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\gdef\cardinalProductSymbol{\inlineProdSymbol} \gdef\vertexProductSymbol{\!\inlineProdSymbol\!} \gdef\edgeProductSymbol{\inlineProdSymbol} \gdef\indexSum#1#2#3{ {\sum_{#2}^{#3} #1}} \gdef\indexProd#1#2#3{ {\prod_{#2}^{#3} #1}} \gdef\indexMax#1#2#3{ {\max_{#2}^{#3} #1}} \gdef\indexMin#1#2#3{ {\min_{#2}^{#3} #1}} \gdef\oneTo#1{1..{#1}} \gdef\zeroTo#1{0..{#1}} \gdef\qstring#1{\mathtt{"}{#1}\mathtt{"}} \gdef\qchar#1{\mathtt{'}{#1}\mathtt{'}} \gdef\lstr#1{\mathtt{#1}} \gdef\lchar#1{\mathtt{#1}} \gdef\string#1{ {#1}} \gdef\character#1{ {#1}} \gdef\homomorphismMapping#1{\assocArray{#1}} \gdef\translationPresentation#1{\textrm{Z}_{#1}} \gdef\starTranslationPresentation#1{\textrm{Z}^*_{#1}} \gdef\translationPathValuation#1{\mathcal{\overrightharpoon Z}_{#1}} \gdef\starTranslationPathValuation#1{\overrightharpoon{\mathcal{Z}^*_{#1}}} \gdef\translationWordHomomorphism#1{\mathcal{Z}_{#1}} \gdef\starTranslationWordHomomorphism#1{\mathcal{Z}^*_{#1}} \gdef\translationCardinalValuation#1{\textrm{T}_{#1}} \gdef\starTranslationCardinalValuation#1{\textrm{T}^*_{#1}} \)
Introduction

Introduction

This site describes an incomplete and ongoing project to build a discrete, computational framework for geometry, using "off the shelf" components from a range of mathematical fields, including graph theory, group theory, abstract algebra, representation theory, theoretical computer science, differential geometry, calculus, and others.

It asks the questions: how are we to think about geometries that have a "smallest scale"? And how do we represent geometries that are produced by computational processes?

Motivation

For thousands of years we have encoded our intuitions of physical space into our mathematical and mental models for geometry. These continuous geometries – which assume infinite subdivisibility of space – are the foundation for much of modern mathematics and its applied branches. The resulting abstractions are now highly refined and immensely successful in their applications.

Nevertheless, much of our internal and external world is of a discrete and distributed nature, whether we consider the structures of our computer networks, our social networks, our economies, our software, our languages, our genetic lineages, even the symbolic operations of our thoughts. I strongly I believe there is a "missing theory" to express and model these "discrete worlds": a theory of geometry that rebases our intuitions on a computational footing and that can unlock new perspectives on all these phenomena.

Creating this theory will be slow work, since it must metabolize the ideas of continuous goemetry piecemeal – creating new intuition, notation, and software as it goes. Rather than working privately for years, I am inspired by the spirit of open source to maintain this site as a kind of "continuous beta", updated as ideas are clarified and elaborated.

What is geometry?

From an abstract point of view, we may ask what geometry is fundamentally about.

One appealing answer is that it is the logic of movement: the rules by which an object or agent may be moved from place to place, or a more abstract system may transition from state to state, and the rules used to reason about these transitions. There are only two fundamental ingredients: the states / places that an agent or system can be in, and the movements / transitions available at each such state or place.

Both of these ingredients are discrete, taking only a finite number of values in any given instantiation: an agent may move from state \(\vert{x}\) to \(\vert{y}\) to \(\vert{z}\), but it does so in discrete jumps, and describes these with discrete labels: \(\tde{\vert{x}}{\vert{y}}{\card{a}}\), \(\tde{\vert{y}}{\vert{z}}{\card{b}}\). Time, if it is involved in such descriptions, is also discrete, passing like the ticking of a CPU clock.

Novelty

None of the conceptual tools developed in this work are on their own very novel, although their assembly and emphasis into a unifying whole might prove to be for some readers. A theory is as much a way of seeing as it is a body of theorems, and so the approach I am pursuing hopes to offer a way of recasting and re-interpreting existing structures to emphasize a more computational and geometric perspective.

Visualization

One of my aims is to render intuitive this approach to discrete geometry through careful visualization. Visualization is fundamental to the theory and practice of geometry, and many arguments or explanations are simpler, or indeed redundant, in a visual medium.

Prerequisites

The perspective I advocate is cross-disciplinary, and this makes an "ideal reader" hard to gauge. Instead, I have assumed my reader to be myself at age 17: a curious undergraduate interested in various computer science and pure mathematics fields, but not deeply versed in any, and willing to read other references where necessary.

I use an informal style that attempts to be more engaging, rather than maximizing the information density of the text as is typical of academic writing. Eventually there will be a more terse, academic-style summaries of relevant "chunks" of work as they are completed and worthy of publication outside of this medium.

Format

This site represents a "living document" of a work in progress, detailing the topics within quiver geometry that I am confident enough to explain and illustrate.

Some pages contain missing subsections marked "under construction" – these are conceptually complete but require writeup that I haven't got around to yet. Gray sections on the navigation menu on the left represent ideas that are, at best, partially developed, and require future work.

Prior work and references

Unfortunately, this document doesn't currently have inline references. A full index and bibliography is forthcoming.

The closest existing body of ideas lies in geometric group theory, which roughly speaking uses Cayley graphs of groups as way to view groups as geometric objects. In a different direction, discrete differential geometry aims to discrete continuous manifolds for use in computational geometry. Lastly, topological crystallography studies graphs associated with crystallographic lattices.

Physics

While quiver geometry doesn’t make immediate connections to physics, it does aim to populate the toolbox of ideas and methods we might use to build and analyze models of fundamental physics that are discrete, finite, and computable, such as that proposed by e.g. the Wolfram Physics Project, Gerard t'Hooft, and others.

If the universe does turn out to be operating on fundamentally discrete, computational principles, we will in retrospect see continuous geometry as an enticing and useful illusion, but one that never had the physical basis that was its historical justification and inspiration.

The continuum

From a philosophical point of view, I believe we should attempt to construct continuous geometries as the limiting cases of particular discrete geometries. I'm not yet certain how to accomplish this. But I believe it will yield more insight than trying to construct discrete geometries by "discretizing" continuous geometries, by forcing us to embrace combinatorial and computational approaches from the very beginning.