ H(2g -2) (respectively in PH(2g -2)). An open neighborhood of this point is of the form is of the form U/Aut(C, α) for some open set U in H 1 (C, C) (respectively in PH 1 (C, C)). This open set is included in the positive cone C = {v ∈ H 1 (C, C), s.t. h(v) > 0} (respectively in PC).
The volume forms ν h and ν can be defined on PC and we will prove that the relation ( 6) holds in PC.

In H 1 (C, C) we have the coordinates (z Ai , z Bi ) 1≤i≤g obtained from the symplectic structure on C. We introduce the following coordinates

z ai = 1 2 (z Ai -iz Bi ), z bi = 1 2 (z Ai + iz Bi ).
With these coordinates, the hermitian metric h is given by

h(z ai , z bi ) = g i=1 z ai z ai -z bi z bi
and it has signature (g, g). We will also consider the standard metric on H 1 (C, C) given by

h st (z ai , z bi ) = g i=1 |z ai | 2 + |z bi | 2 .
Using this hermitian metric, we can define the curvature of O(-1) on PH 1 (C, C) using the Laplace-Beltrami operator and the top intersection of this form ν st . Besides one can also construct ν st by the same procedure as ν. The volume forms ν st and ν st are proportional (see [START_REF] Voisin | Hodge theory and complex algebraic geometry. I[END_REF], Chapter 3 for example). Let us define the two following functions on PH 1 (C, C) with value in R >0 .

f = ν h ν st , and f = ν ν st .
We have two groups U (g, g) and U (2g) (for the standard metric) acting on PH 1 (C, C). The functions f and f are invariant under the action of γ ∈ U (g, g) ∩ U (2g). Indeed ν h is U (g, g)-equivariant and ν st is U (2g)-equivariant thus f is invariant under U (g, g) ∩ U (2g). Besides the function f is equal to

f (v) = h(v) h st (v) 2g
for any vector v = 0 which is invariant under the action of U (g, g)∩U (2g). Therefore ν h / ν is invariant under the action of U (g, g) ∩ U (2g).

The group U (g, g) ∩ U (g) contains the matrices of the form

U 0 0 U , with U and U in U (g).
For all a and b in R ≥0 , these matrices act transitively on the subspace

H 1 (C, C) ⊃ E a,b = (z a1 , z b1 , . . .), g i=1 |z ai | 2 = a and g i=1 |z bi | 2 = b .
Therefore, we only need to compare the volume forms ν h and ν at the points of the form (z a1 , z b1 , 0, 0, . . . , 0). We consider the chart U a1 ⊂ PH 1 (C, C) defined by z a1 = 0 with its natural identification U a1 C 2g-1 . The line bundle O(-1) has a natural section σ a1 over U a1 given by σ a1 (z b1 , z a2 , z b2 , . . .) = (1, z b1 , z a2 , z b2 , . . .). In this chart, the volume form ν is given by

2(2π) dim R (H(2g -2)) • h(σ a1 ) 2g i 2g-1 dz b1 ∧ dz b1 ∧ i>1 (dz ai ∧ dz ai ∧ dz bi ∧ dz bi ) .
Note the factor 2 in this formula, it comes from the choice of coordinates (z ai , z bi ): in coordinates (z Ai , z Bi ) the expression of ν is the same without this factor 2. At the point (1, z b1 , 0, . . . , 0), the 2-form ω is given by

ω = 1 2iπ (1 -|z b1 | 2 ) -dz b1 ∧ dz b1 + i>1 dz ai ∧ dz ai -dz bi ∧ dz bi -|z b1 | 2 dz b1 ∧ dz b1 (1 -|z b1 | 2 ) 2 = 1 2iπ i>1 dz ai ∧ dz ai -dz bi ∧ dz bi h(σ a1 ) - dz b1 ∧ dz b1 h(σ a1 ) 2 .
We get

ν h = (2g -1)! (2iπ) 2g-1 h(σ a1 ) 2g • (-1) g dz b1 ∧ dz b1 ∧ i>1 (dz ai ∧ dz ai ∧ dz bi ∧ dz bi ) = - (2g -1)! 2(2iπ) 2g dim R (H(2g -2)) ν.
End of the proof of Proposition 3.9 under the assumption 1.2. Let φ : X → PH(2g-2) be a desingularization of PH(2g -2) such that the hermitian metric on φ * (O(1)) is good. Then the intersection number X φ * (ξ 2g-1 ) is equal to the total volume of PH(2g -2) for ν h = ω 2g-1 . Now using the projection formula and the fact that φ is birational (thus of degree 1) we get

PH(2g-2) ξ 2g-1 = ν h (PH(2g -2)) .
Therefore Propostion 3.9 follows from Lemma 2.1.

Induction formula for integrals of canonical classes

Let Z = (k 1 , . . . , k n ) be a vector of non-negative integers. From now on in the text, for all vectors of integers we set

(Z) = length(Z) and |Z| = n i=1 k i .
Definition 3.1. The projectivized marked stratum of type Z is the locus A g (Z) ⊂ PH g,n defined as

{(C, α, x 1 , . . . , x n ), s.t. x i is a zero of order k i for all 1 ≤ i ≤ n} .
It is a smooth substack of PH g,n codimension |Z|. We denote by A g (Z) the Zariski closure of A g (Z) in PH g,n .

Note that in this definition Z does not need to satisfy |Z| = 2g -2. However, if |Z| = 2g -2, then we have

PHg ξ 2g-2+n [PH(Z)] = 1 |Aut(Z)| PHg,n ξ 2g-2+n [A g (Z)]
where Aut(Z) is the group of permutation of [ [1, n]] preserving Z (this follows from the projection formula). The purpose of this section is to compute the intersection number on the right-hand side.

3.1. Vanishing for n > 1. First let us recall the following classical result.

Lemma 3.2 (Mumford,[START_REF] Mumford | Towards an enumerative geometry of the moduli space of curves[END_REF]). We have the following equality in H * (PM g,n ):

s * (H g,n ) = c * (H g,n ) -1 = c * (H ∨ g,n )
where s * and c * stand for the total Segre and Chern classes. In particular λ 2 g = 0.

We use this identity here to simplify the computation of

ξ 2g-2+n [A g (Z)]. The class [A g (Z)] is equal to 2g-2 k=0 ξ 2g-2-k α k g (Z)
where the classes α k g (Z) are pull-back from H 2k (M g,n ). Therefore the push-forward of ξ 2g-2+n [A g (Z)] under p is given by

2g-2 k=0 s g-1+n+k α k g (Z) = 2g-2 k=0 (-1) g-1+n+k λ g-1+n+k α k g (Z)
(this follows from the projection formula and Lemma 3.2). However λ 2g-2+n+k = 0 for 2g -2 + n + k > g. Therefore we get

Proposition 3.3. The class ξ 2g-2+n [A g (Z)] vanishes if n > 1.
For n = 1 we have:

PHg,1 ξ 2g-1 [A g (2g -2)] = Mg,1
(-1) g λ g α 0 g (Z).

3.2. λ g -Conjecture. We recall the following important result of Faber and Pandharipande.

Proposition 3.4 (λ g -conjecture, [START_REF] Faber | Hodge integrals and Gromov-Witten theory[END_REF]). Let b g = Mg,1 λ g ψ 2g-2

1

. Then we have

b g = 2 2g-1 -1 2 2g-1 |B 2g | (2g)! ,
where B 2g is the (2g)th Bernouilli number.

In particular, we have

S(t) = 1 + g>0 b g t 2g
(where we recall that S(t) = t/2 sin(t/2) ). 3.3. Stable differentials. The main tool to prove Theorem 1.6 will be the induction formula established in [START_REF] Sauvaget | Cohomology classes of strata of abelian differentials[END_REF] to compute the cohomology classes [A g (Z)] in H * (PH g,n ). We will state a simplified version of this induction formula because we only need to compute the class λ g α 0 g (Z) ∈ H * (M g,n ). The notation of the present text will also be slightly lighter than the one of [START_REF] Sauvaget | Cohomology classes of strata of abelian differentials[END_REF]. Definition 3.5. Let P = (p 1 , . . . , p m ) be a vector of positive integers. The space of stable differentials H g,n,P is the space whose geometric points are tuples (C, x 1 , . . . , x n+m , α) where • (C, x 1 , . . . , x n+m ) is a pre-stable curve (i.e. a nodal curve with distinct marked points in the smooth locus); • α is a meromorphic differentials with poles of order (p i + 1) at x n+i for all 1 ≤ i ≤ m; • there are finitely many automorphisms of C preserving α.

