Let G be a symmetrizable Kac–Moody group over a field of characteristic zero, let T be a split maximal torus of G. By using a completion of the algebra of strongly regular functions on G, and its restriction on T, we give a formal Chevalley restriction theorem. Specializing to the affine case, and to the field of complex numbers, we obtain a convergent Chevalley restriction theorem, by choosing the formal functions, which are convergent on the semi-groups of trace class elements Gtr⊂G resp. Ttr⊂T.

The classical Schläfli formula relates the variations of the dihedral angles of a smooth family of polyhedra in a space-form to the variation of the enclosed volume. We give higher analogues of this formula: for each p, we prove a simple formula relating the variation of the volumes of the codimension p faces to the variation of the ‘curvature’ – the volumes of the duals of the links in the convex case – of codimension p+2 faces. It is valid also for ideal polyhedra, or for polyhedra with some ideal vertices. This extends results of Suárez-Peiró. The proof is through analoguous smooth formulas. Some applications are described.

We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image Rπ* (π is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf ${\cal L}_\Phi$, called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if Φ is rational). Using ${\cal L}_\Phi$ we define the intersection cohomology space IH(Φ). It is conjectured that a strictly convex piecewise linear function on Φ acts as a Lefschetz operator on IH(Φ). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.

We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. Our proof uses virtual localization on the moduli space of stable maps. We describe how the proof could be simplified by the proper algebro-geometric definition of a ‘relative space’. Such a space has recently been defined by J. Li.

We study Kähler surfaces with harmonic anti-self-dual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kähler metrics, including Kähler–Einstein metrics and conformally Einstein Kähler metrics. We also extend some of our results to almost Kähler 4-manifolds, providing new examples of Ricci-flat almost Kähler metrics which are not Kähler.

It is shown that the Fourier–Whittaker coefficients of Eisenstein series on the n-fold cover of GL(n) are L-functions, improving prior results of T. Suzuki.

In his celebrated paper ‘Generic Projections’, Mather obtained the key result that a generic projection to an affine subspace of a smooth submanifold in Euclidean space is jet-transverse to any ‘modular’ submanifold of (multi-) jet space. He also gave an explicit stratification by modular submanifolds, and used it to conclude that the projection, if in the nice dimensions, is generically (C∞-) stable. In this article, we extend the result to the semi-nice dimensions (where only C0-stability is obtained), using the stratification given in our book. We first recall the definitions of the nice and semi-nice dimensions, review the main known results which involve them, and proceed to the statement of our main results. Next we discuss the condition of modularity, and present a number of methods for establishing modularity of particular strata. Finally, we show that all the strata needed for the main result are covered by these methods.

We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some nontrivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, etc.) our proof only uses the classical theory of linear systems on Riemann surfaces.

Let F be a non-archimedean local field with residue class field k. Put G=GL2(F), Γ=PGL2(k) and let X denote the Bruhat–Tits tree of G. We construct a one-dimensional simplicial complex $\tilde X$, equipped with an action of G × Γ and with a G × Γ-equivariant simplicial projection $\pi: \tilde X\to X$ (for the trivial action of Γ on X). We prove that the cohomology with compact support $H^1_c(\tilde X\open C)$ contains nontrivial representations of G (in particular positive level supercuspidal representations).

The purpose of this paper is to determine the b-functions of all the prehomogeneous vector spaces associated to nilpotent orbits in the Dynkin–Kostant theory for all the complex simple algebraic groups of exceptional type. Our method to calculate b-functions is based on the structure theorem for b-functions of several variables and the functional equations for them.

In this paper, we consider a family of elliptic curves over ${\open Q}$ with 2-torsion part ${\open Z}_2$. We prove that, for every such elliptic curve, a positive proportion of quadratic twists have Mordell–Weil rank 0.

The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,f×g) and L(s, sym2(g)× sym2(g)), where f,g are newforms, holomorphic or of Maass type, on the upper half plane, and sym2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym2(g).

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of C-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples of contact 3-manifolds where Vassiliev invariants of Legendrian and of framed knots are different. Moreover in these examples Vassiliev invariants of Legendrian knots distinguish Legendrian knots that are isotopic as framed knots and homotopic as Legendrian immersions. This raised the question what information about Legendrian knots can be captured using Vassiliev invariants. Here we answer this question by showing that for any contact 3-manifold with a cooriented contact structure the groups of Vassiliev invariants of Legendrian knots and of knots that are nowhere tangent to a vector field that coorients the contact structure are canonically isomorphic.