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The coframe field model is known as a viable model for gravity. The principle problem is an interpretation of six additionaldegrees of freedom. We construct a general family of connections which includes the connections of Levi-Civita and…

We study deformations of 2D Integrable Quantum Field Theories (IQFT) which preserve integrability (the existence of infinitely many local integrals of motion). The IQFT are understood as "effective field theories", with finite ultraviolet…

We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry.…

The author has previously constructed a class of admissible vector fields on the path space of an elliptic diffusion process $x$ taking values in a closed compact manifold. In this Note the existence of flows for this class of vector fields…

We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we…

We show that Rieffel's deformation sends covariant C(T)-algebras into C(T)-algebras. We also treat the lower semi-continuity issue, proving that Rieffel's deformation transforms covariant continuous fields of C*-algebras into continuous…

Consider a finite-dimensional real vector space equipped with a finite group acting unitarily on it. We address the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our approach is based on…

Mostly self-contained script on functorial topological quantum field theories. These notes give a slow introduction to the basic notions of category theory which serve a closer investigation of cobordisms and (commutative) Frobenius…

A classical result in quantum topology is that oriented 2-dimensional topological quantum field theories (2-TQFTs) are fully classified by commutative Frobenius algebras. In 2006, Turaev and Turner introduced additional structure on…

A new notion of integrability called the multi-dimensional consistency for the integrable systems with the Lagrangian 1-form structure is captured in the geometrical language for quantum level. A zero-curvature condition, which implies the…

The moduli space of isometry classes of Riemannian structures on a smooth manifold was emphasized by J.A.Wheeler in his superspace formalism of quantum gravity. A natural question concerning it is: What is a natural topology on such moduli…

This paper details the lesser known conditions on ${\mathbb {R}}^{n}$ for the integrability of pfaffian forms, or 1-forms. Emphasis is given to locality of these conditions, and proofs in some additional detail are provided for theorems due…

This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.

We present an explicit formula for the mean curvature of a unit vector field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal unit vector fields. We also give an…

We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_{n}$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the…

We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian…

We review and extend the results of [1] that gives a condition for reducibility of quantum representations of mapping class groups constructed from Reshetikhin-Turaev type topological quantum field theories based on modular categories. This…

This paper is part of a series of papers on differential geometry of $C^\infty$-ringed spaces. In this paper, we study vector fields and their flows on a class of singular spaces. Our class includes arbitrary subspaces of manifolds, as well…

Let $(M,g)$ be a smooth connected Riemannian manifold. We show an improvement of flatness theorem for hypersurfaces of $M$ of bounded nonlocal mean curvature in the viscosity sense. It implies local $ C^{1,\alpha}$ regularity of these…

For a conformal vector field on a closed, real-analytic, Lorentzian manifold we prove that the flow is locally isometric -- that it preserves a metric in the conformal class on a neighborhood of any point -- or the metric is everywhere…