Abstract:
In this survey we present the interpretation of isomondromy preserving equations on Riemann surfaces with marked points as reduced Hamiltonian systems. The upstairs space is the space of smooth connections of GL(N) bundles with simple poles in the marked points. We discuss relations of these equations with the Whitham quantization of the Hitchin systems and with the classical limit of the Knizhnik-Zamolodchikov-Bernard equations. The main example is the one-parameter family of Painlev\'{e} VI equation and its multicomponent generalization.

Abstract:
In this review we explain interrelations between the Elliptic Calogero-Moser model, integrable Elliptic Euler-Arnold top, monodromy preserving equations and the Knizhnik-Zamolodchikov-Bernard equation on a torus.

Abstract:
We consider the hydrodynamics of the ideal fluid on a 2-torus and its Moyal deformations. The both type of equations have the form of the Euler-Arnold tops. The Laplace operator plays the role of the inertia-tensor. It is known that 2-d hydrodynamics is non-integrable. After replacing of the Laplace operator by a distinguish pseudo-differential operator the deformed system becomes integrable. It is an infinite rank Hitchin system over an elliptic curve with transition functions from the group of the non-commutative torus. In the classical limit we obtain an integrable analog of the hydrodynamics on a torus with the inertia-tensor operator $\bar\partial^2$ instead of the conventional Laplace operator $\partial\bar\partial$.

Abstract:
We consider a large $N$ limit of the Hitchin type integrable systems. The first system is the elliptic rotator on $GL_N$ that corresponds to the Higgs bundle of degree one over an elliptic curve with a marked point. This system is gauge equivalent to the $N$-body elliptic Calogero-Moser system, that is derived from the Higgs bundle of degree zero over the same curve. The large $N$ limit of the former system is the integrable rotator on the group of the non-commutative torus. Its classical limit leads to the integrable modification of 2d hydrodynamics on the two-dimensional torus. We also consider the elliptic Calogero-Moser system on the group of the non-commutative torus and consider the systems that arise after the reduction to the loop group.

Abstract:
We introduce new times in the monodromy preserving equations. While the usual times related to the moduli of complex structures of Riemann curves such as coordinates of marked points, we consider the moduli of generalized complex structures ($W$-structures) as the new times. We consider linear differential matrix equations depending on $W$-structures on an arbitrary Riemann curve. The monodromy preserving equations have a Hamiltonian form. They are derived via the symplectic reduction procedure from a free gauge theory as well as the associate linear problems. The quasi-classical limit of isomonodromy problem leads to integrable hierarchies of the Hitchin type. In this way the generalized complex structures parametrized the moduli of these hierarchies.

Abstract:
In this short review we compare different ways to construct solutions of the periodic Toda lattice. We give two recipes that follow from the projection method and compare them with the algebra-geometric construction of Krichever.

Abstract:
Knizhnik-Zamolodchikov-Bernard (KZB) equation on an elliptic curve with a marked point is derived by the classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on cotangent bundle to the loop group $L(GL(N,{\bf C}))$ extended by the shift operators, to be related to the elliptic module. After the reduction we obtain the Hamiltonian system on cotangent bundle to the moduli of holomorphic principle bundles and the elliptic module. It is a particular example of generalized Hitchin systems (GHS), which are defined as hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present the explicite form of higher quantum Hitchin integrals, which upon on reducing from GHS phase space to the Hitchin phase space gives a particular example of the Belinson-Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.

Abstract:
We describe non-autonomous Hamiltonian systems coming from the Hitchin integrable systems. The Hitchin integrals of motion depend on the W-structures of the basic curve. The parameters of the W-structures play the role of times. In particular, the quadratic integrals dependent on the complex structure (W_2-structure) of the basic curve and times are coordinate on the Teichmuller space. The corresponding flows are the monodromy preserving equations such as the Schlesinger equations, the Painleve VI equation and their generalizations. The equations corresponding to the highest integrals are monodromy preserving conditions with respect to changing of the W_k-structures (k>2). They are derived by the symplectic reduction from the gauge field theory on the basic curve interacting with W_k-gravity. As by product we obtain the classical Ward identities in this theory.

Abstract:
Starting with a Lie algebroid ${\cal A}$ over a space $M$ we lift its action to the canonical transformations on the affine bundle ${\cal R}$ over the cotangent bundle $T^*M$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is a Hamiltonian algebroid ${\cal A}^H$ over ${\cal R}$ with the anchor map from $\G({\cal A}^H)$ to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize Lie algebras of canonical transformations. We prove that the BRST operator for ${\cal A}^H$ is cubic in the ghost fields as in the Lie algebra case. The Poisson sigma model is a natural example of this construction. Canonical transformations of its phase space define a Hamiltonian algebroid with the Lie brackets related to the Poisson structure on the target space. We apply this scheme to analyze the symmetries of generalized deformations of complex structures on Riemann curves $\Si_{g,n}$ of genus $g $with $n$ marked points .We endow the space of local $\GL$-opers with the Adler-Gelfand-Dikii (AGD) Poisson brackets. It allows us to define a Hamiltonian algebroid over the phase space of $W_N$-gravity on $\Si_{g,n}$. The sections of the algebroid are Volterra operators on $\Si_{g,n}$ with the Lie brackets coming from the AGD bivector. The symplectic reduction defines the finite-dimensional moduli space of $W_N$-gravity and in particular the moduli space of the complex structures $\bp$ on $\Si_{g,n}$ deformed by the Volterra operators.

Abstract:
The space of generalized projective structures on a Riemann surface $\Sigma$ of genus g with n marked points is the affine space over the cotangent bundle to the space of SL(N)-opers. It is a phase space of $W_N$-gravity on $\Sigma\times\mathbb{R}$. This space is a generalization of the space of projective structures on the Riemann surface. We define the moduli space of $W_N$-gravity as a symplectic quotient with respect to the canonical action of a special class of Lie algebroids. This moduli space describes in particular the moduli space of deformations of complex structures on the Riemann surface by differential operators of finite order, or equivalently, by a quotient space of Volterra operators. We call these algebroids the Adler-Gelfand-Dikii (AGD) algebroids, because they are constructed by means of AGD bivector on the space of opers restricted on a circle. The AGD-algebroids are particular case of Lie algebroids related to a Poisson sigma-model. The moduli space of the generalized projective structure can be described by cohomology of a BRST-complex.