Let p : H g,n,P → M g,n+m be the forgetful map of the differential. The space H g,n,P is naturally equipped with a structure of cone over M g,n+m . In particular it has a projectivization PH g,n,P . The rank of this cone is m + |P | + g -1 (if P is not empty). Definition 3.6. Let Z = (k 1 , . . . , k n ) be a vector of non-negative integers. We denote by A g (Z, P ) ⊂ PH g,n,P the locus of differentials with zeros of order k i at x i for all 1 ≤ i ≤ n and residues equal to zero at the poles. We denote by A g (Z, P ) its Zariski closure.

Later, we will need the following lemma in genus 0. Lemma 3.7. We suppose that g = 0 and Z = (k). If k + 1 = |P |, then we have

p * A 0 ((k), P ) = 0 ∈ H * (M 0,m+1 ). If k + 1 = |P |, then this class is equal to 1 ∈ H 0 (M 0,m+1 ). Proof. The locus A 0 ((k), P ) is of codimension k + m -1 in PH 0,1,P which is of relative dimension m + |P | -2 over M 0,1+m . Thus the class p * [A 0 ((k), P )] vanishes if k + 1 < |P |.
Besides, any meromorphic differential without residues on a genus 0 curve is exact. In other words any such differential can be integrated to get a meromorphic function of degree d = |P |.

Thus A 0 ((k), P ) is empty if k -1 > d = |P |.
Finally, if k-1 = |P | then p * [A 0 ((k), P )] = 1. Indeed, for every curve in M 0,1+m there exists exactly one differential up to scale with a zero of order k at x 1 and poles (without residue) of orders prescribed by P at the other marked points: it is the derivative of the function

1 m i=1 (z -z i+1 ) pi (the first marked point has coordinate z 1 = ∞).

Twisted graphs.

Let m be a positive integer. We denote by Part(g) m the set of vectors g = (g 0 , g 1 . . . , g m ) of nonnegative integers such that |g| = g and g i = 0 for all 1 ≤ i ≤ m. Given such vector g = (g 0 , g 1 . . . , g m ) we denote by H g ⊂ H g,1 the space of tuples (C, x 1 , α) such that:

• the dual graph of C is the following

g 1 g 2 . . . g m g 0
and the marked point x 1 lies on the component of genus g 0 . • α is identically 0 on the component of genus g 0 and non identically 0 on all the other ones.

We denote by H g the closure of H g . We introduce the following space

H g = H g0,m+1 × m j=1 H gi,1 and M g = M g0,m+1 × m j=1 M gi,1 .
We have two natural gluing maps ζ g : M g → M g,1 and ζ # g : H g → H g,1 (in fact the space H g is the pull-back of the Hodge bundle under ζ g ). Now, we fix a choice of g = (g 0 , g 1 . . . , g m ) ∈ Part(g) m and an integer k ≥ 0. Definition 3.8. A twist for the pair (g, k) is a vector I = (i 1 , . . . , i m ) of positive integers such that

k ≥ g 0 -1 + |I|.
The multiplicity m(I) of the twist I is the product of its entries.

Given g, k, and I we construct the locus A g,k,I ⊂ PH g of tuples (C, x 1 , α) such that:

• the differential α has a zero of order (i i -1) at the node of the component of genus g i for all 1 ≤ i ≤ m; • the component C 0 of genus g 0 of the curve lies in p(A g0,(k),I ) where p :

H g,1,I → M g,1 is the forgetful map (i.e. in the image of differentials with poles prescribed by I at the nodes and a zero of order k at the marked point).

We denote by A g,k,I the Zariski closure of A g,k,I in PH g,1 .

Proposition 3.9 ([15], Proposition 3.21). The Poincaré-dual class of A g,k,I is equal to

(ξ g0 + λ 0 1 ξ g0-1 + . . . + λ 0 g0 ) • ζ # g *   p * p * [A g0,(k),I ], m j=1 [A gj (i j -1)]   where λ 0 i = p * (ζ g * (λ i , 1, 1, . . . , 1 m×
)) (we recall that ζ g * goes from the cohomology ring

H * (M g ) H * (M g0,m+1 ) m j=1 H * (M gj ,1 ) to H * (M g,n )).
We denote by α 0 g,k,I ∈ H * (M g,1 ) the degree 0 coefficient (in ξ) of A g,k,I . The above lemma implies the following expression for α 0 g,k,I .

Lemma 3.10. The following equality holds in H 2k+2 (M g,1 ):

α 0 g,k,I = ζ g *   λ g0 p * [A g0,(k),I ], m j=1 α 0 gj (i j -1)   . As a consequence λ g • α 0 g,k,I = 0 if g 0 = 0.
Proof. The first part of the lemma follows from Proposition 3.9. The second part is a consequence of the decomposition

λ g • ζ g * (1, . . . , 1) = ζ g * m j=0 λ gj .
Indeed, this implies the equality (7)

λ g • α 0 g,k,I = ζ g *   λ 2 g0 p * [A g0,(k),I ], m j=1 λ gj α 0 gj (i j -1)   .
Now the second part of the lemma follows from Lemma 3.2: the class λ 2 g0 = 0 if g 0 > 0.

3.5. Induction formula for a g 's. The main tool to compute the a g 's and d g 's is the following proposition. Proposition 3.11 ([15], Theorem 4). For all g, k ≥ 0, the following equality holds in H * (M g,1 ):

(8) (k + 1)ψ 1 • α 0 g (k) = α 0 g (k + 1) + m≥1 1 m!   g∈Part(g)m I m(I)α 0 g,k,I   ,
where the right hand sum runs over all g and all possible twists for the pair (g, k) as in Definition 3.8.

Remark 3.12. Note that our definition of twisted graph only includes trees with a unique vertex where the differentials vanishes identically. In the induction formula of [START_REF] Sauvaget | Cohomology classes of strata of abelian differentials[END_REF] the set of twisted graphs is larger (bicolored graphs). However, bicolored graphs that are not trees do not contribute to the degree 0 part of the induction. Indeed the cohomology class associated to any bi-colored graph Γ is of the form ξ b1(Γ) a with a ∈ H * (H g,1 ) where b 1 (Γ) is the number of loops of Γ.

The induction formula of Propostion 3.11, together with Lemmas 3.4 and 3.10 implies Theorem 3.13. We have

b g = k≥1 1 k!(2g -k)! d1+...+d k =g   k j=1 (2d j -1)a dj   .
Remark 3.14. This theorem is a reformulation of Formula (3) of Theorem 1.6. Let us outline the strategy of the proof.

• The starting point of the proof is the induction formula of Proposition 3.11 for the classes α 0 g (k): for all k, we multiply this formula by λ g to reduce the summation on the RHS.

• Then we identify the numerical contribution of each term in the reduced sum for all k. • We conclude by summing the contributions for all values of k between 0 and 2g -3.

Proof. We have seen that a g = Mg,1 λ g α 0 g (2g -2). Thus we begin by multiplying equation ( 8) by λ g to obtain

(9) (k + 1)ψ 1 • λ g α 0 g (k) = λ g α 0 g (k + 1) + g,I,g0=0 m(I) ( (g) -1)! λ g α 0 g,k,I ,
where the right-hand sum runs over all g with g 0 = 0 and all possible twists. Indeed Lemma 3.10 implies that the terms with g 0 = 0 vanish. Therefore we need to compute the intersection numbers

Mg,1

ψ 2g-3-k 1 λ g α 0 g,k,I .
We have seen (Formula ( 7)) that

λ g • α 0 g,k,I = ζ g *   p * [A 0,(k),I ], m j=1 λ gj α 0 gj (i j -1)   .

Besides the class p

* [A g0,(k),I ] is equal to 1 if k + 1 = |I| and 0 otherwise. Therefore, if k + 1 = |I| we have Mg,1 ψ 2g-1-k 1 λ g α 0 g,k,I = M0,m+1 ψ 2g-3-k 1 ×   m j=1 Mg j ,1 λ gj α 0 gj (i j -1)   .
For all 1 ≤ j ≤ n, the class λ gj α 0 gj (i j -1) is not in top degree if i j -1 = 2g j -2 thus we get [START_REF] Kontsevich | Connected components of the moduli spaces of Abelian differentials with prescribed singularities[END_REF] Mg j ,1

λ gj α 0 gj (i j -1) = 0 if i j -1 = 2g j -2 (-1) g0 a g0 otherwise.
Therefore for all g with g 0 = 0, we denote by I(g) the twist (2g 1 -1, 2g 2 -1, . . . , 2g m -1).

