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Usability

Usability can be described as the capacity of a system to provide a condition for its users to perform the tasks safely, effectively, and efficiently while enjoying the experience. In software engineering, usability is the degree to which a software can be used by specified consumers to achieve quantified objectives with effectiveness, efficiency, and satisfaction in a quantified context of use.

The object of use can be a software application, website, book, tool, machine, process, vehicle, or anything a human interacts with. A usability study may be conducted as a primary job function by a usability analyst or as a secondary job function by designers, technical writers, marketing personnel, and others. It is widely used in consumer electronics, communication, and knowledge transfer objects (such as a cookbook, a document or online help) and mechanical objects such as a door handle or a hammer.

Usability includes methods of measuring usability, such as needs analysis and the study of the principles behind an object's perceived efficiency or elegance. In human-computer interaction and computer science, usability studies the elegance and clarity with which the interaction with a computer program or a web site (web usability) is designed. Usability considers user satisfaction and utility as quality components, and aims to improve user experience through iterative design.

Introduction
The primary notion of usability is that an object designed with a generalized users' psychology and physiology in mind is, for example:
 * More efficient to use—takes less time to accomplish a particular task
 * Easier to learn—operation can be learned by observing the object
 * More satisfying to use

Complex computer systems find their way into everyday life, and at the same time the market is saturated with competing brands. This has made usability more popular and widely recognized in recent years, as companies see the benefits of researching and developing their products with user-oriented methods instead of technology-oriented methods. By understanding and researching the interaction between product and user, the usability expert can also provide insight that is unattainable by traditional company-oriented market research. For example, after observing and interviewing users, the usability expert may identify needed functionality or design flaws that were not anticipated. A method called contextual inquiry does this in the naturally occurring context of the users own environment. In the user-centered design paradigm, the product is designed with its intended users in mind at all times. In the user-driven or participatory design paradigm, some of the users become actual or de facto members of the design team.

The term user friendly is often used as a synonym for usable, though it may also refer to accessibility. Usability describes the quality of user experience across websites, software, products, and environments. There is no consensus about the relation of the terms ergonomics (or human factors) and usability. Some think of usability as the software specialization of the larger topic of ergonomics. Others view these topics as tangential, with ergonomics focusing on physiological matters (e.g., turning a door handle) and usability focusing on psychological matters (e.g., recognizing that a door can be opened by turning its handle). Usability is also important in website development (web usability). According to Jakob Nielsen, "Studies of user behavior on the Web find a low tolerance for difficult designs or slow sites. People don't want to wait. And they don't want to learn how to use a home page. There's no such thing as a training class or a manual for a Web site. People have to be able to grasp the functioning of the site immediately after scanning the home page—for a few seconds at most." Otherwise, most casual users simply leave the site and browse or shop elsewhere.

Usability can also include the concept of prototypicality, which is how much a particular thing conforms to the expected shared norm, for instance, in website design, users prefer sites that conform to recognised design norms.

Definition
ISO defines usability as "The extent to which a product can be used by specified users to achieve specified goals with effectiveness, efficiency, and satisfaction in a specified context of use." The word "usability" also refers to methods for improving ease-of-use during the design process. Usability consultant Jakob Nielsen and computer science professor Ben Shneiderman have written (separately) about a framework of system acceptability, where usability is a part of "usefulness" and is composed of:
 * Learnability: How easy is it for users to accomplish basic tasks the first time they encounter the design?
 * Efficiency: Once users have learned the design, how quickly can they perform tasks?
 * Memorability: When users return to the design after a period of not using it, how easily can they re-establish proficiency?
 * Errors: How many errors do users make, how severe are these errors, and how easily can they recover from the errors?
 * Satisfaction: How pleasant is it to use the design?

Usability is often associated with the functionalities of the product (cf. ISO definition, below), in addition to being solely a characteristic of the user interface (cf. framework of system acceptability, also below, which separates usefulness into usability and utility). For example, in the context of mainstream consumer products, an automobile lacking a reverse gear could be considered unusable according to the former view, and lacking in utility according to the latter view. When evaluating user interfaces for usability, the definition can be as simple as "the perception of a target user of the effectiveness (fit for purpose) and efficiency (work or time required to use) of the Interface". Each component may be measured subjectively against criteria, e.g., Principles of User Interface Design, to provide a metric, often expressed as a percentage. It is important to distinguish between usability testing and usability engineering. Usability testing is the measurement of ease of use of a product or piece of software. In contrast, usability engineering (UE) is the research and design process that ensures a product with good usability. Usability is a non-functional requirement. As with other non-functional requirements, usability cannot be directly measured but must be quantified by means of indirect measures or attributes such as, for example, the number of reported problems with ease-of-use of a system.

Intuitive interaction or intuitive use
The term intuitive is often listed as a desirable trait in usable interfaces, sometimes used as a synonym for learnable. In the past, Jef Raskin discouraged using this term in user interface design, claiming that easy to use interfaces are often easy because of the user's exposure to previous similar systems, thus the term 'familiar' should be preferred. As an example: Two vertical lines "||" on media player buttons do not intuitively mean "pause"—they do so by convention. This association between intuitive use and familiarity has since been empirically demonstrated in multiple studies by a range of researchers across the world, and intuitive interaction is accepted in the research community as being use of an interface based on past experience with similar interfaces or something else, often not fully conscious, and sometimes involving a feeling of "magic" since the course of the knowledge itself may not be consciously available to the user. Researchers have also investigated intuitive interaction for older people, people living with dementia, and children.

Some have argued that aiming for "intuitive" interfaces (based on reusing existing skills with interaction systems) could lead designers to discard a better design solution only because it would require a novel approach and to stick with boring designs. However, applying familiar features into a new interface has been shown not to result in boring design if designers use creative approaches rather than simple copying. The throwaway remark that "the only intuitive interface is the nipple; everything else is learned." is still occasionally mentioned. Any breastfeeding mother or lactation consultant will tell you this is inaccurate and the nipple does in fact require learning on both sides. In 1992, Bruce Tognazzini even denied the existence of "intuitive" interfaces, since such interfaces must be able to intuit, i.e., "perceive the patterns of the user's behavior and draw inferences." Instead, he advocated the term "intuitable," i.e., "that users could intuit the workings of an application by seeing it and using it". However, the term intuitive interaction has become well accepted in the research community over the past 20 or so years and, although not perfect, it should probably be accepted and used.

ISO/TR 16982:2002 standard
ISO/TR 16982:2002 ("Ergonomics of human-system interaction—Usability methods supporting human-centered design") is an International Standards Organization (ISO) standard that provides information on human-centered usability methods that can be used for design and evaluation. It details the advantages, disadvantages, and other factors relevant to using each usability method. It explains the implications of the stage of the life cycle and the individual project characteristics for the selection of usability methods and provides examples of usability methods in context. The main users of ISO/TR 16982:2002 are project managers. It therefore addresses technical human factors and ergonomics issues only to the extent necessary to allow managers to understand their relevance and importance in the design process as a whole. The guidance in ISO/TR 16982:2002 can be tailored for specific design situations by using the lists of issues characterizing the context of use of the product to be delivered. Selection of appropriate usability methods should also take account of the relevant life-cycle process. ISO/TR 16982:2002 is restricted to methods that are widely used by usability specialists and project managers. It does not specify the details of how to implement or carry out the usability methods described.

ISO 9241 standard
ISO 9241 is a multi-part standard that covers a number of aspects of people working with computers. Although originally titled Ergonomic requirements for office work with visual display terminals (VDTs), it has been retitled to the more generic Ergonomics of Human System Interaction. As part of this change, ISO is renumbering some parts of the standard so that it can cover more topics, e.g. tactile and haptic interaction. The first part to be renumbered was part 10 in 2006, now part 110.

IEC 62366
IEC 62366-1:2015 + COR1:2016 & IEC/TR 62366-2 provide guidance on usability engineering specific to a medical device.

Designing for usability
Any system or device designed for use by people should be easy to use, easy to learn, easy to remember (the instructions), and helpful to users. John Gould and Clayton Lewis recommend that designers striving for usability follow these three design principles

 * Early focus on end users and the tasks they need the system/device to do
 * Empirical measurement using quantitative or qualitative measures
 * Iterative design, in which the designers work in a series of stages, improving the design each time

Early focus on users and tasks
The design team should be user-driven and it should be in direct contact with potential users. Several evaluation methods, including personas, cognitive modeling, inspection, inquiry, prototyping, and testing methods may contribute to understanding potential users and their perceptions of how well the product or process works. Usability considerations, such as who the users are and their experience with similar systems must be examined. As part of understanding users, this knowledge must "...be played against the tasks that the users will be expected to perform." This includes the analysis of what tasks the users will perform, which are most important, and what decisions the users will make while using your system. Designers must understand how cognitive and emotional characteristics of users will relate to a proposed system. One way to stress the importance of these issues in the designers' minds is to use personas, which are made-up representative users. See below for further discussion of personas. Another more expensive but more insightful method is to have a panel of potential users work closely with the design team from the early stages.

Empirical measurement
Test the system early on, and test the system on real users using behavioral measurements. This includes testing the system for both learnability and usability. (See Evaluation Methods). It is important in this stage to use quantitative usability specifications such as time and errors to complete tasks and number of users to test, as well as examine performance and attitudes of the users testing the system. Finally, "reviewing or demonstrating" a system before the user tests it can result in misleading results. The emphasis of empirical measurement is on measurement, both informal and formal, which can be carried out through a variety of evaluation methods.

Iterative design
Iterative design is a design methodology based on a cyclic process of prototyping, testing, analyzing, and refining a product or process. Based on the results of testing the most recent iteration of a design, changes and refinements are made. This process is intended to ultimately improve the quality and functionality of a design. In iterative design, interaction with the designed system is used as a form of research for informing and evolving a project, as successive versions, or iterations of a design are implemented. The key requirements for Iterative Design are: identification of required changes, an ability to make changes, and a willingness to make changes. When a problem is encountered, there is no set method to determine the correct solution. Rather, there are empirical methods that can be used during system development or after the system is delivered, usually a more inopportune time. Ultimately, iterative design works towards meeting goals such as making the system user friendly, easy to use, easy to operate, simple, etc.

