Joseph Corneli's publications

The purpose of this paper is to show how we can combine and adapt methods from elite training, future studies, and collaborative design, and apply them to address significant problems in social networks. We focus on three such methods: we use Project Action Reviews to implement social perception, Causal Layered Analysis to implement social cognition, and Design Pattern Languages to implement social action. We present the results of two studies: firstly, we use Causal Layered Analysis to explore the ways in which the design pattern discourse has been evolving. Secondly, to illustrate the three methods in combination, we develop a case study, showing how we applied the methods to bootstrap a distributed cross-disciplinary research seminar. Building on these analyses, we elaborate several scenarios for the future use of design patterns in large-scale distributed collaboration. Our case study suggests ways in which progress could be made towards realising these scenarios. We conclude that the combination of methods is robust to uncertainty, insofar as they support adaptations as circumstances change, and incorporate diverse perspectives. In particular, we show how methods drawn from other domains enrich and are enriched by design patterns; we believe the analysis will be of interest to all of the communities whose methods we draw upon.

VirtuosA (‘virtuous algorithm’) is introduced, a model in which artificial intelligence (AI) systems learn ethical behaviour based on a framework adapted from Christian philosopher Dallas Willard and brought together with associated neurobiological structures and broader systems thinking. To make the inquiry concrete, the authors present a simple example scenario that illustrates how a robot might acquire behaviour akin to the virtue of kindness that can be attributed to humans. References to philosophical work by Peter Sloterdijk help contextualise Willard’s virtue ethics framework. The VirtuosA architecture can be implemented using state-of-the-art computing practices and plausibly redescribes several concrete scenarios implemented from the computing literature and exhibits broad coverage relative to other work in ethical AI. Strategies are described for using the model for systems evaluation —particularly the role of ‘embedded evaluation’ within the system—and its broader application as a meta-ethical device is discussed.

To adequately model mathematical arguments the analyst must be able to represent the mathematical objects under discussion and the relationships between them, as well as inferences drawn about these objects and relationships as the discourse unfolds. We introduce a framework with these properties, which has been used to analyse mathematical dialogues and expository texts. The framework can recover salient elements of discourse at, and within, the sentence level, as well as the way mathematical content connects to form larger argumentative structures. We show how the framework might be used to support computational reasoning, and argue that it provides a more natural way to examine the process of proving theorems than do Lamport’s structured proofs.

This chapter develops a meta-evaluation of progress markers in Computational Creativity. We rely on an analysis of interview data with people who have applied several standard metrics. We use an existing meta-evaluation framework to distil findings in a format that will support comparison with future research on this topic.

This chapter surveys frameworks that describe social creativity. It focuses on mathematics, but includes work which heads in a more general direction: first, by examining social creativity in music, and then turning to a description of several new results that use formal, qualitative, and simulation techniques to theorise social creativity on computers. The chapter includes a pilot study examining the salience of various frameworks for the analysis of mathematical text.

In a straightforward meta-level shift of focus, we use design patterns as a medium and process for capturing insight about the process of design. We survey mainstream design genres, and draw conclusions about how they can help inform the design of intelligent systems.

This article reflects on current challenges we encounter in teaching data science to graduate students. A common critique of data science classes is that examples are static and student group work is embedded in an ‘artificial’ and ‘academic’ context. We look at how we can make teaching data science classes more relevant to real-world problems. Student engagement with real problems---and not just ‘real-world data sets’---has the potential to stimulate learning, exchange, and serendipity on all sides, and on different levels: noticing unexpected things in the data, developing surprising skills, finding new ways to communicate, and, lastly, in the development of new strategies for teaching, learning and practice.

Whereas formal mathematical theories are well studied, computers cannot yet adequately represent and reason about mathematical dialogues and other informal texts. To address this gap, we have developed a representation and reasoning strategy that draws on contemporary argumentation theory and classic AI techniques for representing and querying narratives and dialogues. In order to make the structures that these modelling tools produce accessible to computational reasoning, we encode representations in a higher-order nested semantic network. This system, for which we have developed a preliminary prototype in LISP, can represent both the content of what people say, and the dynamic reasoning steps that move from one step to the next.