Finally, the string equation implies that M0,m+1 ψ 2g-3-k 1 = 1 if 2g -3-k = m-2 and 0 otherwise. Thus we set m = 2g -1 -k and we multiply equation ( 9) by ψ m-2 1 to obtain

ψ m-2 1 λ g (k + 1)ψ 1 α 0 g (k) -α 0 g (k + 1) = 1 m! g∈Part(g)m g0=0 m(I(g))λ g ψ m-1 1 α 0 g,k,I(g) = (-1) g m! g1+...+gm=g   m j=1 (2g j -1)a gj   .
Finally we get

(2g -2)!ψ 2g-2 1 λ g = a g + k>1 (2g -2)(2g -3) . . . (2g -k + 1) k! g1+...+g k =g   k j=1 (2g j -1)a gj   .
This implies

ψ 2g-2 1 λ g = k>0 1 k!(2g -k)! g1+...+g k =g   k j=1 (2g j -1)a gj   .
We use Proposition 3.4 to conclude

b g = k>0 1 k!(2g -k)! g1+...+g k =g   k j=1 (2g j -1)a gj   .
3.6. Induction formula for the d g 's. First let us recall that λ g • δ 0 = 0 (see [START_REF] Mumford | Towards an enumerative geometry of the moduli space of curves[END_REF] for example). Besides, we have the following equality

Mg,1 δ 0 • λ g-1 ψ 2g-2 1 = 1 2 Mg-1,3 λ g-1 ψ 2g-2 1 = 1 2 b g-1
(see [START_REF] Faber | Hodge integrals and Gromov-Witten theory[END_REF] for example). Moreover, if we still denote [A g (2g

-2)] = 2g-2 k=0 ξ k α k then we have Ag(2g-2) ξ 2g-2 δ 0 = Mg,1 (-1) g-1 δ 0 λ g-1 α 0 .
Using these equalities, we can prove the following theorem (which is equivalent to Formula (4) in Theorem 1.6). Theorem 3.15. For all g > 0, we have

b g-1 2 = g g =1 g-g k=0 (2g -1)d g k!(2g -1 -k)!   g1+...+g k =g-g (2g i -1)a gi   .
Proof. We follow the same strategy as in the proof of Theorem 3.13: we will multiply the induction formula of Proposition 3.11 by λ g-1 δ 0 and then identify the numerical contributions of the sum in the RHS.

First we fix some g ∈ Part m (g). We recall that we have

(δ 0 λ g-1 ) • ζ g (1, . . . , 1) = m j=0
ζ g (λ g0 , . . . , δ 0 λ gj -1 , . . . , λ gm ). Now we fix k ≥ 0 and a twist I for g. We study the intersection number

ψ 2g-3-k 1 δ 0 λ g-1 α 0 g,k,I .
By the same arguments as before, this intersection number vanishes if g 0 = 0, or if

I = I(g), or if 2g -3 -k = m -2.
Besides if g 0 = 0, I = I(g), and m = 2g -1 -k then

ψ 2g-3-k 1 δ 0 λ g-1 α 0 g,k,I = m j=1 (2g j -1)d gj   j =j (2g j -1)a g j   .
Thus, using the induction Formula (8) for the classes α 0 g (k) we get

ψ m-2 1 δ 0 λ g-1 (k + 1)ψ 1 α 0 g (k) -α 0 g (k + 1) = 1 m! g1+...+gm=g m j=1 (2g j -1)d gj   j =j (2g j -1)a g j   = 1 (m -1)! g g =1 (2g j -1)d gj g1+...+gm-1=g-g   m j=1 (2g j -1)a gj   , if m = 2g -k -1.
Therefore, if we sum over all 0 ≤ k ≤ 2g -3, then we get

(2g -2)!ψ 2g-2 1 δ 0 λ g-1 = d g + k>0 1≤g ≤g (2g -2)! (k)!(2g -k -1)! (2g -1)d g g1+...+g k =g   k j=1 (2g j -1)a gj   .
Finally we use the equality ψ 2g-2 1 δ 0 λ g-1 = b g-1 /2 to conclude.

Large genus asymptotics

In this section, we prove Theorem 1.9 using the induction formulas (3), and (4) and the relations (1) and (2).

First, let us recall that the Euler-Maclaurin formula gives the following expression for absolute values of Bernouilli numbers

|B 2g | = 2(2g)! (1 -2 -g )π 2g ∞ k=-∞ (4k + 1) -2g .
Therefore, for all R ∈ N * and for all i ∈ N * we have

(11) (2π) 2g 2 b g = 1+O 1 g R , and (
2g)!b g (2g -2i)!b g-i = (2π) 2i 2i-1 j=0 (2g-j)+O 1 g R
In particular the dominating term in the right hand side is a polynomial in Q[π 2 ][g]. Theorem 1.9 is a consequence of the following lemma.

Lemma 4.1. For all R ∈ N, we have

(2g)!b g = R i=0 c (1) g,i (2g -2i -1)a g-i + O (2g)!b g g R+1 , (2g -1)!b g-1 = R i=0 c (2) g,i (2g -2i -1)d g-i + R i=1 c (3) g,i (2g -2i -1)a g-i + O (2g -1)!b g-1 g R+1 (the last sum is 0 if R = 0) where Q (i) g (x) = k≥0 c (i)
g,k x k are the formal series in x defined by

Q (1) g (x) = 2gF(x) 2g-1 , Q (2) g (x) = 2F(x) 2g-1 , Q (3) g (x) = 2(2g -1)∆(x)F(x) 2g-2 .
End of the proof of Theorem 1.9 under the assumption of Lemma 4.1. The formal series Q

(i) g have coefficients in Q[g]
. Besides, we have c

(1) g = 2g and c

(2)

g = 2. We fix R ≥ 0. For all 0 ≤ k ≤ R we have (2g -2k)!b g-k = R-k i=0 c (1) g-k,i (2(g -k -i) -1)a g-k-i + O (2g)!b g g R+1 , ( 12
) (2g -2k -1)!b g-k-1 = R-k i=0 c (2) g-k,i (2(g -k -i) -1)d g-k-i (13) + R-k i=1 c (3) g-k,i (2(g -k -i) -1)a g-k-i + O (2g -1)!b g-1 g R+1 .
The first set of equations [START_REF] Mirzakhani | Towards large genus asymptotics of intersection numbers on moduli spaces of curves[END_REF] implies that the vector ((2g -2k)!b g-k ) 0≤k≤R-1 is the image of the vector (a g-k ) 0≤k≤R-1 under an upper triangular matrix with coefficients in Q[g] modulo a term in O (2g)!b g /g R+1 . The coefficients on the diagonal of this matrix are equal to (2g -2k)(2g -2k -1) for all 0 ≤ k ≤ R -1. Therefore this linear system has an inverse in the space of matrices with coefficients in Q(g):

a g = R-1 i=0 c (1) g,i (2g -2i)!b g-i + O (2g)!b g g R+1 .
Now we use the asymptotic behavior of the b g given in [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF] to obtain

a g = 2(2g -2)! (2π) 2g 1 - π 2 12g + 24π 2 -π 4 288g 2 + . . . + c R g R + O 1 g R+1
where the coefficients c k lie in Q[π 2 ]. Using Proposition 1.3, we deduce the first part of Theorem 1.9. Now to compute the asymptotic expansion of the Siegel-Veech constants we turn to the second set of linear equations. Once again the coefficients c

(2) g-k,i define a triangular matrix with coefficients in Q[g]. All coefficients on the diagonal of this matrix are equal to 2. Therefore we have

d g = R-1 i=0 c (2) g,i (2g -2i -1)!b g-i + R-1 i=1 c (3) g,i a g-i + O 1 g R+1
where the coefficients c

d g = (2g -2)! (2π) 2g-2 1 - 3 + π 2 12g + . . . + c R g R + O 1 g R+1
where the coefficients c k lie in Q[π 2 ]. Therefore we deduce the second part of Theorem 1.9 by using c area (2g -2) = d g /(4π 2 a g ).

The proof of Lemma 4.1 is essentially borrowed from [START_REF] Bender | An asymptotic expansion for the coefficients of some formal power series[END_REF] but we repeat most arguments for completeness. We begin by proving two preliminary lemmas. Proof. We have seen that B g is equivalent to 2(2g -1)!/(2π) 2g as g goes to infinity. Thus there exist positive constants C and C such that We will prove that the inequalities ( 14) hold for all g ≥ k ≥ 2 with this constant C. We work by induction on k.

