Langlands program

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Langlands_program 

In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by the Canadian mathematician Robert Langlands (1967, 1970). More precisely, it seeks to relate the structure of Galois groups in algebraic number theory to automorphic forms and, more generally, the representation theory of algebraic groups over local fields and adeles.

Background

The Langlands program is built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra and Gelfand, the work and Harish-Chandra's approach on semisimple Lie groups, and in technical terms the trace formula of Selberg and others.

What was new in Langlands' work, besides technical depth, was the proposed connection to number theory, together with its rich organisational structure hypothesised (so-called functoriality).

Harish-Chandra's work exploited the principle that what can be done for one semisimple (or reductive) Lie group can be done for all. Therefore, once the role of some low-dimensional Lie groups such as GL(2) in the theory of modular forms had been recognised, and with hindsight GL(1) in class field theory, the way was open to speculation about GL(n) for general n > 2.

The "cusp form" idea came out of the cusps on modular curves but also had a meaning visible in spectral theory as "discrete spectrum", contrasted with the "continuous spectrum" from Eisenstein series. It becomes much more technical for bigger Lie groups, because the parabolic subgroups are more numerous.

In all these approaches technical methods were available, often inductive in nature and based on Levi decompositions amongst other matters, but the field remained demanding.

From the perspective of modular forms, examples such as Hilbert modular forms, Siegel modular forms, and theta-series had been developed.

Objects

The conjectures have evolved since Langlands first stated them. Langlands conjectures apply across many different groups over many different fields for which they can be stated, and each field offers several versions of the conjectures. Some versions are vague, or depend on objects such as Langlands groups, whose existence is unproven, or on the L-group that has several non-equivalent definitions.

Objects for which Langlands conjectures can be stated:

• Representations of reductive groups over local fields (with different subcases corresponding to archimedean local fields, p-adic local fields, and completions of function fields)
• Automorphic forms on reductive groups over global fields (with subcases corresponding to number fields or function fields).
• Analogues for finite fields.
• More general fields, such as function fields over the complex numbers.

Conjectures

The conjectures can be stated variously in ways that are closely related but not obviously equivalent.

Reciprocity

The starting point of the program was Emil Artin's reciprocity law, which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of an algebraic number field whose Galois group is abelian; it assigns L-functions to the one-dimensional representations of this Galois group, and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin's reciprocity law.

For non-abelian Galois groups and higher-dimensional representations of them, L-functions can be defined in a natural way: Artin L-functions.

Langlands' insight was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in Langland's more general setting. Hecke had earlier related Dirichlet L-functions with automorphic forms (holomorphic functions on the upper half plane of the complex number plane ${\displaystyle \mathbb{C}}$ that satisfy certain functional equations). Langlands then generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group GL(n) over the adele ring of ${\displaystyle \mathbb{Q}}$ (the rational numbers). (This ring tracks all the completions of ${\displaystyle \mathbb{Q},}$ see p-adic numbers.)

Langlands attached automorphic L-functions to these automorphic representations, and conjectured that every Artin L-function arising from a finite-dimensional representation of the Galois group of a number field is equal to one arising from an automorphic cuspidal representation. This is known as his reciprocity conjecture.

Roughly speaking, this conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an L-group. This offers numerous variations, in part because the definitions of Langlands group and L-group are not fixed.

Over local fields this is expected to give a parameterization of L-packets of admissible irreducible representations of a reductive group over the local field. For example, over the real numbers, this correspondence is the Langlands classification of representations of real reductive groups. Over global fields, it should give a parameterization of automorphic forms.

Functoriality

The functoriality conjecture states that a suitable homomorphism of L-groups is expected to give a correspondence between automorphic forms (in the global case) or representations (in the local case). Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.

Generalized functoriality

Langlands generalized the idea of functoriality: instead of using the general linear group GL(n), other connected reductive groups can be used. Furthermore, given such a group G, Langlands constructs the Langlands dual group ^LG, and then, for every automorphic cuspidal representation of G and every finite-dimensional representation of ^LG, he defines an L-function. One of his conjectures states that these L-functions satisfy a certain functional equation generalizing those of other known L-functions.

He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a (well behaved) morphism between their corresponding L-groups, this conjecture relates their automorphic representations in a way that is compatible with their L-functions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of an induced representation construction—what in the more traditional theory of automorphic forms had been called a 'lifting', known in special cases, and so is covariant (whereas a restricted representation is contravariant). Attempts to specify a direct construction have only produced some conditional results.