This paper outlines a strategy for building semantically meaningful representations and carrying out effective reasoning in technical knowledge domains such as mathematics. Our central assertion is that the semi-structured Q&A format, as used on the popular Stack Exchange network of websites, exposes domain knowledge in a form that is already reasonably close to the structured knowledge formats that computers can reason about. The knowledge in question is not only facts – but discursive, dialectical, argument for purposes of proof and pedagogy. We therefore assert that modelling the Q&A process computationally provides a route to domain understanding that is compatible with the day-to-day practices of mathematicians and students. This position is supported by a small case study that analyses one question from Mathoverflow in detail, using concepts from argumentation theory. A programme of future work, including a rigorous evaluation strategy, is then advanced.

The simulation of mathematical reasoning has been a driving force throughout the history of Artificial Intelligence research. However, despite significant successes in computer mathematics, computers are not widely used by mathematicians apart from their quotidian applications. An oft-cited reason for this is that current computational systems cannot do mathematics in the way that humans do. We draw on two areas in which Automated Theorem Proving (ATP) is currently unlike human mathematics: firstly in a focus on soundness, rather than understandability of proof, and secondly in social aspects. Employing techniques and tools from argumentation to build a framework for mixed-initiative collaboration, we develop three complementary arcs. In the first arc – our theoretical model – we interpret the informal logic of mathematical discovery proposed by Lakatos, a philosopher of mathematics, through the lens of dialogue game theory and in particular as a dialogue game ranging over structures of argumentation. In our second arc – our abstraction level – we develop structured arguments, from which we induce abstract argumentation systems and compute the argumentation semantics to provide labelings of the acceptability status of each argument. The output from this stage corresponds to a final, or currently accepted proof artefact, which can be viewed alongside its historical development. Finally, in the third arc – our computational model – we show how each of these formal steps is available in implementation. We demonstrate our approach with a formal, implemented example of real-world mathematical collaboration. Finally, we offer reflections on our mixed-initiative collaborative approach.

In this paper, we present our vision on a collaborative semi-formal, language independent math(s) encyclopedia. We describe how we are going to derive that new plattform based on our previous works and how the public will benefit from it.

Tim Berners-Lee defines social machines to be a class of systems “in which the people do the creative work and the machine does the administration.” Here, we note that the ansatz in computational creativity – a subfield of artificial intelligence – is that, in the future if not already, computational systems and agents will also contribute directly to creative work. But it can be presumed that we will use standard social machines to develop the next generation ‘target’ systems, whatever they may be. The domain of mathematics is a particularly interesting and practical place to explore a range of issues in AI and other aspects of computing. Contemporary mathematical practice intersects several areas of computing practice and theory, ranging from computer-mediated communication to formal computational modelling of mathematical argument. We ask: How will we use the various existing social machines to bootstrap the next edition? How can the idea of programming languages help?

Our software system simulates the classical collaborative Japanese poetry form, renga, made of linked haikus. We used NLP methods wrapped up as web services. This approach is suitable for collaborative human-AI generation, as well as purely computer-generated poetry. Evaluation included a blind survey comparing AI and human haiku. To gather ideas for future work, we examine related research in semiotics, linguistics, and computing.

"Natural Language," whether spoken and attended to by humans, or processed and generated by computers, requires networked structures that reflect creative processes in semantic, syntactic, phonetic, linguistic, social, emotional, and cultural modules. Being able to produce novel and useful behavior following repeated practice gets to the root of both artificial intelligence and human language. This paper investigates the modalities involved in language-like applications that computers -- and programmers -- engage with, and aims to fine tune the questions we ask to better account for context, self-awareness, and embodiment.

Modelling the creativity that takes place in social settings presents a range of theoretical challenges. Mel Rhodes’s classic "4Ps" of creativity, the "Person, Process, Product, and Press," offer an initial typology. Here, Rhodes’s ideas are connected with Elinor Ostrom’s work on the analysis of economic governance to generate several "creativity design principles." These principles frame a survey of the shared concepts that structure the contexts that support creative work. The concepts are connected to the idea of computational "tests" to foreground the relationship with standard computing practice, and to draw out specific recommendations for the further development of computational creativity culture.

Conceptual blending is a powerful tool for computational creativity where, for example, the properties of two harmonic spaces may be combined in a consistent manner to produce a novel harmonic space. However, deciding about the importance of property features in the input spaces and evaluating the results of conceptual blending is a nontrivial task. In the specific case of musical harmony, defining the salient features of chord transitions and evaluating invented harmonic spaces requires deep musicological background knowledge. In this paper, we propose a creative tool that helps musicologists to evaluate and to enhance harmonic innovation. This tool allows a music expert to specify arguments over given transition properties. These arguments are then considered by the system when defining combinations of features in an idiomblending process. A music expert can then assess whether the new harmonic idiom makes musicological sense and re-adjust the arguments (selection of features) to explore alternative blends that can potentially produce better harmonic spaces. We conclude with a discussion of future work that would further automate the harmonisation process.