We have seen that the inequality ( 14) holds for k = 2. Suppose that k ≥ 3. Then for all g ≥ k we have Proof of Lemma 4.3. The numbers a g and d g are positive for all g. Therefore, using Theorems 3.13 and 3.15, we deduce the following inequalities for all g ≥ 1 a g ≤ (2g -2)!b g , 2d g ≤ (2g -2)!b g-1 .

We use these two sets of inequalities to get also lower bounds where the coefficients c

a g ≥ (2g -
(1) g,i are defined above. This finishes the proof of the first part of Lemma 4.1. The proof of the second part is identical.

Figure 1 .Conjecture 1 . 11 .

 1111 Figure 1. Absolute value of g. 1 (µ) in function of g. The broken lines correspond respectively to µ = (2g -2), (2, 2, . . . , 2), and (1, . . . , 1).

Figure 2 .

 2 Figure 2. Absolute value of g. 1 (µ) in function of len(µ)/(2g -2) for g = 6, 7, . . . , 12. In black, the graph of y = π 2 /12(1 + x).

  2) whose fibers are isomorphic to C * . Let D ⊂ PH(2g -2) be an open domain. Then the volume of D for ν is the total volume of p -1 (D) ∩ H ≤1 (2g -2).

  are in Q[g]. Therefore we get

Lemma 4 . 2 .

 42 Let us denote by B g = (2g -1)!b g (with B 0 = 0). There exists a positive constant C such that for all g ≥ k ≥ 2, we have[START_REF] Mumford | Towards an enumerative geometry of the moduli space of curves[END_REF] g1+...+g k =g k j=1 B gi ≤ C k-1 B g-k+1 .

2 (

 2 2n -3)! (2π) 2g-2 ≤ CB g-1 .

B 2 . 4 . 3 .

 243 g1 B g-g1-k+2 ≤ C k-1 B g-k+1 ,thus the inequality (14) holds for any g ≥ k ≥ Lemma We have the following asymptotic resultsa g = 2d g + O (2g -2)!b g g = (2g -2)!b g + O (2g -2)!b g g .

≥=(2g 1 - 1

 11 2)!b g -k>1 (2g -2)! k!(2g -k)! g1+...+g k =g   k j=1 (2g j -1)!b gj   , 2d g ≥ (2g -2)!b g -k>0 g0>0 (2g -2)!(2g 0 -1)!b g0-1 k!(2g -1 -k)! g1+...+g k =g-g0 (2g -2)!b g -k>1 (2g -2)! (k -1)!(2g -k)! g1+...+g k =g   k j=1 (2g j -1)!b gj   , Now we use Lemma 4.2 to deduce 0 ≤(2g -2)!b g -a g ≤ g k=2 (2g -2)! k!(2g -k)! C k-1 B g-k+1By the same argument we have0 ≤(2g -2)! b g 2 -d g ≤ k>1 (2g -2)! (k -1)!(2g -k)! C k-1 B g-k+1 = O (2g -2)!b g g .Proof of Lemma 4.1. For all R ∈ N, we have (2g)!b g = Using Lemma 4.3, the second sum is O (2g)!bg g R+1 and the first sum is equal to g g1=1 (2g 1 -1)a g1 + O (2g)!b g g R+1

IMJ-PRG, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France E-mail address: adrien.sauvaget@imj-prg.fr


Nouhad Rizek

Elias M Choueiri

Changing the Education Culture through Technology

Keywords:

Technology has a potentially rich, but largely unrealised, role in teaching and learning. This role is defined variously by what the teacher has available, has had time to learn, or can find an appropriate use for, and by what students have access to, are familiar with, and are willing to use. In all of these ways, technology usually plays an adjunct role to others, more traditional modalities for teaching and learning, including lecture, laboratory, library, textbook, tutorial, and practicum.

Researchers perceive problems arising from the significant cultural and organisational differences affecting the management and leadership environment of the modern higher education institutions integrating technology in relation to traditional higher education institutions. This present study describes and analyzes the culture effects in higher education organisations that are challenging the future pre-eminence of the use of technological tools in teaching in higher education. This study tries to explain that, in order to understand opportunities for change in higher education institutions, one must understand that the external environment is by far the most powerful source of internal change. The findings support the view that transformational leadership, collaboration and the classroom culture are three major characteristics of change in higher education institutions, where integrating technology is considered as a way to help the university to be a more effective learning organisation.

Introduction

Every person carries within himself or herself patterns of thinking, feeling, and potential acting which have been learned throughout their lifetime. Hofstede [START_REF] Hofstede | Cultures and Organisations: Software of the Mind[END_REF] defines culture as a collective phenomenon, because it is at least partly shared with people who live or lived within the same social environment, which is where it was learned. It is the collective programming of the mind which distinguishes the members of one group or category of people from another [START_REF] Robbins | Bourdieu and Culture[END_REF]. Hollins [START_REF] Hollins | Culture in[END_REF] defines culture as the underlying phenomenon guiding humanity. Hollins cites a definition of culture by Barrett [4], as "the body of learned beliefs, traditions, and guides for behaviour that are shared among members of any human society". Gollnick and Chinn [START_REF] Gollnick | Multicultural education in a pluralistic society[END_REF] suggest that culture is a way of perceiving, believing, evaluating, and behaving. Culture has many different characteristics: culture is learned, not inherited; it is shared through values and behaviors; it is an adaptation of accommodated natural resources and environment conditions {5]; and it is a dynamic system which changes continuously because of new developments or new information.

The concept of culture came to education from the corporate workplace with the notion that it would provide direction for a more efficient and stable learning environment [START_REF] Deal | The principal's role in shaping school culture[END_REF]. Higher 2 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] education (HE) institutions are learning organisations that provide explicit opportunities to reflect on and evaluate past actions and decisions [START_REF] Bennis | Learning to lead: A workbook on becoming a leader[END_REF]. Cheng [START_REF] Cheng | Profiles of organisational culture and effective schools[END_REF] looked at profiled effective and ineffective organisational cultures. He found that educational institution culture correlates with teachers' attitudes toward their work in such a way that stronger educational institution cultures had better-motivated teachers. Although moral purpose is natural, it will flourish only if leaders cultivate it [START_REF] Fullan | Leading in a Culture of Change[END_REF]. Fullan argues that moral purpose has a tendency to become stronger as human kind evolves. However, in traditional HE institutions, as in all learning organisations, innovation is introduced with difficulty. Teachers are, on an individualist basis, free to innovate within their own classrooms, provided either that this does not break the system's rules and regulations or that breaches can be protected from the knowledge of superiors. In pure collegial systems the institutional focus is supported by collaborative relationships through which members work together to implement any change designed to realise shared goals compatible with the institution's mission. The best teachers integrate the intellectual, emotional, and spiritual aspects of teaching to create a powerful learning environment [START_REF] Palmer | Measuring contextual error in the diagnosis of alternative conceptions in science[END_REF]. Therefore, in an environment with strong organisational ideology, shared participation, charismatic leadership, and intimacy, teachers experience higher job satisfaction and have increased productivity. In order to stay updated in a fast-changing environment, everyone in an organisation needs to be learning continuously. An organisation that can learn invests in its capacity to embrace the future, because its members are continually focused on enhancing and expanding their collective awareness and capabilities, and on continuously improving the quality of what they do [START_REF] Freed | Quality principles and practices in higher education: Different questions for different times[END_REF].

Organisational learning is facilitated in a climate of openness and mutual trust that allows people to embrace experimentation and change without feeling personally threatened. However, Goodman [START_REF] Goodman | Technology enhanced learning: Opportunities for Change[END_REF] acknowledges the complexity and difficulty of change in higher education institutions which do not have good mechanisms to read and adapt to environmental change. The quality of work relationships in an education environment has a great deal to do with the institution's ability to improve [START_REF] Adler | International Dimensions of Organisational Behavior, 3rd Ed[END_REF]. The degree of openness, trust, communication and support that staff share, encourages not only learning but also work satisfaction and improved productivity and performance.

Learning is also encouraged if the people live with integrity, openness, commitment, and collective intelligence [START_REF] Senge | The fifth discipline fieldbook: Strategies and tools for building a learning organisation[END_REF]. The organisational culture in a learning organisation is based on shared responsibility, thus enhancing members' ability to understand interdependency [START_REF] Sparks | Designing powerful professional developments for teachers and principals[END_REF]. In a higher education institution where staff members feel free to ask questions in order to enhance their understanding, and to question existing practices and structures, members' ability to understand interdependency is enhanced. They understand how their individual roles contribute towards reaching the overall vision of the institution. The only way of getting the best from academics is through leadership by democratic, participative structures with broad representative membership. In this sense, management is effective through facilitating and co-ordinating everyone's best efforts, and not through controlling.