All these conjectures can be formulated for more general fields in place of ${\displaystyle \mathbb{Q}}$: algebraic number fields (the original and most important case), local fields, and function fields.

Geometric conjectures

Main article: Geometric Langlands correspondence

The geometric Langlands program, suggested by Gérard Laumon following ideas of Vladimir Drinfeld, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates l-adic representations of the étale fundamental group of an algebraic curve to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.

In 2024, a 9-person collaborative project led by Dennis Gaitsgory announced a proof of the (categorical, unramified) geometric Langlands conjecture leveraging Hecke eigensheaves as part of the proof.

Status

The Langlands correspondence for GL(1, K) follows from (and is essentially equivalent to) class field theory.

Langlands proved the Langlands conjectures for groups over the archimedean local fields ${\displaystyle \mathbb{R}}$ (the real numbers) and ${\displaystyle \mathbb{C}}$ (the complex numbers) by giving the Langlands classification of their irreducible representations.

Lusztig's classification of the irreducible representations of groups of Lie type over finite fields can be considered an analogue of the Langlands conjectures for finite fields.

Andrew Wiles' proof of modularity of semistable elliptic curves over rationals can be viewed as an instance of the Langlands reciprocity conjecture, since the main idea is to relate the Galois representations arising from elliptic curves to modular forms. Although Wiles' results have been substantially generalized, in many different directions, the full Langlands conjecture for ${\displaystyle \text{GL}(2,\mathbb{Q})}$ remains unproved.

In 1998, Laurent Lafforgue proved Lafforgue's theorem verifying the global Langlands correspondence for the general linear group GL(n, K) for function fields K. This work continued earlier investigations by Drinfeld, who previously addressed the case of GL(2, K) in the 1980s.

In 2018, Vincent Lafforgue established one half of the global Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields.

Local Langlands conjectures

Main article: Local Langlands conjectures

Philip Kutzko proved the local Langlands correspondence for the general linear group GL(2, K) over local fields.

Gérard Laumon, Michael Rapoport, and Ulrich Stuhler proved the local Langlands correspondence for the general linear group GL(n, K) for positive characteristic local fields K. Their proof uses a global argument, realizing smooth admissible representations of interest as the local components of automorphic representations of the group of units of a division algebra over a curve, then using the point-counting formula to study the properties of the global Galois representations associated to these representations.

Michael Harris and Richard Taylor proved the local Langlands conjectures for the general linear group GL(n, K) for characteristic 0 local fields K. Guy Henniart gave another proof. Both proofs use a global argument of a similar flavor to the one mentioned in the previous paragraph. Peter Scholze gave another proof.

Fargues–Scholze program

Laurent Fargues and Peter Scholze recast the local Langlands correspondence in geometric terms using the Fargues–Fontaine curve. In their approach, the key automorphic object is the stack ${\displaystyle {\mathrm{Bun}}_G}$ of ${\displaystyle G}$-bundles on the Fargues–Fontaine curve, viewed as a geometric replacement for the representation theory of ${\displaystyle G(E)}$ over a non-archimedean local field ${\displaystyle E}$.

Fargues and Scholze develop a theory of ${\displaystyle \ell }$-adic sheaves on ${\displaystyle {\mathrm{Bun}}_G}$, prove a version of the geometric Satake equivalence for the Fargues–Fontaine curve, and define a geometric moduli stack of Langlands parameters.

Their work also formulates a conjectural categorical form of local Langlands, in which the usual correspondence between representations and parameters is expected to arise from a deeper relationship between categories of sheaves on ${\displaystyle {\mathrm{Bun}}_G}$ and coherent sheaves on the stack of Langlands parameters. This is thus not a complete proof of the local Langlands correspondence for all reductive groups, but a geometric reformulation that yields new results and a broadened conceptual framework for the subject.

Fundamental lemma

Main article: Fundamental lemma (Langlands program)

In 2008, Ngô Bảo Châu proved the "fundamental lemma", which was conjectured initially by Langlands and Shelstad in 1983 and is required in the proof of some essential conjectures in the Langlands program.

Pure mathematics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Pure_mathematics 

Pure mathematics is an informal term to describe the study of the properties and structure of abstract objects, such as the Mandelbrot set. This may be done without focusing on concrete applications of concepts in the physical world.