Evaluation methods
There are a variety of usability evaluation methods. Certain methods use data from users, while others rely on usability experts. There are usability evaluation methods for all stages of design and development, from product definition to final design modifications. When choosing a method, consider cost, time constraints, and appropriateness. For a brief overview of methods, see Comparison of usability evaluation methods or continue reading below. Usability methods can be further classified into the subcategories below.

Cognitive modeling methods
Cognitive modeling involves creating a computational model to estimate how long it takes people to perform a given task. Models are based on psychological principles and experimental studies to determine times for cognitive processing and motor movements. Cognitive models can be used to improve user interfaces or predict problem errors and pitfalls during the design process. A few examples of cognitive models include:

Parallel design
With parallel design, several people create an initial design from the same set of requirements. Each person works independently, and when finished, shares concepts with the group. The design team considers each solution, and each designer uses the best ideas to further improve their own solution. This process helps generate many different, diverse ideas, and ensures that the best ideas from each design are integrated into the final concept. This process can be repeated several times until the team is satisfied with the final concept.

GOMS
GOMS stands for goals, operator, methods, and selection rules. It is a family of techniques that analyzes the user complexity of interactive systems. Goals are what the user must accomplish. An operator is an action performed in pursuit of a goal. A method is a sequence of operators that accomplish a goal. Selection rules specify which method satisfies a given goal, based on context.

Human processor model
Sometimes it is useful to break a task down and analyze each individual aspect separately. This helps the tester locate specific areas for improvement. To do this, it is necessary to understand how the human brain processes information. A model of the human processor is shown below.

Many studies have been done to estimate the cycle times, decay times, and capacities of each of these processors. Variables that affect these can include subject age, aptitudes, ability, and the surrounding environment. For a younger adult, reasonable estimates are:

Long-term memory is believed to have an infinite capacity and decay time.

Keystroke level modeling
Keystroke level modeling is essentially a less comprehensive version of GOMS that makes simplifying assumptions in order to reduce calculation time and complexity.

Inspection methods
These usability evaluation methods involve observation of users by an experimenter, or the testing and evaluation of a program by an expert reviewer. They provide more quantitative data as tasks can be timed and recorded.

Card sorts
Card sorting is a way to involve users in grouping information for a website's usability review. Participants in a card sorting session are asked to organize the content from a Web site in a way that makes sense to them. Participants review items from a Web site and then group these items into categories. Card sorting helps to learn how users think about the content and how they would organize the information on the Web site. Card sorting helps to build the structure for a Web site, decide what to put on the home page, and label the home page categories. It also helps to ensure that information is organized on the site in a way that is logical to users.

Tree tests
Tree testing is a way to evaluate the effectiveness of a website's top-down organization. Participants are given "find it" tasks, then asked to drill down through successive text lists of topics and subtopics to find a suitable answer. Tree testing evaluates the findability and labeling of topics in a site, separate from its navigation controls or visual design.

Ethnography
Ethnographic analysis is derived from anthropology. Field observations are taken at a site of a possible user, which track the artifacts of work such as Post-It notes, items on desktop, shortcuts, and items in trash bins. These observations also gather the sequence of work and interruptions that determine the user's typical day.

Heuristic evaluation
Heuristic evaluation is a usability engineering method for finding and assessing usability problems in a user interface design as part of an iterative design process. It involves having a small set of evaluators examining the interface and using recognized usability principles (the "heuristics"). It is the most popular of the usability inspection methods, as it is quick, cheap, and easy. Heuristic evaluation was developed to aid in the design of computer user-interface design. It relies on expert reviewers to discover usability problems and then categorize and rate them by a set of principles (heuristics.) It is widely used based on its speed and cost-effectiveness. Jakob Nielsen's list of ten heuristics is the most commonly used in industry. These are ten general principles for user interface design. They are called "heuristics" because they are more in the nature of rules of thumb than specific usability guidelines.
 * Visibility of system status: The system should always keep users informed about what is going on, through appropriate feedback within reasonable time.
 * Match between system and the real world: The system should speak the users' language, with words, phrases and concepts familiar to the user, rather than system-oriented terms. Follow real-world conventions, making information appear in a natural and logical order.
 * User control and freedom: Users often choose system functions by mistake and will need a clearly marked "emergency exit" to leave the unwanted state without having to go through an extended dialogue. Support undo and redo.
 * Consistency and standards: Users should not have to wonder whether different words, situations, or actions mean the same thing. Follow platform conventions.
 * Error prevention: Even better than good error messages is a careful design that prevents a problem from occurring in the first place. Either eliminate error-prone conditions or check for them and present users with a confirmation option before they commit to the action.
 * Recognition rather than recall: Minimize the user's memory load by making objects, actions, and options visible. The user should not have to remember information from one part of the dialogue to another. Instructions for use of the system should be visible or easily retrievable whenever appropriate.
 * Flexibility and efficiency of use: Accelerators—unseen by the novice user—may often speed up the interaction for the expert user such that the system can cater to both inexperienced and experienced users. Allow users to tailor frequent actions.
 * Aesthetic and minimalist design: Dialogues should not contain information that is irrelevant or rarely needed. Every extra unit of information in a dialogue competes with the relevant units of information and diminishes their relative visibility.
 * Help users recognize, diagnose, and recover from errors: Error messages should be expressed in plain language (no codes), precisely indicate the problem, and constructively suggest a solution.
 * Help and documentation: Even though it is better if the system can be used without documentation, it may be necessary to provide help and documentation. Any such information should be easy to search, focused on the user's task, list concrete steps to be carried out, and not be too large.

Thus, by determining which guidelines are violated, the usability of a device can be determined.

Usability inspection
Usability inspection is a review of a system based on a set of guidelines. The review is conducted by a group of experts who are deeply familiar with the concepts of usability in design. The experts focus on a list of areas in design that have been shown to be troublesome for users.

Pluralistic inspection
Pluralistic Inspections are meetings where users, developers, and human factors people meet together to discuss and evaluate step by step of a task scenario. As more people inspect the scenario for problems, the higher the probability to find problems. In addition, the more interaction in the team, the faster the usability issues are resolved.

Consistency inspection
In consistency inspection, expert designers review products or projects to ensure consistency across multiple products to look if it does things in the same way as their own designs.

Activity Analysis
Activity analysis is a usability method used in preliminary stages of development to get a sense of situation. It involves an investigator observing users as they work in the field. Also referred to as user observation, it is useful for specifying user requirements and studying currently used tasks and subtasks. The data collected are qualitative and useful for defining the problem. It should be used when you wish to frame what is needed, or "What do we want to know?"

Inquiry methods
The following usability evaluation methods involve collecting qualitative data from users. Although the data collected is subjective, it provides valuable information on what the user wants.

Task analysis
Task analysis means learning about users' goals and users' ways of working. Task analysis can also mean figuring out what more specific tasks users must do to meet those goals and what steps they must take to accomplish those tasks. Along with user and task analysis, a third analysis is often used: understanding users' environments (physical, social, cultural, and technological environments).

Focus groups
A focus group is a focused discussion where a moderator leads a group of participants through a set of questions on a particular topic. Although typically used as a marketing tool, focus groups are sometimes used to evaluate usability. Used in the product definition stage, a group of 6 to 10 users are gathered to discuss what they desire in a product. An experienced focus group facilitator is hired to guide the discussion to areas of interest for the developers. Focus groups are typically videotaped to help get verbatim quotes, and clips are often used to summarize opinions. The data gathered is not usually quantitative, but can help get an idea of a target group's opinion.

Questionnaires/surveys
Surveys have the advantages of being inexpensive, require no testing equipment, and results reflect the users' opinions. When written carefully and given to actual users who have experience with the product and knowledge of design, surveys provide useful feedback on the strong and weak areas of the usability of a design. This is a very common method and often does not appear to be a survey, but just a warranty card.

Prototyping methods
It is often very difficult for designers to conduct usability tests with the exact system being designed. Cost constraints, size, and design constraints usually lead the designer to creating a prototype of the system. Instead of creating the complete final system, the designer may test different sections of the system, thus making several small models of each component of the system. Prototyping is an attitude and an output, as it is a process for generating and reflecting on tangible ideas by allowing failure to occur early. prototyping helps people to see what could be of communicating a shared vision, and of giving shape to the future. The types of usability prototypes may vary from using paper models, index cards, hand drawn models, or storyboards. Prototypes are able to be modified quickly, often are faster and easier to create with less time invested by designers and are more apt to change design; although sometimes are not an adequate representation of the whole system, are often not durable and testing results may not be parallel to those of the actual system.

The Tool Kit Approach
This tool kit is a wide library of methods that used the traditional programming language and it is primarily developed for computer programmers. The code created for testing in the tool kit approach can be used in the final product. However, to get the highest benefit from the tool, the user must be an expert programmer.

The Parts Kit Approach
The two elements of this approach include a parts library and a method used for identifying the connection between the parts. This approach can be used by almost anyone and it is a great asset for designers with repetitive tasks.

Animation Language Metaphor
This approach is a combination of the tool kit approach and the part kit approach. Both the dialogue designers and the programmers are able to interact with this prototyping tool.

Rapid prototyping
Rapid prototyping is a method used in early stages of development to validate and refine the usability of a system. It can be used to quickly and cheaply evaluate user-interface designs without the need for an expensive working model. This can help remove hesitation to change the design, since it is implemented before any real programming begins. One such method of rapid prototyping is paper prototyping.

