title: Succession to Muhammad

text: Among Sunni Muslims, the Ghadir Khumm is not associated with the succession to Muhammad. Instead, the event is often connected with Ali's earlier campaign in Yemen, where he is said to have strictly imposed the Islamic guidelines for a fair distribution of booty. The Sunni Ibn Kathir () suggests that the Ghadir Khumm sermon was simply intended as a public declaration of Muhammad's love and esteem for Ali in response to criticism of some soldiers. Accepting this explanation as such, that Muhammad equated Ali with himself in an extraordinary announcement at the Ghadir Khumm still provides a strong basis for the Shia claims, suggests Jafri.


title: Succession to Muhammad

text: For the Sunni, it is also unimaginable that most companions would act wrongly and ignore a clear appointment of Ali at the Ghadir Khumm. The Shia response is that numerical strength cannot be a factor in a tribal community, adding that majority does not imply legitimacy in the Qur'an.


title: Succession to Muhammad

text: The most notable event that supports Abu Bakr's right to succession reportedly occurred towards the end of Muhammad's life. According to Walker, too ill to lead the prayers himself, Muhammad instructed Abu Bakr to take his place, ignoring concerns that he was too emotionally delicate for the role. In particular, when Muhammad entered the prayer hall one morning during the "fajr" prayer, Abu Bakr attempted to step back to let Muhammad lead the prayer but the prophet allowed Abu Bakr to continue.


title: Succession to Muhammad

text: Jafri finds the related traditions often contradictory, many of which are attributed by Ibn Sa'd () to Abu Bakr's daughter Aisha, whose rivalry and dislike for Ali and Fatima are well-documented. Mentioning three different versions of the story, Madelung defers to Caetani, who considers it fabricated. Shia authors similarly dispute the authenticity of these reports, adding that Muhammad had earlier instructed his companions, including Abu Bakr and Umar, to leave Medina on a military campaign against the Byzantines under Usama ibn Zayd.


title: Succession to Muhammad

text: Sunni authors often refer to this event as evidence of Abu Bakr's right to succeed Muhammad, though their accounts of the event might be colored by later Shia-Sunni polemics. Ayoub adds that the prayer argument was likely not a consideration in the early caliphal debate. Lecomte writes that Muhammad respected Abu Bakr but considers the prayer story inconclusive because it does not formally relate to the political leadership of the community. Walker notes that this and similar traditions suggest friendship and trust but are hardly related to succession because Muhammad regularly delegated this task and other positions of authority to others. For the same reason, Shaban goes further and assigns no significance to the prayer story.


title: Succession to Muhammad

text: Shortly before his death, Muhammad asked for writing materials, so that he would "write something, after which you will not be led into error," as reported in the canonical Sunni collection "Sahih al-Bukhari". Of those present at Muhammad's bedside, this report continues that his companion Umar protested, "The illness has overwhelmed the prophet. We have the book of God [Quran] and that is enough for us." In some reports by the Sunni Ibn Sa'd (), Umar instead says that Muhammad is raving. A quarrel then broke out at Muhammad's bedside, with some suggesting that his orders should be followed and some siding with Umar to disregard the prophet's request. The argument is said to have saddened Muhammad, who asked them to leave and did not write anything. Some sources write that Muhammad instead gave oral recommendations, which have been recorded differently by various authors.


title: Succession to Muhammad

text: The disobedience to Muhammad in this incident has been downplayed by some Sunni scholars, whereas others view this incident as a missed opportunity to formally designate Abu Bakr as the successor. Ibn Kathir () goes further, claiming that Muhammad had publicly appointed Abu Bakr before his final illness. In contrast, the incident is viewed as a calamity and a missed opportunity to designate Ali in Shia sources.


title: Succession to Muhammad

text: In relation to this episode, Madelung quotes an exchange between Ibn Abbas and Umar in which the latter claimed that Muhammad intended to name Ali as his successor and that he prevented this out of the conviction that Arabs would revolt against Ali. A tradition to this effect is also cited by the Shia Tababatai (). This view has been echoed by Hazleton.


title: Succession to Muhammad

text: The general Sunni belief is that Muhammad had not chosen anyone to succeed him, instead reasoning that he had intended for the community to decide on a leader amongst themselves. However, some specific hadiths are used to justify that Muhammad intended Abu Bakr to succeed, but that he had shown this decision through his actions rather than doing so verbally.