In the context of the philosophy of mathematics, pure mathematics is an informal term to describe the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but research is not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of defining new mathematical objects or working out the mathematical consequences of basic principles.

While the distinction between pure and applied mathematics has existed since at least ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods.

Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure Internet communications.

It follows that, currently, the distinction between pure and applied mathematics is more a philosophical point of view or preference rather than a rigid subdivision of mathematics.

History

Ancient Greece

Mathematical Platonism

Ancient Greek mathematicians were amongst the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory, and "logistic", now called arithmetic. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being." In this wise Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must make gain of what he learns." The Greek mathematician Apollonius of Perga, asked about the usefulness of some of his theorems in Book IV of Conics, asserted that

They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.

And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of Conics that the subject is one of those that "...seem worthy of study for their own sake."

Aristotle

It is worth noting that in ancient Greek philosophy, an experimental approach of mathematics and no real distinction between mathematics and physics was already present in authors such as Thales and Archimedes, and for example in Aristotle and his school. This duality of platonism vs Aristotle is still present in modern mathematics.

Different historical records of the school of Pythagoras show this contradiction too from elements of mysticism similar to Plato to the proof irrationality which leads to the paradox that an infinite non repetitive set of digits cannot belong to the "finite" and "well defined" world of Euclidean geometry. Last but not least Zeno paradox shows again this duality between the pure logical reasoning (splitting distance by two is always possible) and the applied realm (the fact that the traveler never arrives).

19th century

Pure mathematics invented

The term "pure mathematics" itself is enshrined in the full title of the Sadleirian Chair, "Sadleirian Professor of Pure Mathematics", founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Carl Friedrich Gauss (1777 to 1855) made no sweeping distinction of the kind between pure and applied. In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent.

The problem of infinities

After Weierstrass, by the end of 19th century, an important discussion about the role of infinities came from authors like Gregor Cantor and early examples of fractals and chaos. Ludwig Wittgenstein considered Cantor's approach with uncountable sets the cancer of mathematics. Henri Poincare seems to have compared set theory to a temporary disease Infinities in general are difficult to treat axiomatically, and therefore have been always considered on the fringe of pure mathematics, and this complexity became evident in the works of Bertrand Russell and Gödel on paradoxes in the early 20th century.

20th century

At the start of the twentieth century, mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof.

Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. "Pure mathematician" became a recognized vocation, achievable through training. That said, the case was made pure mathematics is useful in engineering education:

There is a training in habits of thought, points of view, and intellectual comprehension of ordinary engineering problems, which only the study of higher mathematics can give.

Major advances in the beginning of 20th century was the formalization of abstract algebra and topology; these two fields were deeply influenced by the pure mathematics philosophy.

Finiteness, Number theory and Algebraic geometry

Historically areas often considered attached to pure mathematics are number theory, where these infinities are typically countable and algebraic geometry where functions are typically tamed functions (i.e. piecewise polynomial or rational functions). The success in the proof of Weil conjectures and the unification of these two fields ultimately in the geometric Langlands program gave momentum to the concept of pure mathematics as a research activity that can self sustain independently. On the other hand, the fields of functional analysis, partial differential equations, statistics and dynamical systems were often considered in the applied mathematics camp, and this is reflected in the typical organization of mathematics curriculum.

The advance of computers

By the end of the century numerical proofs of the four color theorem were primary examples of the advance of computers. Famous mathematicians such as Paul Cohen already in the 1970s challenged the concept of pure mathematics, stating that there will be a moment in the future that most mathematicians will be replaced by computers.

French vs Russian schools

The French school of mathematics and the western authors in general were deeply influenced by the Bourbaki group and therefore by the philosophical idea to detach mathematics from natural sciences. The Russian school of mathematics instead (e.g., Kolmogorov, Gelfand and Vladimir Arnold) believed that mathematics is actually fully grounded in experimental sciences such as physics and biology, this was reflected in the separation of the two schools during the Cold War era due to geopolitical motivations and by the works of authors like Robert Langlands which started to piece wise reconnect work from both schools.

21st century

At the beginning of the 21st century, the application of artificial intelligence to pure mathematics has attracted the attention of such luminaries as Ken Ono and François Charton.

Looking at recent research fields such as the Analytic Langlands conjecture one can see that the distinction between pure and applied blurs again. Another example is the emergence of chaos in number theory and authors like Robert Langlands advocate for the unification of mathematics with Physics through the Langlands program.