Testing methods
These usability evaluation methods involve testing of subjects for the most quantitative data. Usually recorded on video, they provide task completion time and allow for observation of attitude. Regardless to how carefully a system is designed, all theories must be tested using usability tests. Usability tests involve typical users using the system (or product) in a realistic environment [see simulation]. Observation of the user's behavior, emotions, and difficulties while performing different tasks, often identify areas of improvement for the system.

Metrics
While conducting usability tests, designers must use usability metrics to identify what it is they are going to measure, or the usability metrics. These metrics are often variable, and change in conjunction with the scope and goals of the project. The number of subjects being tested can also affect usability metrics, as it is often easier to focus on specific demographics. Qualitative design phases, such as general usability (can the task be accomplished?), and user satisfaction are also typically done with smaller groups of subjects. Using inexpensive prototypes on small user groups provides more detailed information, because of the more interactive atmosphere, and the designer's ability to focus more on the individual user.

As the designs become more complex, the testing must become more formalized. Testing equipment will become more sophisticated and testing metrics become more quantitative. With a more refined prototype, designers often test effectiveness, efficiency, and subjective satisfaction, by asking the user to complete various tasks. These categories are measured by the percent that complete the task, how long it takes to complete the tasks, ratios of success to failure to complete the task, time spent on errors, the number of errors, rating scale of satisfactions, number of times user seems frustrated, etc. Additional observations of the users give designers insight on navigation difficulties, controls, conceptual models, etc. The ultimate goal of analyzing these metrics is to find/create a prototype design that users like and use to successfully perform given tasks. After conducting usability tests, it is important for a designer to record what was observed, in addition to why such behavior occurred and modify the model according to the results. Often it is quite difficult to distinguish the source of the design errors, and what the user did wrong. However, effective usability tests will not generate a solution to the problems, but provide modified design guidelines for continued testing.

Remote usability testing
Remote usability testing (also known as unmoderated or asynchronous usability testing) involves the use of a specially modified online survey, allowing the quantification of user testing studies by providing the ability to generate large sample sizes, or a deep qualitative analysis without the need for dedicated facilities. Additionally, this style of user testing also provides an opportunity to segment feedback by demographic, attitudinal and behavioral type. The tests are carried out in the user's own environment (rather than labs) helping further simulate real-life scenario testing. This approach also provides a vehicle to easily solicit feedback from users in remote areas. There are two types, quantitative or qualitative. Quantitative use large sample sized and task based surveys. These types of studies are useful for validating suspected usability issues. Qualitative studies are best used as exploratory research, in small sample sizes but frequent, even daily iterations. Qualitative usually allows for observing respondent's screens and verbal think aloud commentary (Screen Recording Video, SRV), and for a richer level of insight also include the webcam view of the respondent (Video-in-Video, ViV, sometimes referred to as Picture-in-Picture, PiP)

Remote usability testing for mobile devices
The growth in mobile and associated platforms and services (e.g.: Mobile gaming has experienced 20x growth in 2010–2012) has generated a need for unmoderated remote usability testing on mobile devices, both for websites but especially for app interactions. One methodology consists of shipping cameras and special camera holding fixtures to dedicated testers, and having them record the screens of the mobile smart-phone or tablet device, usually using an HD camera. A drawback of this approach is that the finger movements of the respondent can obscure the view of the screen, in addition to the bias and logistical issues inherent in shipping special hardware to selected respondents. A newer approach uses a wireless projection of the mobile device screen onto the computer desktop screen of the respondent, who can then be recorded through their webcam, and thus a combined Video-in-Video view of the participant and the screen interactions viewed simultaneously while incorporating the verbal think aloud commentary of the respondents.

Thinking aloud
The Think aloud protocol is a method of gathering data that is used in both usability and psychology studies. It involves getting a user to verbalize their thought processes (i.e. expressing their opinions, thoughts, anticipations, and actions) as they perform a task or set of tasks. As a widespread method of usability testing, think aloud provides the researchers with the ability to discover what user really think during task performance and completion.

Often an instructor is present to prompt the user into being more vocal as they work. Similar to the Subjects-in-Tandem method, it is useful in pinpointing problems and is relatively simple to set up. Additionally, it can provide insight into the user's attitude, which can not usually be discerned from a survey or questionnaire.

RITE method
Rapid Iterative Testing and Evaluation (RITE) is an iterative usability method similar to traditional "discount" usability testing. The tester and team must define a target population for testing, schedule participants to come into the lab, decide on how the users behaviors will be measured, construct a test script and have participants engage in a verbal protocol (e.g., think aloud). However it differs from these methods in that it advocates that changes to the user interface are made as soon as a problem is identified and a solution is clear. Sometimes this can occur after observing as few as 1 participant. Once the data for a participant has been collected the usability engineer and team decide if they will be making any changes to the prototype prior to the next participant. The changed interface is then tested with the remaining users.

Subjects-in-tandem or co-discovery
Subjects-in-tandem (also called co-discovery) is the pairing of subjects in a usability test to gather important information on the ease of use of a product. Subjects tend to discuss the tasks they have to accomplish out loud and through these discussions observers learn where the problem areas of a design are. To encourage co-operative problem-solving between the two subjects, and the attendant discussions leading to it, the tests can be designed to make the subjects dependent on each other by assigning them complementary areas of responsibility (e.g. for testing of software, one subject may be put in charge of the mouse and the other of the keyboard.)

Component-based usability testing
Component-based usability testing is an approach which aims to test the usability of elementary units of an interaction system, referred to as interaction components. The approach includes component-specific quantitative measures based on user interaction recorded in log files, and component-based usability questionnaires.

Cognitive walkthrough
Cognitive walkthrough is a method of evaluating the user interaction of a working prototype or final product. It is used to evaluate the system's ease of learning. Cognitive walkthrough is useful to understand the user's thought processes and decision making when interacting with a system, specially for first-time or infrequent users.

Benchmarking
Benchmarking creates standardized test materials for a specific type of design. Four key characteristics are considered when establishing a benchmark: time to do the core task, time to fix errors, time to learn applications, and the functionality of the system. Once there is a benchmark, other designs can be compared to it to determine the usability of the system. Many of the common objectives of usability studies, such as trying to understand user behavior or exploring alternative designs, must be put aside. Unlike many other usability methods or types of labs studies, benchmark studies more closely resemble true experimental psychology lab studies, with greater attention to detail on methodology, study protocol and data analysis.

Meta-analysis
Meta-analysis is a statistical procedure to combine results across studies to integrate the findings. This phrase was coined in 1976 as a quantitative literature review. This type of evaluation is very powerful for determining the usability of a device because it combines multiple studies to provide very accurate quantitative support.

Persona
Personas are fictitious characters created to represent a site or product's different user types and their associated demographics and technographics. Alan Cooper introduced the concept of using personas as a part of interactive design in 1998 in his book The Inmates Are Running the Asylum, but had used this concept since as early as 1975. Personas are a usability evaluation method that can be used at various design stages. The most typical time to create personas is at the beginning of designing so that designers have a tangible idea of who the users of their product will be. Personas are the archetypes that represent actual groups of users and their needs, which can be a general description of person, context, or usage scenario. This technique turns marketing data on target user population into a few physical concepts of users to create empathy among the design team, with the final aim of tailoring a product more closely to how the personas will use it. To gather the marketing data that personas require, several tools can be used, including online surveys, web analytics, customer feedback forms, and usability tests, and interviews with customer-service representatives.

Benefits
The key benefits of usability are:
 * Higher revenues through increased sales
 * Increased user efficiency and user satisfaction
 * Reduced development costs
 * Reduced support costs

Corporate integration
An increase in usability generally positively affects several facets of a company's output quality. In particular, the benefits fall into several common areas:
 * Increased productivity
 * Decreased training and support costs
 * Increased sales and revenues
 * Reduced development time and costs
 * Reduced maintenance costs
 * Increased customer satisfaction

Increased usability in the workplace fosters several responses from employees: "Workers who enjoy their work do it better, stay longer in the face of temptation, and contribute ideas and enthusiasm to the evolution of enhanced productivity." To create standards, companies often implement experimental design techniques that create baseline levels. Areas of concern in an office environment include (though are not necessarily limited to):
 * Working posture
 * Design of workstation furniture
 * Screen displays
 * Input devices
 * Organization issues
 * Office environment
 * Software interface

By working to improve said factors, corporations can achieve their goals of increased output at lower costs, while potentially creating optimal levels of customer satisfaction. There are numerous reasons why each of these factors correlates to overall improvement. For example, making software user interfaces easier to understand reduces the need for extensive training. The improved interface tends to lower the time needed to perform tasks, and so would both raise the productivity levels for employees and reduce development time (and thus costs). Each of the aforementioned factors are not mutually exclusive; rather they should be understood to work in conjunction to form the overall workplace environment. In the 2010s, usability is recognized as an important software quality attribute, earning its place among more traditional attributes such as performance, robustness and aesthetic appearance. Various academic programs focus on usability. Several usability consultancy companies have emerged, and traditional consultancy and design firms offer similar services.

There is some resistance to integrating usability work in organisations. Usability is seen as a vague concept, it is difficult to measure and other areas are prioritised when IT projects run out of time or money.

Professional development
Usability practitioners are sometimes trained as industrial engineers, psychologists, kinesiologists, systems design engineers, or with a degree in information architecture, information or library science, or Human-Computer Interaction (HCI). More often though they are people who are trained in specific applied fields who have taken on a usability focus within their organization. Anyone who aims to make tools easier to use and more effective for their desired function within the context of work or everyday living can benefit from studying usability principles and guidelines. For those seeking to extend their training, the User Experience Professionals' Association offers online resources, reference lists, courses, conferences, and local chapter meetings. The UXPA also sponsors World Usability Day each November. Related professional organizations include the Human Factors and Ergonomics Society (HFES) and the Association for Computing Machinery's special interest groups in Computer Human Interaction (SIGCHI), Design of Communication (SIGDOC) and Computer Graphics and Interactive Techniques (SIGGRAPH). The Society for Technical Communication also has a special interest group on Usability and User Experience (UUX). They publish a quarterly newsletter called Usability Interface.

title: Usability


text:
User talk:Filet

Hi, I believe whilst is still used in the U.K. That's not a criticism, just an observation. There's been some discussion about whether to use British or U.S. English, but I don't remember where. I believe the consensus was to try to use whatever the article was started in so as to avoid ruffling feathers.

