相关论文: The notion of dimension in geometry and algebra

This talk is an introduction to ideas of non-commutative geometry and star products. We will discuss consequences for physics in two different settings: quantum field theories and astrophysics. In case of quantum field theory, we will…

In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One…

Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic…

Convergence theory is an extension of general topology. In contrast with topology, it is closed under some important operations, like exponentiation. With all its advantages, convergence theory remains rather unknown. It is an aim of this…

This note surveys how the exterior algebra and deformations or quotients of it, gives rise to centrally important notions in five domains of mathematics: Combinatorics, Topology, Lie theory, Mathematical physics, and Algebraic geometry.

We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.

Our first experience of dimension typically comes in the intuitive Euclidean sense: a line is one dimensional, a plane is two-dimensional, and a volume is three-dimensional. However, following the work of Mandelbrot \cite{mandelbrot},…

This paper introduces the concept of the dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly…

Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of study area. This phenomenon has been puzzling many researchers. This…

An extended summary of the lecture course given at the V School on Geometry and Physics, Bia\l owe\.za 2016, in which an algebraic approach to differentiation and integration that is characteristic for non-commutative geometry is described.

This article was prepared in connection with the 2009 Barnett lecture at the University of Cincinnati, and deals with various classes of fractal sets and analysis on them.

A new type of sectional curvature is introduced. The notion is purely algebraic and can be located in linear algebra as well as in differential geometry.

Recent numerical results on the fractal structure of two-dimensional quantum gravity coupled to $c=-2$ matter are reviewed. Analytic derivation of the fractal dimensions based on the Liouville theory and diffusion equation is also…

In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a…

We invent the notion of a {\it dimension of a variety} $V$ as the cardinality of all its proper {\it derived} subvarieties (of the same type). The dimensions of varieties of lattices, varieties of regular bands and other general algebraic…

We discuss new approaches to fundamental problems of mathematics and mathematical physics such as mathematical foundation of quantum field theory, the Riemann hypothesis, and construction of noncommutative algebraic geometry.

We develop the notion of crystal in the context of derived algebraic geometry, and to connect crystals to more classical objects such as D-modules.

We present a generalisation of the theory of iterated function systems and associated fractals to the setting of noncommutative geometry. Along the way, we discuss some ideas surrounding locally compact noncommutative metric spaces.

The nature of the change in perspective that accompanies the proposal of a unified physical theory deriving from the single dimension of time is elaborated. On expressing a temporal interval in a multi-dimensional form, via a direct…

Here we look at (collections of) semimetrics and seminorms, including their ultrametric versions. In particular, we are concerned with geometric properties related to connectedness and topological dimension 0.