Pure versus applied mathematics

Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of this debate can be found in G.H. Hardy's 1940 essay A Mathematician's Apology.

It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry, Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express physical truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called "real" mathematics, "which has permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use.

Hardy considered some physicists, such as Einstein and Dirac, to be among the "real" mathematicians, but at the time that he was writing his Apology, he considered general relativity and quantum mechanics to be "useless", which allowed him to hold the opinion that only "dull" mathematics was useful. Moreover, Hardy briefly admitted that—just as the application of matrix theory and group theory to physics had come about unexpectedly—the time may come where some kinds of beautiful, "real" mathematics may be useful as well.

Another insightful view is offered by American mathematician Andy Magid:

I've always thought that a good model here could be drawn from ring theory. In that subject, one has the subareas of commutative ring theory and non-commutative ring theory. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a non-commutative ring is a not-necessarily-commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by the latter we mean not-necessarily-applied mathematics... [emphasis added]

Friedrich Engels argued in his 1878 book Anti-Dühring that "it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and figure have not been invented from any source other than the world of reality". He further argued that "Before one came upon the idea of deducing the form of a cylinder from the rotation of a rectangle about one of its sides, a number of real rectangles and cylinders, however imperfect in form, must have been examined. Like all other sciences, mathematics arose out of the needs of men...But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, become divorced from the real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform."

In education

After the year 2000, the case was made by authors like Cédric Villani that pure and abstract mathematics without a constructivist approach and taking into account other cognitive processes can become detrimental to education. More precisely that in order to reform the school system and the curriculum during 1960s and 1970s it was asked to the leading mathematicians how to teach mathematics and founding the educational system on set theory seemed to be the way to go. This led to a generation of un-prepared teachers, a generation of low-quality educational material and ultimately poor education.

We actually don't know how people learn mathematics, and an experimental approach to understand how to teach mathematics should be promoted.

— Cedric Villani

A few such examples are investment in mathematics education and by the Simons Foundation.

Any exact science contains typically two cognitive processes: an inductive one (i.e. from experimental evidence to abstraction) and a deductive one (i.e. from axioms to theorems and proofs). Both of them are key in the pedagogical process and more precisely the deductive one is typical of pure mathematics whereas the inductive one is key to mathematics education and to experimental mathematics.

The covid pandemic led also to the so called global learning crisis  with adverse affects on all scientific education.

As of 2026, major commercially available LLM can reproduce multiple proofs of classical theorems, other LLMs can have very high scores in mathematical olympiads reflecting average high school mathematics. This technology, even without being able to produce novelty, has the potential to ideally transform education but it is currently not the case.

In artificial intelligence

As of 2026 it may be feasible in some cases to automatically create large databases of mathematical knowledge (i.e. automating aspects of the experimental process) and automatically find patterns across large sets of data with artificial intelligence to derive mathematical conjectures (i.e. automating aspects of the inductive process). Then, putative proofs generated by artificial intelligence can be verified via proof assistant tools such as Lean, and large supporting databases of theorems (automating the deductive process). Some mathematicians believe that these approaches may become mainstream in the 21st century.

Curiosity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Curiosity

Space and telescopes have been a quintessential symbol for curiosity.

Curiosity (from Latin cūriōsitās, from cūriōsus "careful, diligent, curious", akin to cura "care") can be a quality related to inquisitive thinking, such as exploration, investigation, and learning, evident in humans and other animals. It can also refer to something (an even, object, person, etc.) which can cause curiosity. When it is for an object it may be called a curio, and a collection of curios or curiosities may be stored together as a collection called curiosities.

The term curiosity can also denote the behavior, characteristic, or emotion of being curious, in regard to the desire to gain knowledge or information. Curiosity helps human development, from which derives the process of learning and desire to acquire knowledge and skill. Curiosity as a behavior and emotion is the driving force behind human development, such as progress in science, language, and industry.

Curiosity can be considered to be an evolutionary adaptation based on an organism's ability to learn. Certain curious animals (namely, corvids, octopuses, dolphins, elephants, rats, etc.) will pursue information in order to adapt to their surrounding and learn how things work. This behavior is termed neophilia, the love of new things. For animals, a fear of the unknown or the new, neophobia, is much more common, especially later in life.