Anyway, welcome to wikipedia. :-) Koyaanis Qatsi 00:49, 5 Aug 2003 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Filet


text:
Red Queen

Red Queen may refer to:

Literature

 * Red Queen (Through the Looking-Glass), a character in Lewis Carroll's Through the Looking-Glass (1871)
 * "The Red Queen's Race", a 1949 short story by Isaac Asimov
 * The Red Queen: Sex and the Evolution of Human Nature, a popular 1993 science book by Matt Ridley
 * The Red Queen (Drabble novel), a 2004 novel by Margaret Drabble
 * Queen Redd, the main antagonist of Frank Beddor's 2004 The Looking-Glass Wars series, based on the above Lewis Carroll character
 * The Red Queen (Gregory novel), a 2010 historical novel by Philippa Gregory
 * The Red Queen (Obernewtyn Chronicles), a 2012 novel by Isobelle Carmody
 * Red Queen (novel), a 2015 novel by Victoria Aveyard

Entertainment

 * Red Queen (comics), a number of comics characters
 * Red Queen to Gryphon Three, a 1974 concept album by the band Gryphon based on the game of chess
 * Red Queen, an artificial intelligence, a character in the 2002 Resident Evil films
 * The sword wielded by Nero in the 2008 Devil May Cry series
 * Sana Kashimura, otherwise called the Red Queen, a character in the 2012 manga and anime series Alice & Zouroku
 * "Red Queen" (Gotham), a 2016 episode of the third season of Gotham
 * Red Queen (TV series), a 2024 Spanish–Mexican thriller television series on Amazon Prime Video

Other

 * A red Queen (playing card)
 * Elisabeth of Bavaria, Queen of the Belgians, nicknamed "The Red Queen"
 * Red Queen's Hypothesis, an evolutionary hypothesis to the advantage of sex at the level of individuals, and the constant evolutionary arms race between competing species
 * Red Queen (EP), a 2003 album by Funker Vogt
 * The Red Queen's race, a parable in Alice in Wonderland that has served as the basis of a number of scientific and sociological theories
 * The Tomb of the Red Queen, an unidentified Mayan noblewoman
 * Red Queen (anti-aircraft gun), a British Army experimental anti-aircraft gun

title: Red Queen


text: Talk:Worcester Polytechnic Institute

title: Talk:Worcester Polytechnic Institute


text:
Astral projection

Astral projection (also known as astral travel, soul journey, soul wandering, spiritual journey, spiritual travel) is a term used in esotericism to describe an intentional out-of-body experience (OBE) that assumes the existence of a subtle body, known as the astral body or body of light, through which consciousness can function separately from the physical body and travel throughout the astral plane.

The idea of astral travel is ancient and occurs in multiple cultures. The term "astral projection" was coined and promoted by 19th-century Theosophists. It is sometimes associated with dreams and forms of meditation. Some individuals have reported perceptions similar to descriptions of astral projection that were induced through various hallucinogenic and hypnotic means (including self-hypnosis). There is no scientific evidence that there is a consciousness whose embodied functions are separate from normal neural activity or that one can consciously leave the body and make observations of the physical universe, and astral projection has been characterized as a pseudoscience.

Ancient Egyptian
Similar concepts of soul travel appear in various other religious traditions. For example, ancient Egyptian teachings present the soul (ba) as having the ability to hover outside the physical body via the ka, or subtle body.

Amazon
The yaskomo of the Waiwai is believed to have the ability to perform a soul flight that can serve several functions such as healing, flying to the sky to consult cosmological beings (the moon or the brother of the moon) to obtain a name for a newborn baby, flying to the cave of peccaries' mountains to ask the father of peccaries for abundance of game, or flying deep down into a river to seek the aid of other beings.

Inuit
In some Inuit groups, individuals with special capabilities, known as angakkuq, are said to be able to travel to (mythological) remote places, and report their experiences and important matters back to their community. Those abilities would be unavailable to individuals with normal capabilities. Among other things, an angakkuq was said to have the ability to stop bad hunting luck or heal a sick person.

Hindu
Similar ideas such as the Liṅga Śarīra are found in ancient Hindu scriptures such as the Yogavashishta-Maharamayana of Valmiki. Modern Indians who have vouched for astral projection include Paramahansa Yogananda who witnessed Swami Pranabananda doing a miracle through a possible astral projection.

The Indian spiritual teacher Meher Baba described one's use of astral projection:

"In the advancing stages leading to the beginning of the path, the aspirant becomes spiritually prepared for being entrusted with free use of the forces of the inner world of the astral bodies. He may then undertake astral journeys in his astral body, leaving the physical body in sleep or wakefulness. The astral journeys that are taken unconsciously are much less important than those undertaken with full consciousness and as a result of deliberate volition. This implies conscious use of the astral body. Conscious separation of the astral body from the outer vehicle of the gross body has its own value in making the soul feel its distinction from the gross body and in arriving at fuller control of the gross body. One can, at will, put on and take off the external gross body as if it were a cloak and use the astral body for experiencing the inner world of the astral and for undertaking journeys through it, if and when necessary. ... The ability to undertake astral journeys therefore involves considerable expansion of one's scope for experience. It brings opportunities for promoting one's own spiritual advancement, which begins with the involution of consciousness."

Astral projection is one of the siddhis ('magical powers') considered achievable by yoga practitioners through self-disciplined practice. In the epic Mahabharata, Drona leaves his physical body to see if his son is alive.

Japanese
In Japanese mythology, an ikiryō (生霊) is a manifestation of the soul of a living person separately from their body. Traditionally, if someone holds a sufficient grudge against another person, it is believed that a part or the whole of their soul can temporarily leave their body and appear before the target of their hate in order to curse or otherwise harm them, similar to an evil eye. Souls are also believed to leave a living body when the body is extremely sick or comatose; but such ikiryō are not malevolent.

Taoist
Taoist alchemical practice involves creation of an energy body by breathing meditations, drawing energy into a 'pearl' that is then circulated.

"Xiangzi ... with a drum as his pillow fell fast asleep, snoring and motionless. His primordial spirit, however, went straight into the banquet room and said, 'My lords, here I am again.' When Tuizhi walked with the officials to take a look, there really was a Taoist sleeping on the ground and snoring like thunder. Yet inside, in the side room, there was another Taoist beating a fisher drum and singing Taoist songs. The officials all said, 'Although there are two different people, their faces and clothes are exactly alike. Clearly he is a divine immortal who can divide his body and appear in several places at once. ...' At that moment, the Taoist in the side room came walking out, and the Taoist sleeping on the ground woke up. The two merged into one."

Judaic and Christian
Carrington, Muldoon, Peterson, and Williams say that the subtle body is attached to the physical body by means of a psychic silver cord. The final chapter of the Book of Ecclesiastes is often cited in this respect: "Before the silver cord be loosed, or the golden bowl be broken, or the pitcher be shattered at the fountain, or the wheel be broken at the cistern." Rabbi Nosson Scherman, however, contends that the context points to this being merely a metaphor, comparing the body to a machine, with the silver cord referring to the spine.

James Hankins argues that Paul's Second Epistle to the Corinthians refers to the astral planes: "I know a man in Christ who fourteen years ago was caught up to the third heaven. Whether it was in the body or out of the body I do not know—God knows."

Western esoteric
According to the classical, medieval, renaissance Hermeticism, Neoplatonism, and later Theosophist and Rosicrucian thought, the 'astral body' is an intermediate body of light linking the rational soul to the physical body while the astral plane is an intermediate world of light between Heaven and Earth, composed of the spheres of the planets and stars. These astral spheres were held to be populated by angels, demons, and spirits.

In the Neoplatonism of Plotinus, for example, the individual is a microcosm ("small world") of the universe (the macrocosm or "great world"). "The rational soul...is akin to the great Soul of the World" while "the material universe, like the body, is made as a faded image of the Intelligible". Each succeeding plane of manifestation is causal to the next, a world-view known as emanationism; "from the One proceeds Intellect, from Intellect Soul, and from Soul—in its lower phase, or that of Nature—the material universe". The idea of the astral figured prominently in the work of the nineteenth-century French occultist Eliphas Levi, whence it was adopted and developed further by Theosophy, and used afterwards by other esoteric movements.

The subtle bodies, and their associated planes of existence, form an essential part of some esoteric systems that deal with astral phenomena. Often these bodies and their planes of existence are depicted as a series of concentric circles or nested spheres, with a separate body traversing each realm.

Terminology
The expression "astral projection" came to be used in two different ways. For the Hermetic Order of the Golden Dawn and some Theosophists, it retained the classical and medieval philosophers' meaning of journeying to other worlds, heavens, hells, the astrological spheres and other landscapes in the body of light; but outside these circles the term was increasingly applied to non-physical travel around the physical world.

Though this usage continues to be widespread, the term, "etheric travel", used by some later Theosophists, offers a useful distinction. Some experimenters say they visit different times and/or places: etheric, then, is used to represent the sense of being out of the body in the physical world; whereas astral may connote some alteration in time-perception. Robert Monroe describes the former type of projection as "Locale I" or the "Here-Now", involving people and places that exist: Robert Bruce calls it the "Real Time Zone" (RTZ) and describes it as the non-physical dimension-level closest to the physical. This etheric body is usually, though not always, invisible but is often perceived by the experient as connected to the physical body during separation by a silver cord. Some link falling dreams with projection.