We present recent developments in the Flowchart Writer (FloWr) project, where we have built a framework for implementing creative systems as flowcharts of processing nodes. We describe how the system has been migrated from a desktop application to a web portal and document the various features that the portal provides to support Computational Creativity research and development. This includes a node development package and automated chart development assistants. We detail how we have supplemented the online graphical platform with a web service API to enable developers to remotely access the features of FloWr through a programming language of their choice. This encompasses developing systems as flowcharts, together with running flowcharts remotely and also allows developers to publish flowcharts as web services. Importantly, the API allows Computational Creativity researchers to experiment with the automated development of creative software systems. To encourage this, we have also introduced simple models for automated software development into the FloWr API itself, providing a novel system for unsophisticated users to experiment with. We demonstrate the potential benefits of using FloWr, with case studies showing how the web portal has been used for both node and chart development by novice and expert users.

Within the WHIM project, we study fictional ideation, i.e., processes for automatically inventing, assessing and presenting fictional ideas. Here we examine the foundational notion of the plausibility of fictional ideas, by performing an empirical study to surface the factors that affect judgements of plausibility. Our long term aim is to formalise a computational method which captures some intuitive notions of plausibility and can predict how certain types of people will assess the plausibility of certain types of fictional ideas. This paper constitutes a first firm step towards this aim.

Peeragogy is a collection of practical techniques for collaborative learning and collaborative work. Until the advent of the web and widespread access to digital media, traditional systems have effectively monopolized the pursuit of learning. Now, with access to open educational resources and free or inexpensive communication platforms, groups of people can more easily learn together outside as well as inside formal institutions.

We sketch the process of creating a novel video game by blending two video games specified in the Video Game Description Language (VGDL), following the COINVENT computational model of conceptual blending. We highlight the choices that need to be made in this process, and discuss the prospects for a computational game designer based on blending.

We describe nine design patterns that we have developed in our work on the Peeragogy project, in which we aim to help design the future of learning, inside and outside of institutions. We use these patterns to build an ``emergent roadmap'' for the project. The primary audience we envision for the paper are teams of people who aspire to collaboratively manage their own free/open/libre learning and development projects.

One particular challenge in AI is the computational modelling and simulation of creativity. Feedback and learning from experience are key aspects of the creative process. Here we investigate how we could implement feedback in creative systems using a social model. From the field of creative writing we borrow the concept of a Writers Workshop as a model for learning through feedback. The Writers Workshop encourages examination, discussion and debates of a piece of creative work using a prescribed format of activities. We propose a computational model of the Writers Workshop as a roadmap for incorporation of feedback in artificial creativity systems. We argue that the Writers Workshop setting describes the anatomy of the creative process. We support our claim with a case study that describes how to implement the Writers Workshop model in a computational creativity system. We present this work using patterns other people can follow to implement similar designs in their own systems. We conclude by discussing the broader relevance of this model to other aspects of AI.

We model the mathematical process whereby new mathematical theories are invented. Here we explain the use of conceptual blending for this purpose, and show examples to illustrate the process in action. Our longer-term goal is to support machine and human mathematical creativity.

The paper motivates the use of argumentation for evaluating “concept blends” and other forms of combinatorial creativity. We present a basic introduction to argumentation theory and Lakatosian reasoning in the background section. The paper has two core sections: First, we present a semiotic system for icon design, showing how icons can be described in terms of affordances and how they are related by sign patterns. The interpretation of a sign pattern conveys an intended meaning for an icon. This intended meaning is subjective and depends on the way the concept blending was realized. Therefore, argumentation can be used for evaluating the blends, in this case, icons. Second, we show different examples how abstract argumentation and Lakatosian reasoning can be applied to evaluate different blends. In a closing discussion, we talk how the argumentation can play a role at different stages of the concept blending process.

Creativity cannot exist in a vacuum; it develops through feedback, learning, reflection and social interaction with others. However, this perspective has been relatively under-investigated in computational creativity research, which typically examines systems that operate individually. We develop a thought experiment showing how structured dialogues can help develop the creative aspects of computer poetry. Centrally in this approach, we ask questions of a poem, inviting it to tell us in what way it may be considered a “creative making.”