Causes

Children peer over shoulders to see what their friends are reading.

Many species display curiosity including apes, cats, and rodents. It is common in human beings at all ages from infancy through adulthood. Research has shown that curiosity is not a fixed attribute amongst humans but rather can be nurtured and developed.

Early definitions of curiosity call it a motivated desire for information. This motivational desire has been said to stem from a passion or an appetite for knowledge, information, and understanding.

Traditional ideas of curiosity have expanded to consider the difference between perceptual curiosity, as the innate exploratory behavior that is present in all animals, and epistemic curiosity, as the desire for knowledge that is specifically attributed to humans.

Daniel Berlyne recognized three classes of variables playing a role in evoking curiosity: psychophysical variables, ecological variables, and collative variables. Psychophysical variables correspond to physical intensity, ecological variables to motivational significance and task relevance. Collative variables involve a comparison between different stimuli or features, which may be actually perceived or which may be recalled from memory. Berlyne mentioned four collative variables: novelty, complexity, uncertainty, and conflict (though he suggested that all collative variables probably involve conflict). Additionally, he considered three variables supplementary to novelty: change, surprisingness, and incongruity. Finally, curiosity may not only be aroused by the perception of some stimulus associated with the aforementioned variables ("specific exploration"), but also by a lack of stimulation, out of "boredom" ("diversive exploration").

Curiosity-driven behavior

Mystery House (an adventure video game) in the video game museum (a designated space for learning), Computerspielemuseum, Berlin.

Curiosity-driven behavior is often defined as behavior through which knowledge is gained – a form of exploratory behavior. It therefore encompasses all behaviors that provide access to or increase sensory information. Berlyne divided curiosity-driven behavior into three categories: orienting responses, locomotor exploration, and investigatory responses or investigatory manipulation. Previously, Berlyne suggested that curiosity also includes verbal activities, such as asking questions, and symbolic activities, consisting of internally fueled mental processes such as thinking ("epistemic exploration").

Theories

Like other desires and need-states that take on an appetitive quality (e.g. food/hunger), curiosity is linked with exploratory behavior and experiences of reward. Curiosity can be described in terms of positive emotions and acquiring knowledge; when one's curiosity has been aroused it is considered inherently rewarding and pleasurable. Discovering new information may also be rewarding because it can help reduce undesirable states of uncertainty rather than stimulating interest. Theories have arisen in attempts to further understand this need to rectify states of uncertainty and the desire to participate in pleasurable experiences of exploratory behaviors.

Curiosity-drive theory

Curiosity-drive theory posits undesirable experiences of "uncertainty" and "ambiguity". The reduction of these unpleasant feelings is rewarding. This theory suggests that people desire coherence and understanding in their thought processes. When this coherence is disrupted by something that is unfamiliar, uncertain, or ambiguous, an individual's curiosity-drive causes them to collect information and knowledge of the unfamiliar to restore coherent thought processes. This theory suggests that curiosity is developed out of the desire to make sense of unfamiliar aspects of one's environment through exploratory behaviors. Once understanding of the unfamiliar has been achieved and coherence has been restored, these behaviors and desires subside.

Derivations of curiosity-drive theory differ on whether curiosity is a primary or secondary drive and if this curiosity-drive originates due to one's need to make sense of and regulate one's environment or if it is caused by an external stimulus. Causes can range from basic needs that need to be satisfied (e.g. hunger, thirst) to needs in fear-induced situations. Each of these derived theories state that whether the need is primary or secondary, curiosity develops from experiences that create a sensation of uncertainty or perceived unpleasantness. Curiosity then acts to dispel this uncertainty. By exhibiting curious and exploratory behavior, one is able to gain knowledge of the unfamiliar and thus reduce the state of uncertainty or unpleasantness. This theory, however, does not address the idea that curiosity can often be displayed even in the absence of new or unfamiliar situations. This type of exploratory behavior, too, is common in many species. A human toddler, if bored in his current situation devoid of arousing stimuli, will walk about until he finds something interesting. The observation of curiosity even in the absence of novel stimuli pinpoints one of the major shortcomings in the curiosity-drive model.

Optimal-arousal theory

Optimal-arousal theory developed out of the need to explain this desire to seek out opportunities to engage in exploratory behaviors without the presence of uncertain or ambiguous situations. Optimal-arousal suggests that one can be motivated to maintain a pleasurable sense of arousal through such exploratory behaviors.