According to Max Heindel, the etheric double serves as a medium between the astral and physical realms. In his system the ether, also called prana, is the vital force that empowers the physical forms to change. From his descriptions it can be inferred that, to him, when one views the physical during an out-of-body experience, one is not technically in the astral realm at all.

Other experiments may describe a domain that has no parallel to any known physical setting. Environments may be populated or unpopulated, artificial, natural or abstract, and the experience may be beatific, horrific or neutral. A common Theosophical belief is that one may access a compendium of mystical knowledge called the Akashic records. In many accounts the experiencer correlates the astral world with the world of dreams. Some even report seeing other dreamers enacting dream scenarios unaware of their wider environment.

The astral environment may also be divided into levels or sub-planes by theorists, but there are many different views in various traditions concerning the overall structure of the astral planes: they may include heavens and hells and other after-death spheres, transcendent environments, or other less-easily characterized states.

Scientific reception
There is no known scientific evidence that astral projection as an objective phenomenon exists, although there are cases of patients having experiences suggestive of astral projection from brain stimulation treatments and hallucinogenic drugs, such as ketamine, phencyclidine, and DMT. Subjects in parapsychological experiments have attempted to project their astral bodies to distant rooms and see what was happening. However, such experiments have not produced clear results.

Psychologist Donovan Rawcliffe wrote that astral projection can be explained by delusion, hallucination, and vivid dreams. Arthur W. Wiggins wrote that purported evidence of the ability to astrally travel great distances and give descriptions of places visited is predominantly anecdotal and considers astral travel an illusion. He looks to neuroanatomy, prior knowledge, and human belief and imagination to provide prosaic explanations for those who experience it. Robert Todd Carroll writes that the main evidence to support claims of astral travel is anecdotal and comes "in the form of testimonials of those who claim to have experienced being out of their bodies when they may have been out of their minds."

Notable practitioners
Emanuel Swedenborg was one of the first practitioners to write extensively about the out-of-body experience, in his Spiritual Diary (1747–1765). In her book, My Religion, Helen Keller tells of her beliefs in Swedenborgianism and how she once traveled astrally to Athens:

"I have been far away all this time, and I haven't left the room...It was clear to me that it was because I was a spirit that I had so vividly 'seen' and felt a place a thousand miles away. Space was nothing to spirit!"

In occult traditions, practices range from inducing trance states to the mental construction of a second body, called the "body of light" by Aleister Crowley (1875–1947), through visualization and controlled breathing, followed by the transfer of consciousness to the secondary body by a mental act of will.

There are many 20th-century publications on astral projection, although only a few writers continue to be cited. These include Edgar Cayce (1877–1945), Hereward Carrington (1880–1958), Oliver Fox (1885–1949), Sylvan Muldoon (1903–1969), and Robert Monroe (1915–1995).

Robert Monroe's accounts of journeys to other realms (1971–1994) popularized the term "OBE" and were translated into a large number of languages. Though his books themselves only placed secondary importance on descriptions of method, Monroe also founded an institute dedicated to research, exploration and non-profit dissemination of auditory technology for assisting others in achieving projection and related altered states of consciousness.

Carlos Castaneda (1925–1998) discusses his teacher Don Juan's beliefs about "the double" and its abilities in his books Tales of Power (1974), The Second Ring of Power (1977), and The Art of Dreaming (1993). Florinda Donner, a student of Castaneda, further describes methods of using the double to access the physical world while dreaming and access the dream world while in a waking dream state in her 1992 book, Being-in-Dreaming.

Michael Crichton (1942–2008) gives lengthy and detailed explanations and experience of astral projection in his 1988 non-fiction book Travels. Robert Bruce, William Buhlman, Marilynn Hughes, and Albert Taylor have discussed their theories and findings on the syndicated show Coast to Coast AM several times.

title: Astral projection


text:
Talk:Atari 5200

Generation
I think it's really time to fix this longstanding error that's only becoming increasingly entrenched as it's allowed to linger. I've posted on the ColecoVision talk page as well because there was considerable if not universal consensus that the Third Generation (often described as "Third Wave" back then) started with it. And the 5200 launched two or three months later. Moving the 5200, ColecoVision, and Vectrex to Third Generation solves every problem and is consistent with the, frankly, robust primary sources from the era.

https://i357.photobucket.com/albums/oo12/Alison123456789/Third%20Gen/Gamermagazine1983_zpsfbc8c06c.jpg

https://i357.photobucket.com/albums/oo12/Alison123456789/Third%20Gen/TVGamer1983_zps1b9e74ad.jpg

https://i357.photobucket.com/albums/oo12/Alison123456789/Third%20Gen/VideoGamesmagazineJan1983_zps6c3c77eb.jpg

https://i357.photobucket.com/albums/oo12/Alison123456789/Third%20Gen/Colecothirdgen_zps6cfdc993.jpg

https://i357.photobucket.com/albums/oo12/Alison123456789/Third%20Gen/TelespieleReport84CBSColeco_zpsdc2e4001.jpg

https://i357.photobucket.com/albums/oo12/Alison123456789/Third%20Gen/videogamesplayerfall82_zps24b6db23.jpg

https://imgur.com/a/E86Lh (Atari 5200 and ColecoVision described as "third wave")

Wikipedia currently isn't even consistent with itself. The ColecoVision and SG-1000 are for all intents and purposes the same machine released 11 months of each other yet the SG-1000 is in the Third Generation. The 5200's placement is logically inconsistent since it's effectively Atari's Wii U. Everyone is in agreement that the Wii U was "next gen" to the Wii and the Switch was a mid-cycle replacement. Nobody disputes this. We don't retroactively group the Wii U with it's predecessor because it flopped. So why are we doing it to the 5200? The discrepancy of all this is irreconcilable. Bumping the ColecoVision, 5200, and Vectrex to Third Generation cures every problem. 47.16.172.103 (talk) 15:49, 3 January 2024 (UTC)

title: Talk:Atari 5200


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User:RickHeineman

Welcome. Wikipedia is cool, isn't it? I made a few edits to your Red Queen article: our style is to put the subject in the first sentence, and you don't need to keep linking the same term. Vicki Rosenzweig 00:57, 5 Aug 2003 (UTC)

title: User:RickHeineman


text:
André Sainte-Laguë

André Sainte-Laguë (, 20 April 1882, Saint-Martin-Curton – 18 January 1950) was a French mathematician who was a pioneer in the area of graph theory. His research on seat allocation methods (published in 1910) led to one being named after him, the Sainte-Laguë method. Also named after him is the Sainte-Laguë Index for measuring the proportionality of an electoral outcome.

He is also notable for his informal calculation that supposedly demonstrated that a bumblebee could not fly, referred to in the introduction of 'Le Vol des Insectes' (Hermann and Cle, Paris, 1934) by the entomologist Antoine Magnan. This casual calculation was based on a comparison between an aeroplane and a bee, making the wrong assumption that bees' wings were smooth and flat. He, and others, soon corrected this assumption, but the story of the scientist who demonstrated that bee flight was impossible persists to this day.

He published several popular math texts, including "From the known to the unknown" (foreword by biologist Jean Rostand) which has been translated into several languages.

Biography
Born in Casteljaloux (Lot-et-Garonne) in 1882, Sainte-Lague was admitted at once, at the age of 20 years at the Ecole Polytechnique and Ecole Normale Superieure. He chose the latter and became a professor in the provinces, then in Paris. During World War I, having been wounded three times, he was attached to the Department of Inventions of the Normal School from 1917 to 1919, studied long-range artillery shells, and thereafter, the flight of birds and matters relating to aviation (theory test fish).

After the First World War, as a professor in the schools of Paris, he became a lecturer in mathematics at the Conservatoire National des Arts et Metiers. Then he received in 1938, the Chair of Applied Mathematics. He trained generations of engineers and technicians. He was the organizer and host of the Mathematics Section of the Palace of Discovery, where his encyclopedic mind is still present.

Besides his academic career, he led a life of an activist, especially the Confederation of Intellectual Workers, of which he was president in 1929. From the earliest days of the occupation, he took an important part in the resistance and was even imprisoned for a while. He resumed his duties at the Conservatoire National des Arts et Metiers after the Liberation and had a growing number of students. At his death he was teaching three courses totaling two thousand five hundred students.

Officer of the Legion of Honour, Croix de Guerre and Medal of the Resistance, a professor at the School of Special Public Works, chairman of the International Confederation Intellectual of Workers, Vice-President of the Confederation of the Middle Class, former president of the Society of Fellows, former vice-president of National Economic Council, former member of the General Council of the Banque de France, former Deputy Provisional Consultative Assembly.(...)

His sudden death came at the very moment he had just accepted the chairmanship of the Committee of the League of Friends of the Psychic Institute, where he was vice president in 1949 and member since 1934. "(R. Warcollier, Vice- President of IMI, January–February–March 1950)

Honours

 * National Order of the Legion of Honour

 * Croix de guerre 1914-1918

 * Resistance Medal

title: André Sainte-Laguë


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Ricci flow

In the mathematical fields of differential geometry and geometric analysis, the Ricci flow, sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.

The Ricci flow, so named for the presence of the Ricci tensor in its definition, was introduced by Richard Hamilton, who used it through the 1980s to prove striking new results in Riemannian geometry. Later extensions of Hamilton's methods by various authors resulted in new applications to geometry, including the resolution of the differentiable sphere conjecture by Simon Brendle and Richard Schoen.

Following the possibility that the singularities of solutions of the Ricci flow could identify the topological data predicted by William Thurston's geometrization conjecture, Hamilton produced a number of results in the 1990s which were directed towards the conjecture's resolution. In 2002 and 2003, Grigori Perelman presented a number of fundamental new results about the Ricci flow, including a novel variant of some technical aspects of Hamilton's program. Perelman's work is now widely regarded as forming the proof of the Thurston conjecture and the Poincaré conjecture, regarded as a special case of the former. It should be emphasized that the Poincare conjecture has been a well-known open problem in the field of geometric topology since 1904. These results by Hamilton and Perelman are considered as a milestone in the fields of geometry and topology.

