Related papers: Lecture notes on the Fundamental Structures of Dif…
These notes contain a survey of some aspects of the theory of differential modules and complexes as well as of their generalization, that is, the theory of $N$-differential modules and $N$-complexes. Several applications and examples coming…
A brief introduction to exterior differential systems for graduate students familiar with manifolds and differential forms. For complete files, see https://github.com/Ben-McKay/introduction-to-exterior-differential-systems
We present an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.
In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of…
We consider a differential geometric setting on power sets and Borel algebras. Our chosen framework is based on diffeologies, and we make a link between the various diffeological structures that we propose, having in mind set-valued maps,…
We introduce an alternative formalization of curved spaces in which the concept of a pointwise affine space, as defined here, replaces that of a manifold. New or modified definitions of familiar notions from differential geometry such as…
This paper is a documentation of author's reseach, focusing on the topic Grassmann Algebra spanning over July, August 2025 under mentorship provided by DRP Turkiye 2025. Grassmann algebra is a fundamental structure in mathematics with…
Diffeology extends differential geometry to spaces beyond smooth manifolds. This paper explores diffeology's key features and illustrates its utility with examples including singular and quotient spaces, and applications in symplectic…
The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometrical, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on…
The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered.
These notes were compiled as lecture notes for a course developed and taught at the University of the Southern California. They should be accessible to a typical engineering graduate student with a strong background in Applied Mathematics.…
The first part of these notes is a self-contained introduction to generalized complex geometry. It is intended as a `user manual' for tools used in the study of supersymmetric backgrounds of supergravity. In the second part we review some…
These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.
In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in…
This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet-Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes…
This book is expository and is in Russian (sample English translation of two pages is given). It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear different notions of…
Lecture notes of an algebraic geometry graduate course. The topics covered are as follows. Cohomology: ext sheaves and groups, cohomology with support, local cohomology, local duality. Duality: relative duality, Cohen-Macaulay schemes.…
We define vector fields, leaves and trajectories for schemes. With these tools, we are able to give a geometrical interpretation and to generalize several results of differential Galois theory and constructions on differential schemes. We…
In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375],…
These lecture notes in Lie Groups are designed for a 1--semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This landmark theory of the 20th Century mathematics and physics gives a rigorous…