Teachers must be skilful in building trust and establishing rapport with other teachers and administrators, dealing with organisational processes, managing their own work and building skill and confidence in others. Members of the organisation realise that they are part of interrelated systems working together to meet organisations' needs. The organisational culture is expressed through customs, traditions, ceremonies, rituals, norms, heroes and heroines that come to characterise each learning organisation. Culture is a constructed reality, and as such, demands 3 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] considerable thought, skill, integrity and consistency on the part of the leader to build and maintain in a way that connects all of the learning organisation community [START_REF] Dimmock | Designing the Learning-Centered School: A Cross-Cultural Perspective[END_REF]. Clarifying the leader's role in building and maintaining organisational culture is but one aspect to be considered.

This study outlines three major characteristics of adopting change in HE institutions: hiring professional teachers to develop cultures in the classroom, developing collaborative relationships among teachers to develop updated-curricula and encouraging leaders to be transformational to deal with the change effectively. It starts by discussing macro cultures in HE institutions in the next section. Section 3 studies the organisational environment and leadership. Section 4 discusses the managing change and outlines strategies to manage it. Section 5 represents web-based higher education institutions. Finally, section 6 concludes with some observations regarding the integration of technology in social environment as well as in educational institutions.

Macro Cultures in Higher education institution

Abercrombie, Hill and Turner [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] argue that new members of a group have to be socialised into its culture. Therefore, organisations may contain various sub-cultures based on different groups, rather than a single organisational culture. There may be also multiple cultures in an organisation. A constructive approach in this direction is then to identify crucial issues that educators think are important for preparing teachers for cultural diversity. The interactive and context-driven nature of teaching requires educators to take into account the students' socialcultural environments in which their education occurs. Thus, macro-cultures are societal cultures where organisational cultures are nested within. Organisational culture refers to a collection of theories that attempt to explain and predict how organisations and the people in them act in different circumstances [START_REF] Brown | Organisational Culture[END_REF]. Schein [START_REF] Schein | The corporate culture survival guide: sense and nonsense about culture[END_REF] emphasises the influence of homogeneous and stable groups within an organisation which he considers as pivotal in determining strong or weak cultures and the 'character' of that culture.

A higher educational institution culture is based on the belief that college and university faculty members share a common view of the world and of scholarship and have similar understandings of the nature and purposes of higher education and the role of faculty [START_REF] Banks | Cultural diversity and education: Foundations, curriculum, and teaching[END_REF]. Besides, each student brings to the campus special talents and perspectives, as well as challenges and needs. This diversity tremendously enriches the educational environment with students with different cultures. The most common use of the term refers to different ethnic or national backgrounds. This is often referred to as a "macro" culture.

The organisational culture is influenced by the macro-cultures, policy contexts and communities in which an institution is located [START_REF] Busher | The crux of leadership: Shaping school culture by contesting the policy contexts and practices of teaching and learning[END_REF]. Thus, organisational culture in higher education means that there is the shared connection between the goals of the academy and those of the individual. However, those goals are subjects for change. A new leader is able to change the culture of higher administrators, department heads, faculty, and students. An example might be a new president who wants faculty to spend more time on research with external funding potential or he or she may want to integrate technology in all departments. Besides, the culture of administrators may be different from that of faculty as it is increasingly difficult to find qualified candidates for presidential and other senior positions in larger colleges and higher education institutions [START_REF] Budd | The Academic Library: Its Context, Its Purpose, and Its Operation[END_REF]. Moreover, culture influences how people deal with conflict and with participation. According to Bond [START_REF] Bond | Cultural influences on modes of impression management: implications for the culturally diverse organisation[END_REF], the disturbance of interpersonal relations and group harmony through conflict can cause lasting animosity in some cultures. As a result, the society tends to avoid open 4 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] confrontation and assertiveness. In group context, this is manifested by teachers and head teachers tending to avoid open disagreement, with the leader's view apparently being accepted [START_REF] Dimmock | Developing comparative and international educational leadership and management: a cross-cultural model[END_REF].

Organisational climate and culture are closely linked to creating a learning environment and improving learning opportunities in an institution. Wlodkowski and Ginsberg [START_REF] Wlodkowski | A framework for culturally responsive teaching[END_REF] propose an intrinsic motivational framework for a teacher that recognizes the importance of linking content to the cultural backgrounds of students as a way to enhance student involvement while maintaining their cultural integrity. An effective culturally responsive teaching should be characterized by the following: respect for diversity; engage motivation of all learners; create a safe, inclusive, and respectful learning environment; derive teaching practices from principles that cross disciplines and cultures; and promote justice and equity in society [START_REF] Wlodkowski | A framework for culturally responsive teaching[END_REF].

The higher education environment needs to be user-friendly, supportive of a community of learners, and to communicate caring and respect for every learner. In such an environment, learner growth and the potential of academics to stimulate that growth can be maximised [START_REF] Sergiovanni | Leadership: What's in it for schools? London[END_REF]. A higher education institution that will succeed in achieving an academic excellence is one that creates the kind of learning environment and campus community that supports continuous learning of all of its members. Such institutions will, where necessary, take risks and design radically new approaches to embracing the imperative of change [START_REF] Roberts | Creating a learning community on campus[END_REF].

The question that needs to be considered is how higher education institutions can best cope with and prepare for the many uncertainties and complexities, and support learning for a future about which very little can be predicted [START_REF] Hargreaves | School culture, school effectiveness, and school improvement[END_REF]. The next section discusses the fact that transformational leadership is needed in changing academic environment, and that such leadership needs to be distributed in order to deal with the many diverse demands of higher education.

Organisational culture, leadership and management

Organisational culture and leadership and the relationship between them became a focal point for writings on educational institution culture. A wide range of organisational theories were adopted, some of which gained popular acclaim, some focused on organisational culture [START_REF] Schein | The corporate culture survival guide: sense and nonsense about culture[END_REF], some carrying the effective management label [START_REF] Torrington | Effective Management: People and Organisation[END_REF], some educationally based [START_REF] Weick | Educational organisations as loosely coupled systems[END_REF], some exploring the relationship between educational institution culture and change [START_REF] Sarason | The Culture of the School and the Problem of Change[END_REF], and some studying the micro-politics of educational institution organisation [START_REF] Blase | Bringing out the best in teachers: What effective principals do (2nd Ed)[END_REF].

Nevertheless, the success of a learning organisation relies on its organisational culture [START_REF] Yu | The effects of transformational leadership on teachers' commitment to change in Hong Kong[END_REF].

Impressive evidence indicates that healthy and strong learning organisation cultures increase students' motivation and achievement, and teachers' satisfaction and productivity. Schein [START_REF] Schein | The corporate culture survival guide: sense and nonsense about culture[END_REF] contends that the "bottom line for leaders is that if they do not become conscious of the cultures in which they are embedded, those cultures will manage them." He argues: "Leaders create and modify cultures. Culture creation, culture evolution, and culture management are what ultimately define leadership". Leaders need to conduct "the destruction of culture" and to reinforce the right kind of culture [START_REF] Schein | The corporate culture survival guide: sense and nonsense about culture[END_REF]. Schein's approach is deeper and more useful than many; he studies organisational culture from an anthropological perspective and explains the leader's role in shaping it.

Most people would probably agree that organisational culture is holistic, referring to a whole; it is 5 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] historically determined, reflecting the history of the organisation; it is related to the anthropologists' issues such as rituals and values; it is socially constructed; and it is soft and at the same time difficult to change. However, because of the important role of practices in organisational cultures, the latter can be considered somewhat manageable. Changing collective values of adult people in an intended direction is extremely difficult, if not impossible. Values do change, but not according to someone's master plan. Collective practices, however, depend on organisational characteristics like structures and systems, and can be influenced in more or less predictable ways by changing these [START_REF] Hofstede | Cultures and Organisations: Software of the Mind[END_REF]. Collective practices also depend on interpersonal relationships both inside an organisation and outside it. In learning organisations with collaborative cultures, staff and administrators work together for the good of students. Energy, commitment, and motivation are likely to be higher. Change and improvement will be conscious and collegial. Teachers' sense of efficacy and pride will be higher [START_REF] Fullan | Leading in a Culture of Change[END_REF][START_REF] Peterson | Building collaborative cultures: Seeking ways to reshape urban schools[END_REF]. Staff and administration can work together to build collaborative cultures. Therefore, the type of collegial relationships and the development of these relationships may represent the infrastructure of improvement and enhancement in any institution [START_REF] Kent | The importance of collaborative cultures[END_REF].