We define and explore in simulation several rules for the local evolution of generative rules for 1D and 2D cellular automata. Our implementation uses strategies from conceptual blending. We discuss potential applications to modelling social dynamics.

The notion that software could be independently and usefully creative is becoming more commonplace in scientific, cultural, business and public circles. It is not fanciful to imagine creative software embedded in society in the short to medium term, acting as collaborators and autonomous creative agents for much societal benefit. Technologically, there is still some way to go to enable Artificial Intelligence methods to create artefacts and ideas of value, and to get software to do so in interesting and engaging ways. There are also a number of sociological hurdles to overcome in getting society to accept software as being truly creative, and we concentrate on those here. We discuss the various communities that can be considered stakeholders in the perception of computers being creative or not. In particular, we look in detail at three sets of stakeholders, namely the general public, Computational Creativity researchers and fellow creatives. We put forward various philosophical points which we argue will shape the way in which society accepts creative software. We make various claims along the way about how people perceive software as being creative or not, which we believe should be addressed with scientific experimentation, and we call on the Computational Creativity research community to do just that.

The Peeragogy Project facilitates the reflective practice of peer learning and peer production. We are building living projects that continuously evolve, and we have come together with several of our own projects to build a space for mutual aid. In short, we are building a peer produced Peeragogy Accelerator. We present the background and operating principles of the Accelerator, and describe some of the Accelerated projects: SOLE out of the Box, which aims to build a self-sufficient distributed makerspace; PlanetMath.org, an online mathematics community that will soon run its first peeragogical Calculus course; and Managing Innovation and Change, a semester-long academic course on building startups that is using peeragogy concepts. The Peeragogy Accelerator works “transversally” across these and other projects, supporting both peer-to-peer and Project-to-Project collaboration. We are actively seeking to involve others from the OER world.

Interlinking knowledge is one of the cornerstones of online collaboration. While wiki systems typically rely on links supplied by authors, in the early 2000s the mathematics encyclopedia at PlanetMath.org introduced a feature that provides automatic linking for previously defined concepts. The NNexus software suite was developed to support the necessary subtasks of concept indexing, concept discovery and link-annotation. In this paper, we describe our recent reimplementation and revisioning of the NNexus system.

Determining conclusively whether a new version of software creatively exceeds a previous version or a third party system is difficult, yet very important for scientific approaches in Computational Creativity research. We argue that software product and process need to be assessed simultaneously in assessing progress, and we introduce a diagrammatic formalism which exposes various timelines of creative acts in the construction and execution of successive versions of artefactgenerating software. The formalism enables estimations of progress or regress from system to system by comparing their diagrams and assessing changes in quality, quantity and variety of creative acts undertaken; audience perception of behaviours; and the quality of artefacts produced. We present a case study in the building of evolutionary art systems, and we use the formalism to highlight various issues in measuring progress in the building of creative systems.

This research project develops around a technological intervention intended to transform a peer produced reference resource into a peer produced learning environment. Through the work described in this thesis, PlanetMath.org, an early online community devoted to mathematics, has now become a mathematical practicum, and a laboratory for learning science. A new theory that describes the nexus of peer production and peer learning is foundational for the research programme. The candidate theory was initially developed during a pilot study based on online field work at the Peer-2-Peer University. The new theory -- which is given the name "paragogy" -- has implications for designers, researchers, educators, and others whose work relies on peer learning and peer production. Further research and development work in the PlanetMath context helped to refine the theory, and applied it along with a range of mixed methods to develop an anthropologically-inspired study of modern mathematics. A quantitative approach was employed to detect the factors of interaction that influence learning outcomes, using legacy data from PlanetMath. A qualitative, interview-based approach was employed, to understand the desiderata potential users of a new system emphasizing peer learning. The new software system was implemented, informed by paragogy and these stakeholder perspectives, using Drupal and other open source components. Field work with PlanetMath users after the launch of the new system employed an emergent design process to elaborate the theory and develop a viable approach to ongoing development and codesign.

