Related papers: A classical approach to smooth supermanifolds
A notion of differentiability is being proposed for maps between Wasserstein spaces of order 2 of smooth, connected and complete Riemannian manifolds. Due to the nature of the tangent space construction on Wasserstein spaces, we only give a…
The aim of the present text is twofold: to provide a compendium of Lagrangian and Hamiltonian geometries and to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. The fundamental equations (or…
We formulate geodesics on a manifold in terms of a parallel transfer of a particle state vector transformed by local Lorentz and Yang-Mills symmetry groups. This formulation leads to an introduction of a canonical one-form the eigenvalues…
With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral…
These notes, based on a graduate course I gave at Hamburg University in 2003, are intended to students having basic knowledges of differential geometry. Their main purpose is to provide a quick and accessible introduction to different…
This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras…
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…
A complete mathematical framework for coalgebraic formulation of supergeometry and its infinite-dimensional extension is proposed. Within this approach a supermanifold is defined as a graded coalgebra endowed with a smooth structure. The…
This thesis is divided into five chapters. The aim is the study of the effectiveness of a chemical as defined by R. Aris for semilinear elliptic equations. The first chapter focuses on homogenization on quasilinear diffusion-reaction…
Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly…
Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more…
In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely…
We briefly recall a fundamental exterior differential system introduced by the author and then apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemaniann 3-manifolds. The system…
I hesitated for a long time before giving shape to these notes, originally intended for preliminary reading by the attendees to the Summer School "New paths towards quantum gravity" (Holbaek Bay, Denmark, May 2008). At the end, I decide…
Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry.…
This paper develops an analytical approach to the study of the geometry of projective maps using the theory of elliptic differential operators. We construct two elliptic operators of second and fourth order, whose kernels characterize…
Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the…
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…
Differential graded (DG) commutative algebra provides powerful techniques for proving theorems about modules over commutative rings. These notes are a somewhat colloquial introduction to these techniques. In order to provide some motivation…
We introduce and investigate the notion of a $\mathbb Z$-graded covering for a supermanifold. More precisely, Donagi and Witten suggested a construction of the first obstruction class for splitting of a supermanifold via differential…