Mathematical definition
On a smooth manifold $M$, a smooth Riemannian metric $g$ automatically determines the Ricci tensor $Ric^{g}$. For each element $p$ of $M$, by definition $g_{p}$ is a positive-definite inner product on the tangent space $T_{p}M$ at $p$. If given a one-parameter family of Riemannian metrics $g_{t}$, one may then consider the derivative $∂⁄∂t g_{t}$, which then assigns to each particular value of $t$ and $p$ a symmetric bilinear form on $T_{p}M$. Since the Ricci tensor of a Riemannian metric also assigns to each $p$ a symmetric bilinear form on $T_{p}M$, the following definition is meaningful. The Ricci tensor is often thought of as an average value of the sectional curvatures, or as an algebraic trace of the Riemann curvature tensor. However, for the analysis of existence and uniqueness of Ricci flows, it is extremely significant that the Ricci tensor can be defined, in local coordinates, by a formula involving the first and second derivatives of the metric tensor. This makes the Ricci flow into a geometrically-defined partial differential equation. The analysis of the ellipticity of the local coordinate formula provides the foundation for the existence of Ricci flows; see the following section for the corresponding result.
 * Given a smooth manifold $M$ and an open real interval $(a, b)$, a Ricci flow assigns, to each $t$ in the interval $(a,b)$, a Riemannian metric $g_{t}$ on $M$ such that $∂⁄∂t g_{t} = −2 Ric^{g_{t}}|undefined$.

Let $k$ be a nonzero number. Given a Ricci flow $g_{t}$ on an interval $(a,b)$, consider $G_{t} = g_{kt}$ for $t$ between $a⁄k$ and $b⁄k$. Then $∂⁄∂t G_{t} = −2k Ric^{G_{t}}|undefined$. So, with this very trivial change of parameters, the number −2 appearing in the definition of the Ricci flow could be replaced by any other nonzero number. For this reason, the use of −2 can be regarded as an arbitrary convention, albeit one which essentially every paper and exposition on Ricci flow follows. The only significant difference is that if −2 were replaced by a positive number, then the existence theorem discussed in the following section would become a theorem which produces a Ricci flow that moves backwards (rather than forwards) in parameter values from initial data.

The parameter $t$ is usually called, although this is only as part of standard informal terminology in the mathematical field of partial differential equations. It is not physically meaningful terminology. In fact, in the standard quantum field theoretic interpretation of the Ricci flow in terms of the renormalization group, the parameter $t$ corresponds to length or energy, rather than time.

Normalized Ricci flow
Suppose that $M$ is a compact smooth manifold, and let $g_{t}$ be a Ricci flow for $t$ in the interval $(a, b)$. Define $Ψ:(a, b) → (0, ∞)$ so that each of the Riemannian metrics $Ψ(t)g_{t}$ has volume 1; this is possible since $M$ is compact. (More generally, it would be possible if each Riemannian metric $g_{t}$ had finite volume.) Then define $F:(a, b) → (0, ∞)$ to be the antiderivative of $Ψ$ which vanishes at $a$. Since $Ψ$ is positive-valued, $F$ is a bijection onto its image $(0, S)$. Now the Riemannian metrics $G_{s} = Ψ(F^{ −1}(s))g_{F^{ −1}(s)}$, defined for parameters $s ∈ (0, S)$, satisfy $$\frac{\partial}{\partial s} G_s = -2\operatorname{Ric}^{G_s} +\frac{2}{n} \frac{\int_M R^{G_s}\,d\mu_{G_s}}{\int_M d\mu_{G_s}} G_s.$$ Here $R$ denotes scalar curvature. This is called the normalized Ricci flow equation. Thus, with an explicitly defined change of scale $Ψ$ and a reparametrization of the parameter values, a Ricci flow can be converted into a normalized Ricci flow. The converse also holds, by reversing the above calculations.

The primary reason for considering the normalized Ricci flow is that it allows a convenient statement of the major convergence theorems for Ricci flow. However, it is not essential to do so, and for virtually all purposes it suffices to consider Ricci flow in its standard form. Moreover, the normalized Ricci flow is not generally meaningful on noncompact manifolds.

Existence and uniqueness
Let $$M$$ be a smooth closed manifold, and let $$g_0$$ be any smooth Riemannian metric on $$M$$. Making use of the Nash–Moser implicit function theorem, showed the following existence theorem: He showed the following uniqueness theorem: The existence theorem provides a one-parameter family of smooth Riemannian metrics. In fact, any such one-parameter family also depends smoothly on the parameter. Precisely, this says that relative to any smooth coordinate chart $$(U,\phi)$$ on $$M$$, the function $$g_{ij}:U\times(0,T)\to\mathbb{R}$$ is smooth for any $$i,j=1,\dots,n$$.
 * There exists a positive number $$T$$ and a Ricci flow $$g_t$$ parametrized by $$t\in(0,T)$$ such that $$g_t$$ converges to $$g_0$$ in the $$C^\infty$$ topology as $$t$$ decreases to 0.
 * If $$\{g_t:t\in(0,T)\}$$ and $$\{\widetilde{g}_t:t\in(0,\widetilde{T})\}$$ are two Ricci flows as in the above existence theorem, then $$g_t=\widetilde{g}_t$$ for all $$t\in(0,\min\{T,\widetilde{T}\}).$$

Dennis DeTurck subsequently gave a proof of the above results which uses the Banach implicit function theorem instead. His work is essentially a simpler Riemannian version of Yvonne Choquet-Bruhat's well-known proof and interpretation of well-posedness for the Einstein equations in Lorentzian geometry.

As a consequence of Hamilton's existence and uniqueness theorem, when given the data $$(M,g_0)$$, one may speak unambiguously of the Ricci flow on $$M$$ with initial data $$g_0$$, and one may select $$T$$ to take on its maximal possible value, which could be infinite. The principle behind virtually all major applications of Ricci flow, in particular in the proof of the Poincaré conjecture and geometrization conjecture, is that, as $$t$$ approaches this maximal value, the behavior of the metrics $$g_t$$ can reveal and reflect deep information about $$M$$.

Convergence theorems
Complete expositions of the following convergence theorems are given in and. "Let $(M, g_{0})$ be a smooth closed Riemannian manifold. Under any of the following three conditions: the normalized Ricci flow with initial data $g_{0}$ exists for all positive time and converges smoothly, as $M$ goes to infinity, to a metric of constant curvature."
 * $M$ is two-dimensional
 * $M$ is three-dimensional and $(M, g_{0}) × ℝ$ has positive Ricci curvature
 * $t$ has dimension greater than three and the product metric on $g_{0}$ has positive isotropic curvature

The three-dimensional result is due to. Hamilton's proof, inspired by and loosely modeled upon James Eells and Joseph Sampson's epochal 1964 paper on convergence of the harmonic map heat flow, included many novel features, such as an extension of the maximum principle to the setting of symmetric 2-tensors. His paper (together with that of Eells−Sampson) is among the most widely cited in the field of differential geometry. There is an exposition of his result in.

In terms of the proof, the two-dimensional case is properly viewed as a collection of three different results, one for each of the cases in which the Euler characteristic of $M$ is positive, zero, or negative. As demonstrated by, the negative case is handled by the maximum principle, while the zero case is handled by integral estimates; the positive case is more subtle, and Hamilton dealt with the subcase in which $g_{0}$ has positive curvature by combining a straightforward adaptation of Peter Li and Shing-Tung Yau's gradient estimate to the Ricci flow together with an innovative "entropy estimate". The full positive case was demonstrated by Bennett, in an extension of Hamilton's techniques. Since any Ricci flow on a two-dimensional manifold is confined to a single conformal class, it can be recast as a partial differential equation for a scalar function on the fixed Riemannian manifold $(M, g_{0})$. As such, the Ricci flow in this setting can also be studied by purely analytic methods; correspondingly, there are alternative non-geometric proofs of the two-dimensional convergence theorem.

The higher-dimensional case has a longer history. Soon after Hamilton's breakthrough result, Gerhard Huisken extended his methods to higher dimensions, showing that if $g_{0}$ almost has constant positive curvature (in the sense of smallness of certain components of the Ricci decomposition), then the normalized Ricci flow converges smoothly to constant curvature. found a novel formulation of the maximum principle in terms of trapping by convex sets, which led to a general criterion relating convergence of the Ricci flow of positively curved metrics to the existence of "pinching sets" for a certain multidimensional ordinary differential equation. As a consequence, he was able to settle the case in which $M$ is four-dimensional and $g_{0}$ has positive curvature operator. Twenty years later, Christoph Böhm and Burkhard Wilking found a new algebraic method of constructing "pinching sets", thereby removing the assumption of four-dimensionality from Hamilton's result. Simon Brendle and Richard Schoen showed that positivity of the isotropic curvature is preserved by the Ricci flow on a closed manifold; by applying Böhm and Wilking's method, they were able to derive a new Ricci flow convergence theorem. Their convergence theorem included as a special case the resolution of the differentiable sphere theorem, which at the time had been a long-standing conjecture. The convergence theorem given above is due to, which subsumes the earlier higher-dimensional convergence results of Huisken, Hamilton, Böhm & Wilking, and Brendle & Schoen.

Corollaries
The results in dimensions three and higher show that any smooth closed manifold $M$ which admits a metric $g_{0}$ of the given type must be a space form of positive curvature. Since these space forms are largely understood by work of Élie Cartan and others, one may draw corollaries such as So if one could show directly that any smooth closed simply-connected 3-dimensional manifold admits a smooth Riemannian metric of positive Ricci curvature, then the Poincaré conjecture would immediately follow. However, as matters are understood at present, this result is only known as a (trivial) corollary of the Poincaré conjecture, rather than vice versa.
 * Suppose that $M$ is a smooth closed 3-dimensional manifold which admits a smooth Riemannian metric of positive Ricci curvature. If $M$ is simply-connected then it must be diffeomorphic to the 3-sphere.

