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...

Abstract Introduction Ultrasound is used to diagnose pregnancy complications such as miscarriage and fetal health conditions. Within the UK, findings identified during ultrasound examination are delivered by sonographers standard. However, little known about experiences of when delivering unexpected news (DUN), impact this has on them, or their preferences for training delivery. Methods Qualita...

Given a $\sigma $-finite infinite measure space $(\Omega ,\mu )$, it is shown that any Dun\-ford–Schwartz operator $T: \mathcal {L}^1(\Omega )\to )$ can be uniquely extended to the $\mathcal )+\mathcal {L}^\infty (

In this paper, we introduce the “semi-topological K-homology” of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasiprojective complex variety Y coincides with the connective topological Khomology of the associated anal...

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...