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 i We show that wonderful varieties are necessarily spherical (i.e., they are almost homogeneous under any Borel subgroup ofG). This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. Let denote an orthogonal symmetric Lie algbra and let (G, K) be an associated pair, i.e., Lie(G = and Lie(K°) =. Lower degree bounds for modular invariants and a question of I. The aim of the paper is the study of the orbits of the action of PGL4 on the space ?3 of the cubic surfaces of ?3, i.e., the classification of cubic surfaces up to projective motions. We study the modificationA→A' of an affine domainA which produces another affine domainA'=A[I/f] whereI is a nontrivial ideal ofA andf is a nonzero element ofI. We show that Conjectures I and II of Serre about the vanishing ofH1 cannot be strengthened. In this paper we prove that there exists a uniqueGE-equivariant affine mapjE∶IE→I. Representations of the exceptional lie superalgebraE(3, 6) I: Degeneracy conditions We show that the absolute invariants (i.e.,the ${\mbox{\rm GL}}(2, {\mbox{\bf R}})$-invariants in the field of fractions of ${\mathcal P}$) distinguish the isomorphism classes of 2-dimensional non-associative real division algebras. This estimate for multiplicities is uniform, i.e., it depends not on G/H, but only on its complexity. Cohomology of line bundles on Schubert varieties-I All parabolic geometries, i.e., Cartan geometries with homogeneous model a real Let k be an integral domain, n a positive integer, X a generic n × n matrix over k (i.e., the matrix (xij over a polynomial ring k[xij] in n2 indeterminates xij), and adj(X) its classical adjoint. The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant We show that ${\mathcal M}(G,R)$ is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We study the composition of the functor from the category of modules over the Lie algebra $\mathfrak{gl}_m$ to the category of modules over the degenerate affine Hecke algebra of GLN introduced by I. Given an automorphism Φ, we denote by k(X)Φ its field of invariants, i.e., the set of rational functions f on X such that f o Φ = f. Gabor time-frequency lattices are sets of functions of the form $g_{m \alpha , n \beta} (t) =e^{-2 \pi i \alpha m t}g(t-n \beta)$ generated from a given function $g(t)$ by discrete translations in time and frequency.

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