Possible extensions
Given any $n$ larger than two, there exist many closed $n$-dimensional smooth manifolds which do not have any smooth Riemannian metrics of constant curvature. So one cannot hope to be able to simply drop the curvature conditions from the above convergence theorems. It could be possible to replace the curvature conditions by some alternatives, but the existence of compact manifolds such as complex projective space, which has a metric of nonnegative curvature operator (the Fubini-Study metric) but no metric of constant curvature, makes it unclear how much these conditions could be pushed. Likewise, the possibility of formulating analogous convergence results for negatively curved Riemannian metrics is complicated by the existence of closed Riemannian manifolds whose curvature is arbitrarily close to constant and yet admit no metrics of constant curvature.

Li–Yau inequalities
Making use of a technique pioneered by Peter Li and Shing-Tung Yau for parabolic differential equations on Riemannian manifolds, proved the following "Li–Yau inequality". showed the following alternative Li–Yau inequality. Both of these remarkable inequalities are of profound importance for the proof of the Poincaré conjecture and geometrization conjecture. The terms on the right hand side of Perelman's Li–Yau inequality motivates the definition of his "reduced length" functional, the analysis of which leads to his "noncollapsing theorem". The noncollapsing theorem allows application of Hamilton's compactness theorem (Hamilton 1995) to construct "singularity models", which are Ricci flows on new three-dimensional manifolds. Owing to the Hamilton–Ivey estimate, these new Ricci flows have nonnegative curvature. Hamilton's Li–Yau inequality can then be applied to see that the scalar curvature is, at each point, a nondecreasing (nonnegative) function of time. This is a powerful result that allows many further arguments to go through. In the end, Perelman shows that any of his singularity models is asymptotically like a complete gradient shrinking Ricci soliton, which are completely classified; see the previous section.
 * Let $$M$$ be a smooth manifold, and let $$g_t$$ be a solution of the Ricci flow with $$t\in(0,T)$$ such that each $$g_t$$ is complete with bounded curvature. Furthermore, suppose that each $$g_t$$ has nonnegative curvature operator. Then, for any curve $$\gamma:[t_1,t_2]\to M$$ with $$[t_1,t_2]\subset (0,T)$$, one has $$ \frac{d}{dt} \big(R^{g(t)}(\gamma(t))\big)+\frac{R^{g(t)}(\gamma(t))}{t}+\frac{1}{2}\operatorname{Ric}^{g(t)}(\gamma'(t),\gamma'(t))\geq 0.$$
 * Let $$M$$ be a smooth closed $$n$$-manifold, and let $$g_t$$ be a solution of the Ricci flow. Consider the backwards heat equation for $$n$$-forms, i.e. $$\tfrac{\partial}{\partial t}\omega + \Delta^{g(t)}\omega=0$$; given $$p\in M$$ and $$t_0\in(0,T)$$, consider the particular solution which, upon integration, converges weakly to the Dirac delta measure as $$t$$ increases to $$t_0$$. Then, for any curve $$\gamma:[t_1,t_2]\to M$$ with $$[t_1,t_2]\subset (0,T)$$, one has $$ \frac{d}{dt} \big(f(\gamma(t),t)\big) + \frac{f\big(\gamma(t),t\big)}{2(t_0-t)} \leq \frac{R^{g(t)}(\gamma(t))+ |\gamma'(t)|_{g(t)}^2}{2}.$$ where $$\omega=(4\pi(t_0-t))^{-n/2}e^{-f}\text{d}\mu_{g(t)}$$.

See for details on Hamilton's Li–Yau inequality; the books  and  contain expositions of both inequalities above.

Constant-curvature and Einstein metrics
Let $$(M,g)$$ be a Riemannian manifold which is Einstein, meaning that there is a number $$\lambda$$ such that $$\text{Ric}^g=\lambda g$$. Then $$g_t=(1-2\lambda t)g$$ is a Ricci flow with $$g_0=g$$, since then
 * $$\frac{\partial}{\partial t}g_t=-2\lambda g=-2\operatorname{Ric}^g=-2\operatorname{Ric}^{g_t}.$$

If $$M$$ is closed, then according to Hamilton's uniqueness theorem above, this is the only Ricci flow with initial data $$g$$. One sees, in particular, that: In each case, since the Riemannian metrics assigned to different values of $$t$$ differ only by a constant scale factor, one can see that the normalized Ricci flow $$G_s$$ exists for all time and is constant in $$s$$; in particular, it converges smoothly (to its constant value) as $$s\to\infty$$.
 * if $$\lambda$$ is positive, then the Ricci flow "contracts" $$g$$ since the scale factor $$1-2\lambda t$$ is less than 1 for positive $$t$$; furthermore, one sees that $$t$$ can only be less than $$1/2\lambda$$, in order that $$g_t$$ is a Riemannian metric. This is the simplest examples of a "finite-time singularity".
 * if $$\lambda$$ is zero, which is synonymous with $$g$$ being Ricci-flat, then $$g_t$$ is independent of time, and so the maximal interval of existence is the entire real line.
 * if $$\lambda$$ is negative, then the Ricci flow "expands" $$g$$ since the scale factor $$1-2\lambda t$$ is greater than 1 for all positive $$t$$; furthermore one sees that $$t$$ can be taken arbitrarily large. One says that the Ricci flow, for this initial metric, is "immortal".

The Einstein condition has as a special case that of constant curvature; hence the particular examples of the sphere (with its standard metric) and hyperbolic space appear as special cases of the above.

Ricci solitons
Ricci solitons are Ricci flows that may change their size but not their shape up to diffeomorphisms.
 * Cylinders Sk × Rl (for k ≥ 2) shrink self similarly under the Ricci flow up to diffeomorphisms
 * A significant 2-dimensional example is the cigar soliton, which is given by the metric (dx2 + dy2)/(e4t + x2 + y2) on the Euclidean plane. Although this metric shrinks under the Ricci flow, its geometry remains the same. Such solutions are called steady Ricci solitons.
 * An example of a 3-dimensional steady Ricci soliton is the Bryant soliton, which is rotationally symmetric, has positive curvature, and is obtained by solving a system of ordinary differential equations. A similar construction works in arbitrary dimension.
 * There exist numerous families of Kähler manifolds, invariant under a U(n) action and birational to Cn, which are Ricci solitons. These examples were constructed by Cao and Feldman-Ilmanen-Knopf. (Chow-Knopf 2004)
 * A 4-dimensional example exhibiting only torus symmetry was recently discovered by Bamler-Cifarelli-Conlon-Deruelle.

A gradient shrinking Ricci soliton consists of a smooth Riemannian manifold (M,g) and f ∈ C∞(M) such that
 * $$\operatorname{Ric}^g+\operatorname{Hess}^gf=\frac{1}{2}g.$$

One of the major achievements of was to show that, if M is a closed three-dimensional smooth manifold, then finite-time singularities of the Ricci flow on M are modeled on complete gradient shrinking Ricci solitons (possibly on underlying manifolds distinct from M). In 2008, Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu completed the classification of these solitons, showing:
 * Suppose (M,g,f) is a complete gradient shrinking Ricci soliton with dim(M) = 3. If M is simply-connected then the Riemannian manifold (M,g) is isometric to $$\mathbb{R}^3$$, $$S^3$$, or $$S^2\times\mathbb{R}$$, each with their standard Riemannian metrics. This was originally shown by with some extra conditional assumptions. Note that if M is not simply-connected, then one may consider the universal cover $$\pi:M'\to M,$$ and then the above theorem applies to $$(M',\pi^\ast g,f\circ\pi).$$

There is not yet a good understanding of gradient shrinking Ricci solitons in any higher dimensions.

Relationship to uniformization and geometrization
Hamilton's first work on Ricci flow was published at the same time as William Thurston's geometrization conjecture, which concerns the topological classification of three-dimensional smooth manifolds. Hamilton's idea was to define a kind of nonlinear diffusion equation which would tend to smooth out irregularities in the metric. Suitable canonical forms had already been identified by Thurston; the possibilities, called Thurston model geometries, include the three-sphere S3, three-dimensional Euclidean space E3, three-dimensional hyperbolic space H3, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic. (This list is closely related to, but not identical with, the Bianchi classification of the three-dimensional real Lie algebras into nine classes.)

Hamilton succeeded in proving that any smooth closed three-manifold which admits a metric of positive Ricci curvature also admits a unique Thurston geometry, namely a spherical metric, which does indeed act like an attracting fixed point under the Ricci flow, renormalized to preserve volume. (Under the unrenormalized Ricci flow, the manifold collapses to a point in finite time.) However, this doesn't prove the full geometrization conjecture, because of the restrictive assumption on curvature.

Indeed, a triumph of nineteenth-century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane. This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology.

Note that the term "uniformization" suggests a kind of smoothing away of irregularities in the geometry, while the term "geometrization" suggests placing a geometry on a smooth manifold. Geometry is being used here in a precise manner akin to Klein's notion of geometry (see Geometrization conjecture for further details). In particular, the result of geometrization may be a geometry that is not isotropic. In most cases including the cases of constant curvature, the geometry is unique. An important theme in this area is the interplay between real and complex formulations. In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.

Singularities
Hamilton showed that a compact Riemannian manifold always admits a short-time Ricci flow solution. Later Shi generalized the short-time existence result to complete manifolds of bounded curvature. In general, however, due to the highly non-linear nature of the Ricci flow equation, singularities form in finite time. These singularities are curvature singularities, which means that as one approaches the singular time the norm of the curvature tensor $$|\operatorname{Rm}|$$ blows up to infinity in the region of the singularity. A fundamental problem in Ricci flow is to understand all the possible geometries of singularities. When successful, this can lead to insights into the topology of manifolds. For instance, analyzing the geometry of singular regions that may develop in 3d Ricci flow, is the crucial ingredient in Perelman's proof the Poincare and Geometrization Conjectures.