Blackstone [START_REF] Blackstone | Why learn? Higher education in a learning society[END_REF] emphasises that academics in the higher education environment are just as important as those in senior leadership positions for the success of the organisation, as their intelligence, experience and skills are used to address external environmental challenges. The bottom line is that the leader is the primary culture carrier for the organisation," Schweitzer [37] concludes. "If the leader's attitudes and behaviors do not match the culture that you are intending to build, it will not work. The leader and the culture must be in sync."

Senge [START_REF] Senge | The dance of change: the challenges to sustaining momentum in learning organisations[END_REF] argues that leadership for deep change requires replacing the myth of the "hero leader" with the concept of leadership communities. These communities, he believes, enable the building of leadership capacity throughout the organisation so the organisation can continually adapt and reinvent itself. Teacher leadership can take many forms. It may include: (1) advocating the vision for staff development, (2) participating in learning organisation and district improvement teams to help determine goals and strategies, (3) conducting classroom and institution wide action research to determine if changes are improving the learning of all students, (4) mentoring new teachers, serving on peer review panels to provide support and assistance to new and veteran teachers, and (5) working on special assignment as coaches or instructional guides to provide ongoing professional learning for their peers.

Lambert [START_REF] Lambert | Building Leadership capacity in schools[END_REF] argues that leadership is about learning together, and constructing meaning and knowledge collectively and collaboratively. It involves opportunities to surface and mediate perceptions, values, beliefs, information, and assumptions through continuing conversations; to inquire about and generate ideas together, to seek to reflect upon and make sense of work in the light of shared beliefs and new information; and to create actions that grow out of these new understandings. Such is the core of leadership.

Effective leaders are committed to the success of everyone in the organisation, and involve everyone, so that individual members understand how the numerous parts of the organisation are interrelated and why every individual's work is an essential component for success. Members and groups within the organisation learn to understand that the decisions they make affect the entire organisation, and learn to take each other's interests into account. Decisions are made by all in the organisation. The increased involvement of members in making decisions leads to better 6 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] decisions, as people who work within the system have most insight into how it works and the changes necessary to improve it [START_REF] Cartwright | Cultural transformation: Nine factors for continuous business improvement[END_REF].

Thus, what is expected from leaders continues to increase [START_REF] Sergiovanni | Leadership: What's in it for schools? London[END_REF]. Leaders are expected today to create learning communities in their learning organisations and to engage the broader learning organisation community in creating and achieving a compelling vision for its learning organisations, while serving diverse student populations [START_REF] Pennington | the Relationship of Leadership Practices to Culture[END_REF]. The leader's role changes from authoritarian style to instructional. Leaders must steep in curriculum, instruction, and assessment, while they preserve their role of being the key persons in determining whether a learning organisation succeeds. Leader's responsibility for improvement lies in supervising, coaching teachers, and working with them on development plans that support real learning organisation improvement [START_REF] Harris | Behave yourself[END_REF]. In other words, leaders should encourage constructive changes.

Distributed transformational leadership

Transformational leadership is value-driven. The leader sets high aims and purposes for followers, engaging them through inspiration, exemplary practice, collaboration and trust.

Transformational leadership aims at responding to change quickly and at bringing out the best in people. Such leadership is change-oriented and central to the development and survival of organisations in times of environmental chaos when it is necessary to make strategic changes to deal with major threats and opportunities. It derives its power from shared principles, norms and values. Leaders, who encourage and support transformation share power, are willing to learn from others and pay specific attention to intellectual stimulation and to each individual's needs for achievement and growth [START_REF] Ramsden | Learning to lead in higher education[END_REF][START_REF] Caldwell | Beyond the self-managing school[END_REF][START_REF] Bass | Improving organisational effectiveness through transformational leadership[END_REF].

People in different roles and at all levels of organisations are increasingly being called upon to all be leaders in their own right [START_REF] Bergmann | Everyone a leader: A grassroots model for the new workplace[END_REF]. The particular characteristic of higher education institutions as knowledge-intensive organisations is good leadership which is usually naturally distributed across a team or workgroup. Therefore, the distributed character of leading in which the focus is much less on the leader as an individual, seems particularly compatible with higher education contexts [START_REF] Knight | Departmental leadership in higher education[END_REF]. The leader is conceived as first among equals and the characteristics of the group being led is carefully considered. The approach of dispersed leadership takes the spotlight off the heroic leader and the team is highlighted. The focus is on leading others to lead themselves, either through dispersal of power or through liberating team members so that their abilities can be fully utilised [START_REF] Kouzes | The leadership challenge: How to keep getting extraordinary things done in organisations[END_REF]. What emerges is more productive than simply good leadership from the top. Leadership is much more diffuse than is traditionally believed. It can be exercised at all levels within organisations and all participants are capable of practicing it in some way. By focusing only on the behaviour of senior people, one runs the risk of losing sight of those aspects of human behaviour in organisations that lead to effectiveness and consistently high quality. Especially in a higher education institution, which is knowledge-intensive and in which the quality of the institution is largely dependent on the quality of its human resources, it is not possible to excel by relying on good leadership at the top only. Democratic, empowering leadership involves powerwith rather than power over, and the teams within such an empowered community become leaders themselves [START_REF] Knight | Departmental leadership in higher education[END_REF]. Thus, the management of the community is critical.

3.2

Managing the diversity Valuing Diversity is recognising and respecting the value of human differences. Managing Diversity is creating and sustaining an environment where everyone can achieve his or her full 7 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] potential. Despite the substantial numbers of studies, little is known about what distinguishes between good and bad management [START_REF] Hamlin | Towards a Universalistic Model of Leadership: a comparative study of British and American empirically derived criteria of managerial and leadership effectiveness[END_REF]. Furthermore, there is little agreement about what constitutes, and is meant by, managerial and leadership effectiveness. Managing planned change is concerned with activities that are intentional and goal oriented. It seeks to improve the ability of the organisation to adapt to changes and to change organisation members' behaviour [START_REF] Robbins | Organisational Behaviour Concepts, Controversies and Applications[END_REF].

Barker [START_REF] Barker | Do leaders matter?[END_REF] observes that "articles abound on leadership; however, few evaluate the impact of leadership and its effectiveness within the organisation", especially when it comes to higher education institutions. Consequently it is still the case, after many years of leadership research that little is known empirically about what differentiates between highly effective and less effective leaders and managers [START_REF] Martinko | Structured observation of managerial work: a replication and synthesis[END_REF][START_REF] Willcocks | Managerial effectiveness and the public sector: a health service example[END_REF]. The difference between leadership and managership is debatable but it will not be discussed in this study.

Managing the diversity is one of the biggest challenges faced by higher education institutions [START_REF] Scott | Leadership and Management: some thoughts from the UK[END_REF]. Accepting the challenge of diversity implies, among other things, changing the organisational culture, re-conceptualising appropriate leadership styles, restructuring organisations, reformulating what constitutes good teaching, and developing staff and students to work and learn in an organisation that is very different from what it used to be [START_REF] Havenga | Beyond affirmative action there is diversity[END_REF]. Demands for better quality teaching and support services for students, accountability, transparency, quality assurance and cost-effectiveness are made by the state, the business sector and society in general. These demands have put increasing pressure on higher education institutions worldwide to demonstrate the value they are adding to the individual and to society as a whole [START_REF] Altbach | Harsh realities: The professoriate faces a new century[END_REF][START_REF] Benjamin | The implications of the changed environment for governance in higher education[END_REF].

Managing change and developing strategies

Educational institutions are influenced by the societal events constantly occurring around them [START_REF] Hargreaves | School culture, school effectiveness, and school improvement[END_REF]. Recently, several developments in the workplace, changes in student demographics, and the economic trends have been forcing educational institutions to change. The growth of knowledge in any field is rapidly outstripping any individual's ability to remain up-to-date. Knowing how to access information rather than memorizing information is central to coping with this rapid change. Growing numbers of students are working part-time while enrolled in full-time programs; current educational delivery systems do not meet their unique needs. Classroom space within colleges is fully utilised while at the same time enrolments are expected to increase. There is little or no money for expansion. Access to the Internet allows learners to take courses virtually anywhere in the world (accreditation is still a major consideration). Geographic location is no longer relevant; we are in global competition with all other educational providers on the Net.

Thus, educators must demonstrate a history of professional growth and development [START_REF] Levin | Improving educational productivity through a focus on learners[END_REF]. They should be aware of their role in social change and be able to justify to themselves the role they play. Innovations involve the erosion of older practices whenever it takes place, and educators should be sure that they want to erode these traditions before they encourage change [START_REF] Ziguras | Educational technology in transnational higher education in South East Asia[END_REF]. So, educators retain control over content, standards, design, and assessment of student learning while giving up control of delivery of information.

