Abstract. Here, the study of torus actions on topological spaces is presented as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. This link helps in understanding the geometry and topology of a space with torus action by studying the combinatorics of the space of orbits. Conversely, the most subtle properties of a combinatorial object can be recovered by realizing it as the orbit structure for a proper manifold or complex acted on by a torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement, etc., while the combinatorial objects include simplicial and cubical complexes, polytopes, and arrangements. This approach also provides a natural topological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics.The exposition centers around the theory of moment-angle complexes, providing an e ective way to study triangulations by methods of equivariant topology. The book includes many new and well-known open problems and would be suitable as a textbook. We hope that it will be useful for specialists both in topology and in combinatorics and will help to establish even tighter connections between the subjects involved.

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We develop the theory of hyperelliptic Kleinian functions. As applications we consider construction of the explicit matrix realization of the hyperelliptic Kummer varieties, differential operators to have the hyperelliptic curve as spectral variety, solution of the KdV equations by Kleinian functions.

IntroductionIn the theory of Abelian functions on Jacobians, the key role is played by entire functions that satisfy the Riemann vanishing theorem (see, for instance, [9]). Here we introduce polynomials that satisfy an analog of this theorem and show that these polynomials are completely characterized by this property. By rational aalalogs of Abelian functions we mean logarithmic derivatives of orders /> 2 of tlmse polynomials.We call the polynomials thus obtained the Schur-Weierstrass polynomials because they are constructed from classical Schur polynomials, which, however, correspond to special partitions related to Weierstrass sequences. Recently, in connection with the problem of constructing rational solutions of nonlinear integrable equations [1,7], special attention was focused on Schur polynomials [5,6]. Since a Schur polynomial corresponding to all arbitrary partition leads to a rational solution of the Kadomtsev-Petviashvili hierarchy, tile problem of connecting the above solutions with those defined in terms of Abelian functions on Jacobians naturally arose. Our results open the way toward solving this problem on the basis of the Riemann vanishing theorem.We demonstrate our approach by the example of Weierstrass sequences defined by a pair of coprime numbers n and s. Each of these sequences generates a class of plane curves of genus g ----(n -1)(s -1)/2 defined by equations of the formwhere 0 ~< a < s -1, 0 ~

We obtain an explicit realization of the Jacobi and Kummer varieties for trigonal curves of genus g (gcd(g, 3) --1) of the form y3 = xg+Z + ~ Aaa+(9+z)~x~y~ 0 ~< 3~ + (g + 1)/3 < 3g + 3, as algebraic subvarieties in C 4g+$, where 6 = 2(g-3~q/3]), and in C g(g+l)/2. We unfformize these varieties with the help of p-functions of several variables defined on the universal space of Jacobians of such curves. By way of application, we obtain a system of nonlinear partial differential equations integrable in trigonal [p-functions. This system in particular contains the Boussinesq equation.