When a stimulus is encountered that is associated with complexity, uncertainty, conflict, or novelty, this increases arousal above the optimal point, and exploratory behavior is employed to learn about that stimulus and thereby reduce arousal again. In contrast, if the environment is boring and lacks excitement, arousal is reduced below the optimal point and exploratory behavior is employed to increase information input and stimulation, and thereby increasing arousal again. This theory addresses both curiosity elicited by uncertain or unfamiliar situations and curiosity elicited in the absence of such situations.

Cognitive-consistency theory

Cognitive-consistency theories assume that "when two or more simultaneously active cognitive structures are logically inconsistent, arousal is increased, which activates processes with the expected consequence of increasing consistency and decreasing arousal." Similar to optimal-arousal theory, cognitive-consistency theory suggests that there is a tendency to maintain arousal at a preferred, or expected, level, but it also explicitly links the amount of arousal to the amount of experienced inconsistency between an expected situation and the actually perceived situation. When this inconsistency is small, exploratory behavior triggered by curiosity is employed to gather information with which expectancy can be updated through learning to match perception, thereby reducing inconsistency.

This approach associates curiosity with aggression and fear. If the inconsistency is larger, fear or aggressive behavior may be employed to alter the perception in order to make it match expectancy, depending on the size of the inconsistency as well as the specific context. Aggressive behavior alters perception by forcefully manipulating it into matching the expected situation, while fear prompts flight, which removes the inconsistent stimulus from the perceptual field and thus resolves the inconsistency.

Integration of the reward pathway into theory

Taking into account the shortcomings of both curiosity-drive and optimal-arousal theories, attempts have been made to integrate neurobiological aspects of reward, wanting, and pleasure into a more comprehensive theory for curiosity. Research suggests that desiring new information involves mesolimbic pathways of the brain that account for dopamine activation. The use of these pathways, and dopamine activation, may be how the brain assigns value to new information and interprets this as reward. This theory from neurobiology can supplement curiosity-drive theory by explaining the motivation of exploratory behavior.

Role of neurological aspects and structures

Although curiosity is widely regarded, its root causes are largely empirically unknown. However, some studies have provided insight into the neurological mechanisms that make up what is known as the reward pathway which may influence characteristics associated with curiosity, such as learning, memory, and motivation. Due to the complex nature of curiosity, research that focuses on specific neural processes with these characteristics can help us understand of the phenomenon of curiosity as a whole. The following are descriptions of characteristics of curiosity and their links to neurological aspects that are essential in creating exploratory behaviors:

Motivation and reward

Dopamine pathway in the brain

The drive to learn new information or perform some action may be prompted by the anticipation of reward. So what we learn about motivation and reward may help us to understand curiosity.

Reward is defined as the positive reinforcement of an action, reinforcement that encourages a particular behavior by means of the emotional sensations of relief, pleasure, and satisfaction that correlate with happiness. Many areas in the brain process reward and come together to form what is called the reward pathway. In this pathway many neurotransmitters play a role in the activation of the reward sensation, including dopamine, serotonin, and opioids.

Dopamine is linked to curiosity, as it assigns and retains reward values of information gained. Research suggests higher amounts of dopamine are released when the reward is unknown and the stimulus is unfamiliar, compared to activation of dopamine when stimulus is familiar.

Nucleus accumbens

The nucleus accumbens is a formation of neurons that is important in reward pathway activation—such as the release of dopamine in investigating response to novel or exciting stimuli. The fast dopamine release observed during childhood and adolescence is important in development, as curiosity and exploratory behavior are the largest facilitators of learning during early years.

The sensation pleasure of "liking" can occur when opioids are released by the nucleus accumbens. This helps someone evaluate the unfamiliar situation or environment and attach value to the novel object. These processes of both wanting and liking play a role in activating the reward system of the brain, and perhaps in the stimulation of curious or information-seeking tendencies as well.

Caudate nucleus

The caudate nucleus is a region of the brain that is highly responsive to dopamine, and is another component of the reward pathway. Research suggests that the caudate nucleus anticipates the possibility of and reward of exploratory behavior and gathered information, thus contributing to factors of curiosity.

Anterior cortices

Regions of the anterior insula and anterior cingulate cortex correspond to both conflict and arousal and, as such, seem to reinforce certain exploratory models of curiosity.