Besides, the university culture provides a general identity for all faculties, regardless of disciplinary affiliation, and concerns three basic values generally shared by faculty members: (1) the pursuit and dissemination of knowledge as the purpose of higher education; (2) autonomy in the conduct of academic work; and (3) collegiality [START_REF] Barker | The Discipline of team work[END_REF]. The first value, that of pursuing and 8 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] disseminating knowledge, requires faculty members to be knowledgeable and to convey their learning through teaching, inquiry, and publication. The second value, that of expecting autonomy, gives freedom to advance knowledge without interference, through such structures as peer review, tenure, and relatively independent higher education institutions. The third value, that of collegiality, provides mutual support, social interaction, and faculty governance [START_REF] Smith | Changing University culture through promotion policies[END_REF]. An ideal academic community is a university in which the pursuit of learning, academic freedom, and collegiality are strongly valued.

Thus, managing change in a HE institution can be developed by adopting strategies to manage it. The strategies can be the classroom culture and curriculum construction.

Classroom culture

The class has a culture, or sub-culture, of its own. It has shared values, information, techniques, interpretations and meanings. The teacher has a central and critical role in creating a classroom that respects diversity and ensures the self-worth of all students as conditions essential for culturally responsive teaching. This framework shows the interrelationships between three levels of culture: personal, microculture, and macroculture. It considers the personal cultural identity as a pool of constructs, values, beliefs, and attitudes; many of which may be part of micro-cultural groups one may belong to. Microcultural groups are in turn influenced by the macro-culture. On the other hand, whatever the students learn is likely to be cultural in two senses. In the first place they share the understandings generated and sustained in their own classroom group. In the second, these understandings are also shared with certain groups outside the classroom, that is, they are part of the cultural currency of the broader society. On this view, the teacher generates and fosters a culture within his or her class. Furthermore, a classroom culture can be seen as an interface between two sets of cultures: the teachers and the students, and so an arena of conflict or contestation.

If the class is to make the culture its own, it must come to found its own social life on it. Therefore, the sub-culture of an educational institution must be seen in the context of the total culture of a society or civilisation [START_REF] Cheng | Can educational value be borrowed? Looking into cultural differences[END_REF]. Culture is a dynamic field within and through which individuals make contact with one another [START_REF] Bush | Theories of Educational Management[END_REF]. Within a class two types of culture are transmitted. On the one hand, predominantly social understandings may be inculcated such as morals, manners and attitudes to life. On the other hand, education may transmit the understandings which are implied in curriculum. Educators are responsible for controlling the experience of educational groups and for controlling and directing the learning of the class.

A study done by Steineke and Olsen  [64] on Norwegian higher educational institutions discusses the management of the interface between formal knowledge production and the skills requirements of the local labor market. Anderson [START_REF] Anderson | The complex relations between the academy and industry -views from the Literature[END_REF] argues that HEI-industry relations may include technology transfer programs, research parks as well as consulting. Thus, the most important form of knowledge transposition from the HEIs to their local environment follows from their production of university college candidates. The students' knowledge experienced in classroom, contributes to regional competence development.

Therefore, before introducing any new technology into classrooms, teachers must be able to justify its contribution [66]. In order to use IT constructively, educators may have to use technology when it helps them do their good work better. Moreover, no teacher or classroom 9 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] exists in isolation. Implementing technology is an issue for the entire educational institution. For example, the advent of network technologies and access to the Internet in educational institutions has brought attention to the need for learning organisation-wide technology planning. Thus, many organisations in recent times have changed their working culture to embrace such ideas as group working and self-management teams. As such, the introduction of group computing technology into an organisation, either business or educational, should be considered carefully [START_REF] Hogarth | Managing the social and cultural consequences of introducing groupware technology into the group learning environment[END_REF]. Therefore, the purchase of classroom computers and presentation equipment does not guarantee the effective integration of technology into teaching and learning. Classroom management strategies, facilities design, and long-range technology planning are equally necessary to realise technology's potential for education. Classroom design allows the educators to plan classroom layout and technology resources for the effective integration of computer technologies into a university. Planning considerations are absolutely necessary if these technology expenditures are to make any real educational difference. They offer guidelines for long-range planning that seek to equip and empower teachers for technology-enhanced instruction.

Thus, the use of group computing as software solutions that enable groups to work together by using Information and Communications Technology (ICT) supports the collegial sharing. Such support leads to greater readiness to experiment and take risks, and with it a commitment to continuous improvement among teachers as a recognised part of their professional obligation. In this sense, collaboration and collegiality are seen as forming vital bridges between learning organisation improvement and teacher development [START_REF] Rosenholtz | Teachers' Workplace[END_REF].

On the other hand, teaching at higher education institutions is currently facing a number of challenging developments. Students have to adhere to tight study programs in order to graduate within fixed time limits, and the budgets of higher education institutions are under almost constant pressure [START_REF] Checkland | Soft Systems Methodology in Action[END_REF]. In response to these developments, educators and educational institutions are in search of teaching methods, techniques and technologies that make teaching more efficient and effective. As a result, various forms of Information and Communication Technology (ICT) are often deployed. The potential uses of ICT in education range from providing analytical tools and eliminating distance barriers to replacement of repetitive tasks. In terms of teaching methods, collaborative learning in face-to-face settings has shown to be a highly effective learning strategy [START_REF] Nunamaker | Electronic meeting systems to support group work[END_REF]. In contrast to more traditional forms of education where the primary interaction between teacher and students is where the teacher speaks and students listen, collaborative learning emphasises group or cooperative efforts among students and often focuses on the interaction between students themselves. This process helps students conceptualise, construct and initialise procedures and knowledge [START_REF] Vreede | Group support systems for innovative information science education[END_REF]. Also sharing information helps students deepen understanding. It is in this learning environment that ICT technologies such as group computing could enhance the educational process.

Therefore, collaborative technologies can support educational organisations in the creation of new relationships and in introducing new ways of working. These new approaches challenge the old hierarchical organisational structures and outmoded teaching approaches, which are not flexible enough to meet today's requirements. Baldridge and Deal [START_REF] Baldridge | The Dynamics of Organisational Change in Education[END_REF] argue that to understand opportunities for change in higher education institutions, one must understand that the external environment is by far the most powerful source of internal change. Further, successful change, or successful implementation, is none other than learning, but it is the teachers in the system who are learning along with the students. Thus, the method adopted by teachers to learn and the 10

conditions under which they are most likely to learn, are useful for designing and carrying out strategies for implementation.

Curriculum construction

Collaboration and collegiality are promoting professional growth and they are also widely viewed as ways of securing effective implementation of externally introduced change [START_REF] Fullan | Leading in a Culture of Change[END_REF]. Their contribution to the implementation of centralised curriculum reform is a key factor. Implementation means curriculum change. For teachers in classrooms, new materials are important, but are ineffective by themselves. Change also involves new behaviors and practices, and ultimately new beliefs and understandings. It involves changes in what people know and assume. Therefore, collaboration and collegiality bring teacher development and curriculum development together. Indeed, the failure of many curriculum development initiatives is attributable, at least in part, to the failure to build and sustain the collegial working relationships essential to their success.

A curriculum is brought to life in support of the social interaction that takes place in the classroom. Since the curriculum is regarded as a selection of culture, it must be seen in relation both to the social life of the classroom, of which it is the medium, and to the background of the culture of the society as a whole [START_REF] Quicke | A Curriculum for life: Schools for a Democratic Learning Society[END_REF]. Therefore, one of the principal characteristics of higher education curriculum is a strong vocational emphasis. Traditionally, HE institutions have existed as autonomous institutions characterised by low levels of territorial embeddedness [START_REF] Anderson | The complex relations between the academy and industry -views from the Literature[END_REF]. This is now changing, where HE institutions traditionally have been established in order to accelerate economic development in sparsely populated areas [START_REF]OECD: The response of higher education institutions to regional needs OECD/ Programme on institutional management in higher education (IMHE) OECD[END_REF]. General educational effects may often be sacrificed to the desire to make the teaching in each subject an introduction to, and a preparation for, a later specialisation which is directed towards a vocation. Thus, the practical element in the curriculum can obviously play a part in the education of every student. The training and learning elements of a curriculum are controlled by the teacher. Now, the institutions of higher education are managing the interface between knowledge production and the local labor market.

