Related papers: Existence of Natural and Projectively Equivariant …
A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space $E$, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the…
We classify and construct $SL(n,\mathbb{R})$-intertwining differential operators $\mathcal{D}$ from a line bundle to a vector bundle over the real projective space $\mathbb{RP}^{n-1}$ by the F-method. This generalizes a classical result of…
A noncommutative-geometric generalization of the classical formalism of frame bundles is developed, incorporating into the theory of quantum principal bundles the concept of the Levi-Civita connection. The construction of a natural…
Conformally equivariant quantization is a peculiar map between symbols of real weight $\delta$ and differential operators acting on tensor densities, whose real weights are designed by $\lambda$ and $\lambda+\delta$. The existence and…
We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on $T^*M$ is made into a space of (full) symbols of operators acting on forms on $M$. This gives rise to the composition of symbols,…
We establish a compensated compactness theorem in the microlocal and geometric analytic framework. For a weakly $L^2_{\rm loc}$-convergent sequence of sections of a vector bundle over a semi-Riemannian manifold whose image under a…
In this paper we construct strong exceptional collections of vector bundles on smooth projective varieties that have a prescribed endomorphism algebra. We prove the construction problem always have a solution. We consider some applications…
In the first part of this series of papers we developed the invariant differentiation with respect to a Cartan connection, we described this procedure in the terms of the underlying principal connections, and we discussed applications of…
Recent developments concerning canonical quantisation and gauge invariant quantum mechanical systems and quantum field theories are briefly discussed. On the one hand, it is shown how diffeomorphic covariant representations of the…
Let $M$ be a smooth manifold equipped with a conformal structure, $E[w]$ the space of densities with the the conformal weight $w$ and $D_{w,w+\de}$ the space of differential operators from $E[w]$ to $E[w+\delta]$. Conformal quantization $Q$…
In this expository article, we consider first order elliptic differential operators acting on smooth vector bundles over compact manifolds, and certain invariants derived from the analysis of these operators, namely the eta invariant} and…
We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…
We study differential invariants of the third order linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles on two dimensional manifolds with respect to groups of…
We classify equivariant topological complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most)…
Generalizing the algebra of motion-invariant differential operators on a symmetric space we study invariant operators on equivariant vector bundles. We show that the eigenequation is equivalent to the corresponding eigenequation with…
The analysis of mathematical structure of the method of operator manifold guides our discussion. The latter is a still wider generalization of the method of secondary quantization with appropriate expansion over the geometric objects. The…
The quantization of vector bundles is defined. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. (Coadjoint orbits are symplectic spaces with a transitive, semisimple symmetry…
A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding…
The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also…
We develop a combinatorial theory of vector bundles with connection on locally ordered simplicial complexes. This is a first step towards a discrete exterior calculus for bundle-valued forms. The basic building block is the discrete…