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The covariant derivative capable of differentiating and parallel transporting tangent vectors and other geometric objects induced by a parameter-dependent quantum state is introduced. It is proved to be covariant under gauge and coordinate…
This paper is based on my talk at ICM on recent progress in a number of classical problems of linear algebra and representation theory, based on new approach, originated from geometry of stable bundles and geometric invariant theory.
Particles moving inside a fluid near, and interacting with, invariant manifolds is a common phenomenon in a wide variety of applications. One elementary question is whether we can determine once a particle has entered a neighbourhood of an…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
We study directions along which the norms of vectors are preserved under a linear map. In particular, we find families of matrices for which these directions are determined by integer vectors. We consider the two-dimensional case in detail,…
In this paper, we aim to provide a notion of "relative objects", i.e. objects equipped with some sort of subobjects, in differential topology. In spite of active researches relating them, e.g. knot theory or the theory of manifolds with…
Using the advection-diffusion equation, we analytically study contaminant transport in a sharply contrasting medium with a diffusion barrier due to localization of a contaminant source in a low-permeability medium. Anomalous diffusion…
We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in…
We construct a theory of particles moving in curved both momentum space and spacetime, being a generalization of Relative Locality. We find that in order to construct such theory, with desired symmetries, including the general coordinate…
The behavior of the probability density function (PDF) transport equation at the limits of the probability space is studied from the point of view of fluid mechanics. Different boundary conditions are considered depending on the nature of…
New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can…
A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf…
We find relations between quantities defining geometry and quantities defining the length of a curve in geometries underlying Electromagnetism and unified model of Electromagnetism and Gravitation. We show that the length of a vector…
Defects are a useful tool in the study of quantum field theories. This is illustrated in the example of two-dimensional conformal field theories. We describe how defect lines and their junction points appear in the description of symmetries…
The paper starts with discussion of applications of cycle spaces to transportation cost. After a short survey of the known results on cycle spaces, we turn to the study of minimal projections onto cycle spaces in the corresponding…
We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…
Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…
We derive a generalized deviation equation in Riemann-Cartan spacetime. The equation describes the dynamics of the connecting vector which links events on two general adjacent world lines. Our result is valid for any theory in a…
In analyzing balanced parentheses, we consider a group of related variables in Dyck paths. In the four-dimensional space, the Dyck triangle is constructed, i.e. an integer lattice with Dyck paths.
We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosicky. On elastic spaces there is a natural…