Related papers: Basic differential geometry as a sequence of inter…
We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations.…
Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in…
This is a draft of a textbook on differential forms. The primary target audience is sophmore level undergraduates enrolled in what would traditionally be a course in vector calculus. Later chapters will be of interest to advaced…
The field of partial differential equations (PDEs) is vast in size and diversity. The basic reason for this is that essentially all fundamental laws of physics are formulated in terms of PDEs. In addition, approximations to these…
For more than half a century, dualities have been at the heart of modern physics. From quantum mechanics to statistical mechanics, condensed matter physics, quantum field theory and quantum gravity, dualities have proven useful in solving…
The article treats the geometrical theory of partial differential equations in the absolute sense, i.e., without any additional structures and especially without any preferred choice of independent and dependent variables. The equations are…
In this dissertation we study basic local differential geometry, projective differential geometry, and prolongations of overdetermined geometric partial differential equations. It is simple to prolong an n-th order linear ordinary…
The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor…
The fact that quantum theory is non-differentiable, while general relativity is built on the assumption of differentiability sources an incompatibility between quantum theory and gravity. Higher order geometry addresses this issue directly…
A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…
For a surface immersed in a three-dimensional space endowed with a norm instead of an inner product, one can define analogous concepts of curvature and metric. With these concepts in mind, various questions immediately appear. The aim of…
In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form…
Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these…
This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…
Geometry is essentially a global language, which is fully understood in different times, countries and cultures. The proof of a geometric theorem (e.g. the Pythagorean Theorem) or a geometric construction (e.g. the construction of an…
A generic surface in Euclidean 3-space is determined uniquely by its metric and curvature. Classification of all special surfaces where this is not the case, i.e. of surfaces possessing isometries which preserve the mean curvature, is known…
Since the discovery of differential calculus by Newton and Leibniz and the subsequent continuous growth of its applications to physics, mechanics, geometry, etc, it was observed that partial derivatives in the study of various natural…
This text arises from teaching advanced undergraduate courses in differential topology for the master curriculum in Mathematics at the University of Pisa. So it is mainly addressed to motivated and collaborative master undergraduate…
Comparison of $1$-dimensional distance functions is a basic tool in Alexandrov geometry and it is used to characterize spaces with curvature bounded above or below. For the zero curvature bound there is a differential inequality which…
This primer is intended as an introduction to differential forms, a central object in modern mathematical physics, for scientists and engineers.