Cortisol

Cortisol is a chemical known for its role in stress regulation. However, cortisol may also be associated with curious or exploratory behavior. Studies suggesting a role of cortisol in curiosity support optimal arousal theory. They suggest the release of some cortisol, causing some stress, encourages curious behavior, while too much stress can initiate a "back away" response.

Attention

Attention is important to curiosity because it allows one to selectively focus and concentrate on particular stimuli in the surrounding environment. As there are limited cognitive and sensory resources to understand and evaluate stimuli, attention allows the brain to better focus on what it perceives to be the most important or relevant of these stimuli. Individuals tend to focus on stimuli that are particularly stimulating or engaging. The more attention a stimulus garners, the more frequent one's energy and focus will be directed towards that stimulus. This suggests an individual will focus on new or unfamiliar stimuli in an effort to better understand or make sense of the unknown, rather than on more familiar or repetitive stimuli.

Striatum

The striatum is a part of the brain that coordinates motivation with body movement. The striatum likely plays a role in attention and reward anticipation, both of which are important in provoking curiosity.

Precuneus

The precuneus is a region of the brain that is involved in attention, episodic memory, and visuospatial processing. There is a correlation between the amount of grey matter in the precuneus and levels of curious and exploratory behaviors. This suggests that precuneus density has an influence on levels of curiosity.

Memory and learning

Memory plays an important role in curiosity. Memory is how the brain stores and accesses stored information. If curiosity is the desire to seek out and understand unfamiliar or novel stimuli, memory helps determine if the stimulus is indeed unfamiliar. In order to determine if a stimulus is novel, an individual must remember if the stimulus has been encountered before.

Curiosity may also affect memory. Stimuli that are novel tend to capture more of our attention. Additionally, novel stimuli usually have a reward value associated with them, the anticipated reward of what learning that new information may bring. With stronger associations and more attention devoted to a stimulus, it is probable that the memory formed from that stimulus will be longer lasting and easier to recall, both of which facilitate better learning.

Hippocampus and the parahippocampal gyrus

The hippocampus is important in memory formation and recall and therefore in determining the novelty of various stimuli. Research suggests the hippocampus is involved in generating the motivation to explore for the purpose of learning.

The parahippocampal gyrus (PHG), an area of grey matter surrounding the hippocampus, has been implicated in the amplification of curiosity.

Amygdala

The amygdala is associated with emotional processing, particularly for the emotion of fear, as well as memory. It is important in processing emotional reactions towards novel or unexpected stimuli and the induction of exploratory behavior. This suggests a connection between curiosity levels and the amygdala. However, more research is needed on direct correlation.

Early development

Jean Piaget argued that babies and children constantly try to make sense of their reality and that this contributes to their intellectual development. According to Piaget, children develop hypotheses, conduct experiments, and then reassess their hypotheses depending on what they observe. Piaget was the first to closely document children's actions and interpret them as consistent, calculated efforts to test and learn about their environment.

There is no universally accepted definition for curiosity in children. Most research on curiosity focused on adults and used self-report measures that are inappropriate and inapplicable for studying children.

Exploratory behaviour is commonly observed in children and is associated with their curiosity development. Several studies of children's curiosity simply observe their interaction with novel and familiar toys.

Evidence suggests a relationship between the anxiety children might feel and their curiosity. One study found that object curiosity in 11-year-olds was negatively related to psychological maladjusted so children who exhibit more anxiety in classroom settings engage in less curious behaviour. Certain aspects of classroom learning may depend on curiosity, which can be affected by students' anxiety.

An aptitude for curiosity in adolescents may produce higher academic performance. One study revealed that, of 568 high school students, those who exhibited an aptitude for curiosity, in conjunction with motivation and creativity, showed a 33.1% variation in math scores and 15.5% variation in science scores when tested on a standardized academic exam.

Other measures of childhood curiosity used exploratory behaviour as a basis but differed on which parts of this behaviour to focus on. Some studies examined children's preference for complexity/the unknown as a basis for their curiosity measure; others relied on novelty preference as their basis.

Researchers also examined the relationship between a child's reaction to surprise and their curiosity. Children may be further motivated to learn when dealing with uncertainty. Their reactions to not having their expectations met may fuel their curiosity more than the introduction of a novel or complex object would.

