Abstract This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel’s theorem [5]. We prove as a corollary Bott periodicity, independently Atiyah-Singer theorem.

The perturbative expression of Chern-Simons theory for links in Euclidean 3-space is a linear combination of integrals on configuration spaces. This has successively been studied by Guadagnini, Martellini and Mintchev, Bar-Natan, Kontsevich, Bott and Taubes, D. Thurston, Altschuler and Freidel, Yang. . .We give a self-contained version of this study with a new choice of compactification, and we...

On the total space of a G−complex vector bundle E is defined the gauge transformations. A gauge complex invariant Lagrangian determines a special complex nonlinear connection for which the associated Chern-Lagrange and Bott complex connections are gauge invariant. The complex field equations are determined with respect to these associated gauge complex connections. By complex Legendre transform...

Topology of a manifold is reflected in its diffeomorphism group. It is challenging therefore to understand the diffeomorphism group Diff(M) both as a topological and discrete group. Twenty years ago, some work has been done, in connection with characteristic classes of foliations, in constructing continuous cohomology classes for Diff(M). For M closed oriented n-dimensional manifold, a class in...

We provide a new cohomological obstruction to the existence of astheno-Kähler metrics on compact complex manifolds. Several results independent interests regarding Bott-Chern and Aeppli cohomology groups are presented relevant examples discussed.

in the linear system ax = b the points x are sometimes constrained to lie in a given subspace s of column space of a. drazin inverse for any singular or nonsingular matrix, exist and is unique. in this paper, the singular consistent or inconsistent constrained linear systems are introduced and the effect of drazin inverse in solving such systems is investigated. constrained linear system arise ...

In a career spanning five decades, Raoul Bott has wrought profound changes on the landscape of geometry and topology. It is a daunting task to improve upon his own reminiscences [B3], [B4], [B1] and commentaries on papers [B5], punctuated as they are by insight, colorful turns of phrases and amusing anecdotes. This article, based on his writings and on interviews and correspondences with people...

Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. We produce these missing formulas and obtain several surprising expressions for Schubert polynomials. The above results arise naturally from a new geometric model of Sc...

In the framework of fibred cusp operators on a manifold X associated to a boundary fibration Φ : ∂X → Y , the homotopy groups of the space G−∞ Φ (X;E) of invertible smoothing perturbations of the identity are computed in terms of the K-theory of T ∗Y . It is shown that there is a periodicity, namely the odd and the even homotopy groups are isomorphic among themselves. To obtain this result, one...

The Bott-Samelson varieties Z are a powerful tool in the representation theory and geometry of a reductive group G. We give a new construction of Z as the closure of a B-orbit in a product of flag varieties (G/B). This also gives an embedding of the projective coordinate ring of the variety into the function ring of a Borel subgroup: C[Z] ⊂ C[B]. In the case of the general linear group G = GL(n...