Over the past decade a number of systems have been developed that tell museum stories by constructing digital presentations from cultural objects and their metadata. Our novel approach, informed by museum practice, is built around a formalization of stages of museum storytelling that involve: (i) the collection of events, museum objects and their associated stories, (ii) the construction of story sections that organise the content in different ways, and (iii) the assembly of story sections into a story structure. Here we focus in particular on this final stage of building the story structure. Our approach to providing intelligent assistance to story construction involves: (i) separating overlapping or conflicting story sections into separate candidate storylines, (ii) evaluating candidate storylines according the criteria of coverage, richness and coherence, (iii) assembling storylines into linear, layered or multi-route structures and (iv) ordering the story sections according to their setting within the storyline.

We have been writing the missing manual for peer-produced peer learning: the “Peeragogy Handbook” (peeragogy.org). Throughout this work we have asked and aimed to address questions like these: What would a motivated group of self-learners need to know to agree on a subject or skill to learn, find and qualify the best learning resources about that topic, then select and use appropriate communication media to learn it together? What would these people need to know about learning to put together a successful learning programme?

Wiki-like or crowdsourcing models of collaboration can provide a number of benefits to academic work. These techniques may engage expertise from different disciplines, and potentially increase productivity. This paper presents a model of massively distributed collaborative authorship of academic papers. This model, developed by a collective of 31 authors, identifies key tools and techniques that would be necessary or useful to the writing process. The process of collaboratively writing this paper was used to discover, negotiate, and document issues in massively authored scholarship. Our work provides the first extensive discussion of the experiential aspects of large-scale collaborative research.

This paper considers the problem of peer producing rich online learning environments, a task that appears techno-socially feasible, but is not without challenge. We draw on the self-professedly “utopian” approach developed by Baudrillard in “The Mirror of Production”, to establish and understand our two key dimensions of leverage (language and recycling). We then extend a recent paper by Corneli and Danoff on the topic of peer learning with a set of guidelines for practitioners. Our conclusion supports active peer production of learning environments, against a “provisionist” strategy, but we recognize that our paragogical agenda may be at odds with established educational systems in some respects, but perhaps in a complementary manner.

This paper presents our work in progress on the Planetary system, along with a critical evaluation of the project relative to its stated goals and the goals of one of its main “clients”, PlanetMath.org.

The Mathematics Subject Classification (MSC) is a widely used scheme for classifying documents in mathematics by subject. Its traditional, idiosyncratic conceptualization and representation makes the scheme hard to maintain and requires custom implementations of search, query and annotation support. This limits uptake e.g. in semantic web technologies in general and the creation and exploration of connections between mathematics and related domains (e.g. science) in particular. This paper presents the new official implementation of the MSC2010 as a Linked Open Dataset, building on SKOS (Simple Knowledge Organization System). We provide a brief overview of the dataset’s structure, its available implementations, and first applications.

The purpose of the paper is to design a rubric for assessment of informal learning in undergraduate level mathematics. Our proposed assessment strategy would support and parallel the mathematics learning environment we are developing at PlanetMath. PlanetMath is currently an open Web 2.0 system that consists of peer-produced encyclopaedia articles and forum discussions: we will add facilities for contributing textbook-style problems and solutions, and detailed activity tracking to model and support learning.

Paragogy is a theory of peer learning: we endeavor to say how it works, and how it works best. This paper outlines paragogy’s contemporary relevance and expounds its principles, showing their connections to other theories. We present an extended example of paragogy in practice, where we use it to evaluate our experiences working at the Peer 2 Peer University (P2PU).

This demo shows work in progress on a Web 2.0 infrastructure for mathematical problem solving. Our aim is to make undergraduate-level mathematics easier to learn: our strategy is to link problems and solved examples to prerequisite material drawn from an existing free/open mathematical knowledge repository, the encyclopedia at PlanetMath.org.

The Peer-to-Peer Learning Environment (P2PLE) is a proposed approach to helping learners co-construct their learning environment using recommendations about people, content, and tools. The work draws on current research on PLEs, and participant observation at the Peer-to-Peer University (P2PU). We are particularly interested in ways of eliciting explicit, coded, user feedback, and in monitoring the transitions from state to state within the PLE. We discuss the ways in which these ideas can inform the design of a platform for peer-supported study of university-level mathematics.

Learning online has significantly evolved over the past decade due to the emergence of Web 2.0 and 3.0 technologies that facilitate social learning in adaptive online environments. The open content movement and the associated techniques of crowdsourcing (i.e. assimilating several small contributions into resources of high quality) have further influenced education on the Web. This chapter investigates the concept of crowdsourcing in education through an analysis of case studies dealing with two open online learning communities, Peer 2 Peer University, and PlanetMath.org. The case studies proceed via an analysis of the various roles played by the individuals involved in each organization. The outcomes of this analysis are used to extract general recommendations for building online communities and applying crowdsourcing techniques in educational contexts.