Curiosity as a virtue

Curiosity has been of interest to philosophers. Curiosity has been recognised as an important intellectual (or "epistemic") virtue, due to the role that it plays in motivating people to acquire knowledge and understanding. It has also been considered an important moral virtue, as curiosity can help humans find meaning in their lives and to cultivate a sense of care about others and things in the world. When curiosity in young people leads to knowledge-gathering it is widely seen as a positive.

Due to the importance of curiosity, people debate about whether contemporary societies effectively cultivate the right type of curiosity.

Some believe that children's curiosity is discouraged throughout the process of formal education: "Children are born scientists. From the first ball they send flying to the ant they watch carry a crumb, children use science's tools—enthusiasm, hypotheses, tests, conclusions—to uncover the world's mysteries. But somehow students seem to lose what once came naturally."

Impact from disease

Left: normal brain. Right: Alzheimer's disease afflicted brain. Severe degeneration of areas implicated in curiosity

Neurodegenerative diseases and psychological disorders can affect various characteristics of curiosity. For example Alzheimer's disease's effects on memory or depression affect motivation and reward. Alzheimer's is a neurodegenerative disease that degrades memory. Depression is a mood disorder that is characterized by a lack of interest in one's environment and feelings of sadness or hopelessness. A lack of curiosity for novel stimuli might be a predictor for these and other illnesses.

Social curiosity

Social curiosity is defined as a drive to understand one's environment as it relates to sociality with others. Such curiosity plays a role in one's ability to successfully navigate social interactions by perceiving and processing one's own behavior and the behavior of others. It also plays a role in helping one adapt to varying social situations.

Morbid curiosity

"Morbid curiosity" redirects here. For the magazine, see Morbid Curiosity (magazine).

A crowd mills around the site of a car accident in Czechoslovakia in 1980.

Morbid curiosity is focused on death, violence, or any other event that may cause harm physically or emotionally. It typically is described as having an addictive quality, associated with a need to understand or make sense of topics that surround harm, violence, or death. This can be attributed to one's need to relate unusual and often difficult circumstances to a primary emotion or experience of one's own, described as meta-emotions.

One explanation evolutionary biologists offer for curiosity about death is that by learning about life-threatening situations, death can be avoided. Another suggestion some psychologists posit is that as spectators of gruesome events, humans are seeking to empathize with the victim. Alternatively, people may be trying to understand how another person can become the perpetrator of harm. According to science journalist Erika Engelhaupt, morbid curiosity is not "a desire to be sad", instead it "has the ability to set our minds ... at ease by reassuring us that even death follows the rules of the natural world."

Interest in human curiosity about difficult circumstances dates back to Aristotle in his Poetics, in which he noted, "We enjoy and admire paintings of objects that in themselves would annoy or disgust us." A 2017 paper in the journal PLOS One concluded that people choose to see graphic images even when presented the option to avoid them and look at them for a longer period of time than neutral or positive images.

State and trait curiosity

Curiosity can be a temporary state of being, or a stable trait in an individual. State curiosity is external—wondering why things happen just for the sake of curiousness, for example wondering why most stores open at 8 a.m. Trait curiosity describes people who are interested in learning, for example by trying out a new sport or food, or traveling to an unfamiliar place. One can look at curiosity as the urge that draws people out of their comfort zones and fears as the agents that keep them within those zones.

Curiosity in artificial intelligence

AI agents can exhibit curiosity through intrinsic motivation. This can improve the success of an AI agent at various tasks. In artificial intelligence, curiosity is typically defined quantitatively, as the uncertainty the agent has in predicting its own actions given its current state.

In 2019, a study trained AI agents to play video games, but they were rewarded only for curiosity. The agents reliably learned advantageous game behaviors based solely on the curiosity reward.

Curiosity mementos

Red sandalwood curio box with bamboo-veneered decoration. Qianlong era (1736-1795). National Palace Museum 故雜001284N000000000

People will designate things as a 'curiosity' to share their interest with others.

A collection may be stored in an anthology, curiosities cabinet, museum, or some other way to display curios. The following quote is from an antiquarian's anthology of what he found curious:

An antiquary lights on many a curiosity whilst overhauling the dusty tomes of ancient writers. This little book is a small museum in which I have preserved some of the quaintest relics which have attracted my notice during my labours. The majority of the articles were published in 1869. I have now added some others.

Lew Trenchard,

September 1895.

— Sabine Baring-Gould, CURIOSITIES OF OLDEN TIMES (full text on Wikisource), Preface