Related papers: Wheeled props in algebra, geometry and quantizatio…
This note is an expanded and updated version of our entry with the same title for the 2006 Encyclopedia of Mathematical Physics. We give a brief overview of graded Poisson algebras, their main properties and their main applications, in the…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
We set up a framework for using algebraic geometry to study the generalised cohomology rings that occur in algebraic topology. This idea was probably first introduced by Quillen and it underlies much of our understanding of complex oriented…
This note is supposed to answer some questions on deformation theory in derived algebraic geometry. We show that derived algebraic geometry allows for a geometrical interpretation of the full cotangent complex and gives a natural setting…
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
An informal guide to the history of Heisenberg's matrix mechanics. It is designed for mathematicians with only a minimal background in either physics or geometry, and it is based upon Heisenberg's original arguments.
`How do our ideas about quantum mechanics affect our understanding of spacetime?' This familiar question leads to quantum gravity. The complementary question is also important: `How do our ideas about spacetime affect our understanding of…
We consider the renormalization of general gauge theories on curved space-time background, with the main assumption being the existence of a gauge-invariant and diffeomorphism invariant regularization. Using the Batalin-Vilkovisky (BV)…
This is an introduction to the algebraic aspect of Teichm\"uller dynamics, with a focus on its interplay with the geometry of moduli spaces of curves as well as recent advances in the field.
The aim of the paper is to start to develop the most general theory of localizations/inversion. Several new concepts are introduced and studied.
A general theory based upon 7 postulates is introduced. The basical notions are theoretical variables that are associated with an observer or with a group of communicating observers. These variables may be accessible or inaccessible. From…
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
In recent years, many new developments in theoretical physics, and in practical applications rely on different techniques of noncommutative algebras. In this review, we introduce the basic concepts and techniques of noncommutative physics…
This is an expository paper on the subject of the title. It assumes basic scheme theory, commutative and homological algebra.
I present a method of performing geometric quantization using cohomology groups extended via coefficient groups of different types. This is possible according to the Universal Coefficient Theorem (UTC). I also show that by using this method…
Loop quantum gravity in its Hamiltonian form relies on a connection formulation of the gravitational phase space with three key properties: 1.) a compact gauge group, 2.) real variables, and 3.) canonical Poisson brackets. In conjunction,…
The primary aim of this essay, drawn from the author's MMath dissertation at Oxford, is to present and explain Kontsevich's formality theorem. The first two sections introduce the main topic. Sections 3 and 4 discuss Hochschild…
The theory of modular deformations is generalized for the category of complex analytic polyhedra which includes germs of complex space as well as any compact complex analytic space. The objective of the theory is a construction of fine…
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…
In this overview article we present a formalism suitable for constructing models of QFT's on curved spacetimes. The leading principle is the emphasis on local properties. It turns out that this requires a reformulation of the standard QFT…