In this paper we present the Active Documents Paradigm (semantically annotated documents associated with a content commons that holds the corresponding background ontologies) and the Planetary system (as an active document player). We show that the current Planetary system gives a solid foundation and can be extended modularly to address most of the criteria of the Executable Papers Challenge.

Personal Learning Environments (PLEs) are gradually gaining ground over traditional Learning Management Systems (LMS) by facilitating the lone or collaborative study of user-chosen blends of content and courses from heterogeneous sources, including Open Educational Resources (OER). Our work is focused on the social aspect of PLEs, and particularly on live interactions between learners, real-time annotation of OER and content discovery. This paper describes Gravpad, a service for sharing live annotations, and showcases how it can contribute to social learning within a PLE.

GravPad is the next major step in the evolution of the EtherPad, a real-time collaborative editor that was first developed at AppJet, Inc. (http://appjet.com/) and recently opensourced when AppJet was acquired by Google. EtherPad’s developer community, headed up by Egil Möller and John McLear, has made a plugin engine and added standard wiki features to the editor (http://github.com/ether/pad). My GravPad demonstration will show a hacked EtherPad running in a “sidewiki” format - the basis of a platform for live web annotation and content discovery.

In this paper we present recent developments in content markup for mathematics, and a corresponding software stack that functions as an enabling technology for a social and semantic web for the STEM disciplines. We show the potential of this technology in two eMath 3.0 applications: PlanetMathRedux, a re-implementation of the mathematical encyclopedia PlanetMath.org, and PantaRheiRedux, a community reader for course materials. These applications indicate both present and potential uses for this software as a basis for eLearning applications in Science, Technology, Engineering and Mathematics through the addition of suitable pedagogies.

We prove that a standard Y is an area-minimizing partition of Gauss space into three given volumes, provided that the standard double bubble is an area-minimizing partition of high-dimensional spheres. We prove that the standard double bubble is the area-minimizing partition of spheres of any dimension where the volumes differ by at most 4%.

We prove the double bubble conjecture in the three-sphere S³ and hyperbolic three-space H³ in the cases where we can apply Hutchings theory:

• In S³, when each enclosed volume and the complement occupy at least 10% of the volume of S³.
• In H³, when the smaller volume is at least 85% that of the larger.

A balancing argument and asymptotic analysis reduce the problem in S³ and H³ to some computer checking. The computer analysis has been designed and fully implemented for both spaces.

Commons-based peer production is a term that describes authorship of shared information resources. In this article we examine the technical aspects of writing-in-common. We begin with a simple model: that of text and commentary. This scholia-based model emphasizes ownership of speech and freedom of speech. We then consider what happens when the freedom to create derivative versions is added to the mix. The resulting model proves to be quite sophisticated, and flexible enough to describe many different commons-based peer production systems. We provide an overview of our implementation of this model, and suggest some ideas for subsequent work. We conclude by discussing the implications of our model for distributed authorship and writing.

We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of minimizers on flat two-tori, depending on the areas to be enclosed.

We present a conjecture, based on computational results, on the area-minimizing method of enclosing and separating two arbitrary volumes in the flat cubic three-torus, T³. For comparable small volumes, we prove that the standard double bubble from R³ is area-minimizing.

The aim of this thesis is to be a comprehensive and self-contained treatment of known results on area minimizing double bubbles in constant curvature spaces. Difficult proofs of existence and regularity theorems are omitted, as are several more readible proofs that are available (or soon to be made available) in the literature. Included in the thesis are a number of new results, including the reduction of the double bubble conjecture in spherical and hyperbolic space to a problem of asymptotic analysis, a proof that the least-area way to divide Gauss space (Rⁿ with Gaussian measure) into three equal pieces consists of three half-hyperplanes meeting at the origin at 120°, and results due to the author and various collaborators on double bubbles in two- and three-dimensional tori. The theoretical background of these developments is surveyed in some detail. A proof of the isoperimetric property of spheres in the constant curvature space forms is presented. Various detailed appendices are provided with the hope that persons with little or no background in differential geometry or variational methods will be able to read and enjoy the entire thesis.