Related papers: A Geometric Approach to Differential Forms
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
This paper proposes a new framework and algorithms to address the problem of diffeomorphic registration on a general class of geometric objects that can be described as discrete distributions of local direction vectors. It builds on both…
Geometric mechanics is usually studied in applied mathematics and most introductory texts are hence aimed at a mathematically minded audience. The present note tries to provide the intuition of geometric mechanics and to show the relevance…
It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like `triangulated category', not to mention derived functors. I will give some motivation for such things from algebraic geometry,…
This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…
We introduce the notion of a manifold admitting a simple compact Cartan 3-form $\om^3$. We study algebraic types of such manifolds specializing on those having skew-symmetric torsion, or those associated with a closed or coclosed 3-form…
This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise references are given, with indications of the strategies…
This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric…
This is an expository paper in which we define projective GIT quotients and introduce toric varieties from this perspective. It is intended primarily for readers who are learning either invariant theory or toric geometry for the first time.
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of…
Homological algebra is often understood as the translator between the world of topology and algebra. However, this branch of mathematics is worth studying by itself, given that it provides fascinating perspectives about other disciplines,…
A very brief introduction to tropical and idempotent mathematics is presented. Applications to classical mechanics and geometry are especially examined.
Although this article can be read independently, it is a continuation of the introduction to integrable systems aspects of quantum cohomology given in part 1 (math.DG/0104274). In the same elementary style, i.e. assuming basic properties of…
We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of…
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.…
These are slightly informal lecture notes intended for graduate students about the standard local theory of holomorphic foliations and vector fields. Though the material presented here is well-known some of the proofs differs slightly from…
We give an overview of differential cohomology from a modern, homotopy-theoretic perspective in terms of sheaves on manifolds. Although modern techniques are used, we base our discussion in the classical precursors to this modern approach,…
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of…
This is an expository paper which explores the ideas of the authors' paper "From Affine Geometry to Complex Geometry", arXiv:0709.2290. We explain the basic ideas of the latter paper by going through a large number of concrete, increasingly…