Blow-up limits of singularities
To study the formation of singularities it is useful, as in the study of other non-linear differential equations, to consider blow-ups limits. Intuitively speaking, one zooms into the singular region of the Ricci flow by rescaling time and space. Under certain assumptions, the zoomed in flow tends to a limiting Ricci flow $$ (M_\infty, g_\infty(t)), t \in (-\infty, 0] $$, called a singularity model. Singularity models are ancient Ricci flows, i.e. they can be extended infinitely into the past. Understanding the possible singularity models in Ricci flow is an active research endeavor.

Below, we sketch the blow-up procedure in more detail: Let $$ (M, g_t), \, t \in [0,T), $$ be a Ricci flow that develops a singularity as $$t \rightarrow T$$. Let $$(p_i, t_i) \in M \times [0,T) $$ be a sequence of points in spacetime such that
 * $$K_i := \left|\operatorname{Rm}(g_{t_i})\right|(p_i) \rightarrow \infty $$

as $$i \rightarrow \infty$$. Then one considers the parabolically rescaled metrics
 * $$g_i(t) = K_i g\left(t_i + \frac{t}{K_i}\right), \quad t\in[-K_i t_i, 0]$$

Due to the symmetry of the Ricci flow equation under parabolic dilations, the metrics $$g_i(t)$$ are also solutions to the Ricci flow equation. In the case that
 * $$ |Rm| \leq K_i \text{ on } M \times [0,t_i],$$

i.e. up to time $$t_i$$ the maximum of the curvature is attained at $$p_i$$, then the pointed sequence of Ricci flows $$(M, g_i(t), p_i)$$ subsequentially converges smoothly to a limiting ancient Ricci flow $$ (M_\infty, g_\infty(t), p_\infty)$$. Note that in general $$ M_\infty $$ is not diffeomorphic to $$M$$.

Type I and Type II singularities
Hamilton distinguishes between Type I and Type II singularities in Ricci flow. In particular, one says a Ricci flow $$ (M, g_t), \, t \in [0,T) $$, encountering a singularity a time $$T$$ is of Type I if
 * $$ \sup_{t < T} (T-t)|Rm| < \infty $$.

Otherwise the singularity is of Type II. It is known that the blow-up limits of Type I singularities are gradient shrinking Ricci solitons. In the Type II case it is an open question whether the singularity model must be a steady Ricci soliton—so far all known examples are.

Singularities in 3d Ricci flow
In 3d the possible blow-up limits of Ricci flow singularities are well-understood. From the work of Hamilton, Perelman and Brendle, blowing up at points of maximum curvature leads to one of the following three singularity models: The first two singularity models arise from Type I singularities, whereas the last one arises from a Type II singularity.
 * The shrinking round spherical space form $$ S^3/\Gamma $$
 * The shrinking round cylinder $$ S^2 \times \mathbb{R} $$
 * The Bryant soliton

Singularities in 4d Ricci flow
In four dimensions very little is known about the possible singularities, other than that the possibilities are far more numerous than in three dimensions. To date the following singularity models are known Note that the first three examples are generalizations of 3d singularity models. The FIK shrinker models the collapse of an embedded sphere with self-intersection number −1.
 * $$S^3 \times \mathbb{R} $$
 * $$S^2 \times \mathbb{R}^2 $$
 * The 4d Bryant soliton
 * Compact Einstein manifold of positive scalar curvature
 * Compact gradient Kahler–Ricci shrinking soliton
 * The FIK shrinker
 * The BCCD shrinker

Relation to diffusion
To see why the evolution equation defining the Ricci flow is indeed a kind of nonlinear diffusion equation, we can consider the special case of (real) two-manifolds in more detail. Any metric tensor on a two-manifold can be written with respect to an exponential isothermal coordinate chart in the form
 * $$ ds^2 = \exp(2 \, p(x,y)) \, \left( dx^2 + dy^2 \right). $$

(These coordinates provide an example of a conformal coordinate chart, because angles, but not distances, are correctly represented.)

The easiest way to compute the Ricci tensor and Laplace-Beltrami operator for our Riemannian two-manifold is to use the differential forms method of Élie Cartan. Take the coframe field
 * $$ \sigma^1 = \exp (p) \, dx, \; \; \sigma^2 = \exp (p) \, dy$$

so that metric tensor becomes
 * $$ \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 = \exp(2 p) \, \left( dx \otimes dx + dy \otimes dy \right). $$

Next, given an arbitrary smooth function $$h(x,y)$$, compute the exterior derivative
 * $$ d h = h_x dx + h_y dy = \exp(-p) h_x \, \sigma^1 + \exp(-p) h_y \, \sigma^2.$$

Take the Hodge dual
 * $$ \star d h = -\exp(-p) h_y \, \sigma^1 + \exp(-p) h_x \, \sigma^2 = -h_y \, dx + h_x \, dy.$$

Take another exterior derivative
 * $$ d \star d h = -h_{yy} \, dy \wedge dx + h_{xx} \, dx \wedge dy = \left( h_{xx} + h_{yy} \right) \, dx \wedge dy $$

(where we used the anti-commutative property of the exterior product). That is,
 * $$ d \star d h = \exp(-2 p) \, \left( h_{xx} + h_{yy} \right) \, \sigma^1 \wedge \sigma^2. $$

Taking another Hodge dual gives
 * $$ \Delta h = \star d \star d h = \exp(-2 p) \, \left( h_{xx} + h_{yy} \right)$$

which gives the desired expression for the Laplace/Beltrami operator
 * $$ \Delta = \exp(-2 \, p(x,y)) \left( D_x^2 + D_y^2 \right). $$

To compute the curvature tensor, we take the exterior derivative of the covector fields making up our coframe:
 * $$ d \sigma^1 = p_y \exp(p) dy \wedge dx = -\left( p_y dx \right) \wedge \sigma^2 = -{\omega^1}_2 \wedge \sigma^2$$
 * $$ d \sigma^2 = p_x \exp(p) dx \wedge dy = -\left( p_x dy \right) \wedge \sigma^1 = -{\omega^2}_1 \wedge \sigma^1.$$

From these expressions, we can read off the only independent spin connection one-form
 * $$ {\omega^1}_2 = p_y dx - p_x dy,$$

where we have taken advantage of the anti-symmetric property of the connection ($${\omega^2}_1=-{\omega^1}_2$$). Take another exterior derivative
 * $$ d {\omega^1}_2 = p_{yy} dy \wedge dx - p_{xx} dx \wedge dy = -\left( p_{xx} + p_{yy} \right) \, dx \wedge dy.$$

This gives the curvature two-form
 * $$ {\Omega^1}_2 = -\exp(-2p) \left( p_{xx} + p_{yy} \right) \, \sigma^1 \wedge \sigma^2 = -\Delta p \, \sigma^1 \wedge \sigma^2$$

from which we can read off the only linearly independent component of the Riemann tensor using
 * $$ {\Omega^1}_2 = {R^1}_{212} \, \sigma^1 \wedge \sigma^2.$$

Namely
 * $$ {R^1}_{212} = -\Delta p$$

from which the only nonzero components of the Ricci tensor are
 * $$ R_{22} = R_{11} = -\Delta p.$$

From this, we find components with respect to the coordinate cobasis, namely
 * $$ R_{xx} = R_{yy} = -\left( p_{xx} + p_{yy} \right). $$

But the metric tensor is also diagonal, with
 * $$ g_{xx} = g_{yy} = \exp (2 p)$$

and after some elementary manipulation, we obtain an elegant expression for the Ricci flow:
 * $$ \frac{\partial p}{\partial t} = \Delta p. $$

This is manifestly analogous to the best known of all diffusion equations, the heat equation
 * $$ \frac{\partial u}{\partial t} = \Delta u $$

where now $$\Delta = D_x^2 + D_y^2$$ is the usual Laplacian on the Euclidean plane. The reader may object that the heat equation is of course a linear partial differential equation—where is the promised nonlinearity in the p.d.e. defining the Ricci flow?

The answer is that nonlinearity enters because the Laplace-Beltrami operator depends upon the same function p which we used to define the metric. But notice that the flat Euclidean plane is given by taking $$ p(x,y) = 0$$. So if $$p$$ is small in magnitude, we can consider it to define small deviations from the geometry of a flat plane, and if we retain only first order terms in computing the exponential, the Ricci flow on our two-dimensional almost flat Riemannian manifold becomes the usual two dimensional heat equation. This computation suggests that, just as (according to the heat equation) an irregular temperature distribution in a hot plate tends to become more homogeneous over time, so too (according to the Ricci flow) an almost flat Riemannian manifold will tend to flatten out the same way that heat can be carried off "to infinity" in an infinite flat plate. But if our hot plate is finite in size, and has no boundary where heat can be carried off, we can expect to homogenize the temperature, but clearly we cannot expect to reduce it to zero. In the same way, we expect that the Ricci flow, applied to a distorted round sphere, will tend to round out the geometry over time, but not to turn it into a flat Euclidean geometry.

Recent developments
The Ricci flow has been intensively studied since 1981. Some recent work has focused on the question of precisely how higher-dimensional Riemannian manifolds evolve under the Ricci flow, and in particular, what types of parametric singularities may form. For instance, a certain class of solutions to the Ricci flow demonstrates that neckpinch singularities will form on an evolving $$n$$-dimensional metric Riemannian manifold having a certain topological property (positive Euler characteristic), as the flow approaches some characteristic time $$t_{0}$$. In certain cases, such neckpinches will produce manifolds called Ricci solitons.

For a 3-dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.

Kähler metrics remain Kähler under Ricci flow, and so Ricci flow has also been studied in this setting, where it is called Kähler–Ricci flow.

title: Ricci flow