Besides, the emergence, development and diffusion of information technology (IT) have changed society dramatically, into something which is now sometimes called the information society. IT makes it possible to collect process and transmit information much faster and more cheaply than before. The changes have had effects on the economy, production, services and society as a whole and are used in the area of education [START_REF] Hanna | The Diffusion of Information Technology[END_REF]. Faculty members and administrators should evaluate the curriculum and how it should be related to information technology (IT). Departments can design the curriculum for their majors according to their own distinctive approach, and students can choose the program that fits in best with their values and goals. Higher education institutions can design their core curriculum according to an overall educational philosophy. Because decisions about program philosophy and course content are made by the faculty, the contents of and boundaries between courses are flexible; they can be changed to suit evolving circumstances, not least the interests of the best students [START_REF] Ruddock | Lawrence Stenhouse's vision of the teacher, the student and learning[END_REF].

As a result, culture essentially implies standards or norms: so does curriculum. Individual action within culture gains its coherence from the standards or norms which underlie it: so does the action of the teacher in the face of the curriculum, since both the teacher as a transformational leader and the cultural sociologist are deeply concerned with the standards adopted by groups. Society expects higher education to link its curriculum more relevantly to social and economic 11 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] needs. Society also expects higher education to become more flexible in its courses and degree offerings in order to meet new educational needs. Rapid changes in the discipline areas of knowledge, along with rapid growth in the volume of the overall knowledge base, are fuelling a growing emphasis on life-long learning and learning to learn. Besides, students demand that higher education be at their door, not that they be at its door.

Towards web-based higher education institutions

This analysis views higher education as an open system with advanced learning as its core purpose. The system has evolved into a highly complex set of institutions that have organised to achieve this core purpose. Higher education institutions will change radically or perhaps even cease to exist in the 21st century. This achievement can be explained by leadership adaptation of a vision to new challenges, service oriented higher education institutions, on-line higher education institutions and simulating learning environment.

Adapting a vision to new challenges:

A coherent vision specifies the particular values and beliefs that will guide policy and practice within the HE institution. Academics need to take part into the decision-making process and they should have a shared vision for creating an effective learning organisation [START_REF] Sergiovanni | Leadership: What's in it for schools? London[END_REF]. The creation of a vision is not a static event, because the vision must change as culture changes. The transformational leadership enables the adaptation of a vision to new challenges to be more successful in building strong HE cultures.

Service-oriented higher education institutions: Leaders of colleges and higher education institutions must affirm their basic belief that these organisations exist to serve others, and they must do so by way of acts and deeds, not simply by intensifying their rhetoric. One example of services is by extending classroom tradition. The extended classroom connects learners who are separated from each other and the instructor through the use of educational technologies such as the web page of a course, which is available to anyone on the internet. The program and module web pages can provide students with everything they need to know such as the course outlines, regulations and procedures, independently of time, location, and distance, and allow for students to study together. Besides, WebCT, Blackboard, or any other tools, not only present information but also receive it. Such tools can handle information-giving task much more effectively than traditional methods [START_REF] Biggs | Teaching for a quality learning at university[END_REF]. They offer opportunities for students to learn through asynchronous interaction with each other and a faculty member [START_REF] Passey | Information technology supporting change through teacher education[END_REF]. Thus, technology has the ability to manage learning and administer programs as well as to handle large databases, for example information about the university, the department, different programs and modules. Besides, computer-assisted assessment can be used by posting multiple-choice testing on the material.

Virtual higher education institutions also called on-line higher education institutions:

Growing demand among learners for improved accessibility and convenience, lower costs, and direct application of content to work settings is radically changing the environment for higher education globally. In this rapidly changing environment, which is increasingly based within the context of a global, knowledge-based economy, traditional higher education institutions are attempting to adapt purposes, structures, and programs, and new organisations are emerging in response. Organisational changes and new developments are being fuelled by accelerating advances in digital communications and learning technologies that are sweeping the world [START_REF] Walther | Computer-Mediated communication: Impersonal, interpersonal and hyperpersonal interaction[END_REF].

Growing demand for learning combined with these technical advances is in fact a critical pressure point for challenging the dominant assumptions and characteristics of existing traditionally-12 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] organised higher education institutions in the 21st century. This combination of demand, costs, content application, and new technologies is opening the door to emerging competitors and new organisations that will compete directly for students and learners. The recent developments of the worldwide web, digital satellite technology, and new applications of virtual reality to build simulated learning environments are predicted to have particularly dramatic effects upon learning environments at all levels [START_REF] Goodman | Technology enhanced learning: Opportunities for Change[END_REF]. Higher education institutions are experimenting with improving accessibility to existing programs, designing new programs to take advantage of these emerging technologies, and marketing their programs to new audiences and in new ways. Completely new models for higher education institutions are being formed around the promise of virtual environments. Virtual Learning Organisation are ICT-supported university is an initiative of many institutions of higher education (traditional higher education institutions and higher education institutions for professional development). One of the aims of this initiative is to facilitate and support a transformation of higher education from supply-driven to demand-driven education. In this new education students actively learn, with interest and motivation, taking responsibility for their own learning [START_REF]Business Plan of the Dutch Digital University[END_REF].

Conclusions

Many higher education institutions will have difficulties capitalising on web-based course potential. First, they are not in the product marketing business with appropriate venture capital to support high-end Internet learning. Second, institutions don't have the creative development staff to support such venture. Third, higher education institutions are slow to react to market changes, for example how long it takes to get curriculum changes through the system [START_REF] Cohen | Challenges of Information Technology Education in the 21st Century[END_REF]. Once the curriculum is in place, higher education institutions still face challenges from accelerating program growth, faculty shortages, and the increasing complexities of e-commerce business models and their supporting technologies.

The emerging global economy, characterized by the increased international movement of capital, products, technology, and information, is accentuating economic competition and means that students' knowledge and skills must meet international standards. Organizational downsizing, already common in corporations and government agencies, is spreading to colleges and higher education institutions. The implications of these macro-environmental changes are substantial. Educators need to rethink their basic assumptions about organizational structure and curricular programs. To better serve an information-age society, educational organizations needs to be transformed by requiring teachers to design and manage, not deliver, customized learning experiences that include many options, including stand-alone technology and opportunities to learn outside of school. Educational institutions must become self-transforming organizations to take advantage of what is known about learning and what is known about using technology to enhance teaching [START_REF] O'banion | Gladly Would He Learn[END_REF].

Educational leaders must change the existing culture and structure of their organisation by involving people throughout the organisation in a systematic and ongoing analysis to identify emerging or potential developments in the external environment that could affect their learning organisation's future. Binding the intellectual power of stakeholders (boards of trustees, faculty, staff, students) to identify signals of change, analyse the implications of these signals for the organization, and design actions in light of these implications, organizational leaders are able to develop and implement creative plans. Because these plans incorporate the thinking and support of the majority of stakeholders, they can transform the organizational culture and align it with 13 [START_REF] Abercrombie | The Penguin dictionary of sociology (4th Ed)[END_REF] changing realities [START_REF] Morrison | Transforming a University to Meet the Challenges of the Twenty-First Century[END_REF]. Leadership skills and ability must be refined and strengthened, particularly through training and professional development [START_REF] Varghese | Organisational Leadership and the Roles for Educational Leaders, A Breakfast address for the centre for leadership and management in education and the professional development network[END_REF].

An issue of electronic university is what value is placed on the social and interpersonal interactions between students and faculty in a classroom. Students need to be able to interact socially and understand the differences between personal interactions (roommates, friends …) and business interactions (preparation of potential employees). Many educational experts believe that a new learning paradigm is being created with the integration of technology into learning organisations [START_REF] Von Holzen | A Look at the Future of Higher Education[END_REF].

Higher education institutions are responding to external forces to produce innovation. Internal forces can also affect innovation, but do so in the shape of evolving disciplinary and professional cultures, over which institutional leaders and managers have limited leverage. Therefore, new models of higher education will emerge in the future. The models are designed to create a competitive advantage in a rapidly changing and growing marketplace. With rapidly developing learning technologies creating new possibilities for organising learning for adults, these models are both competing with and causing change in, the traditional residential model of higher education. Benefits of this new competitive environment include removing barriers to existing educational programs, responding more effectively and quickly to emerging educational needs, improving educational quality, and achieving long-term cost efficiencies. However, the challenge for teacher education is to produce teachers who can adapt technology to their needs before the technology makes learning organisations themselves irrelevant. This will enable the electronic community of teachers that could support and encourage long term professional development, from initial teacher education to continuing professional development, to be firmly established as a natural part of teacher's practices.

As a result, higher education institutions have to establish their priorities clearly often through planning but also by articulating their strategic direction in an explicit fashion.

Authors:

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Ms