title: Succession to Muhammad

text: In Sunni Islam, the election of a caliph is ideally a democratic choice made by the Muslim community. As this is difficult to enforce, Sunni Islam recognizes as caliph anyone who seizes power, as long as he is from the Quraysh, the tribe of Muhammad. Even the latter is not a strict requirement, given that Ottoman Caliphs had no familial relation to the Quraysh tribe. In Sunni Islam, caliphs are not viewed as infallible and can be removed from the office if their actions are deemed sinful. At the same time, obedience to a caliph is often regarded as a religious obligation even if the caliph is unjust. Conversely, a judge would be considered competent solely on the basis of his appointment by the government.


title: Succession to Muhammad

text: Historically, Abu Bakr, Umar, Uthman, and Ali are regarded by the Sunni as the most righteous of their generation, with their merit being reflected in their caliphate. The subsequent caliphates of the Umayyads and the Abbasids, while not ideal, are seen as legitimate because they complied with the requirements of the law, kept the borders safe and the community united. While the Umayyads and the Abbasids are viewed as kingships, the Sunni was more willing than others to accommodate these rulers, regardless of their legitimacy and mode of government, and in so doing the Sunni made most of Islamic history their own.


title: Succession to Muhammad

text: In the Twelver Shia view, after a prophet's death, it is deemed as essential that a divinely-appointed successor would guide the faithful towards the righteous path. Without a divinely-appointed successor, according to the Twelver Shia, the prophetic mission and God's favor to the faithful would both remain incomplete. At the same time, in Shia theology, this designated successor would not rule by force if the faithful withhold their support.


title: Succession to Muhammad

text: The Twelver Shia view is that, similar to the past prophets in the Quran, the succession to Muhammad was settled by divine appointment, rather than by consensus. Moreover, as with the past prophets in the Quran, God chose Muhammad's successor from his family. A number of verses in the Quran and some hadiths might be linked to the prominent position of Muhammad's family in Islam, including the Verse of Purification, Verse of Mubahala, and Verse of Mawadda in the Quran, and the well-attested Hadith of the Thaqalayn and the Hadith of the Ark.


title: Succession to Muhammad

text: The view advanced by the Shia is that Muhammad announced his cousin and son-in-law, Ali, as his rightful successor shortly before his death at the Event of Ghadir Khumm and also earlier in his prophetic mission at the Event of Dhul Asheera. After the announcement at Ghadir Khumm, there is evidence that the Verse of Ikmal was revealed to Muhammad, declaring the completion of God's favor to the faithful. Though it is believed that Ali considered himself as the rightful successor of Muhammad, he is said to have turned down proposals to forcefully pursue his claims to the caliphate after the appointment of Abu Bakr, for the sake of preserving the unity of Islam in a critical time.


title: Succession to Muhammad

text: Ali's distinctions are amply attested to in Islamic sources. In Mecca, a young Ali is said to have been the first male to embrace Islam and the only person who offered his support when Muhammad first introduced Islam to his relatives. Later, he facilitated Muhammad's safe escape to Medina by risking his life as the decoy. In Medina, Ali sworn a pact a brotherhood with Muhammad and later took the hand of Muhammad's daughter, Fatimah, in marriage. Ali commonly acted as Muhammad's secretary in Medina and served as his deputy during the Expedition of Tabuk. Saluted as Asadullah (literally, "the lion of God"), Ali has been viewed as the most able warrior in Muhammad's army and the two were the only Muslim men who represented Islam against a Christian delegation from Najran. Ali's role in the collection of the Quran, the central text of Islam, is deemed as one of his key contributions. When, following the revelation of the "surah" at-Tawbah, Abu Bakr was sent to Mecca to give an ultimatum to disbelievers, there is strong evidence that Muhammad might have sent out Ali to take over this responsibility.


title: Succession to Muhammad

text: In Shia theology, while direct revelation ended with Muhammad's death, Ali remained the righteous guide towards God, similar to the successors of the past prophets in the Quran. After Muhammad's death, Ali inherited his divine knowledge and his authority to correctly interpret the Quran, especially its allegorical and metaphorical verses ("mutashabihat"). Often cited here is a well-attested hadith, attributed to Muhammad, which reads as, "I am the city of knowledge and Ali is its gate."


title: Succession to Muhammad

text: In Shia Islam, as the righteous guide after Muhammad, Ali is believed to be infallible. Ali is one of the Ahl al-Kisa, who are addressed by the sahih Hadith of Kisa and the related Verse of Purification in the Quran, which includes the passage, "Indeed God desires to repel all impurity from you, O Ahl al-Bayt, and purify you with a thorough purification."


title: Succession to Muhammad

text: According to the Shia, Ali succeeded Muhammad as the first Imam after Muhammad, that is, the righteous guide towards God and His vicar on the earth. This divine authority, known as imamate, is central to the Shia belief and appears in multiple verses of the Quran. In particular, verse 21:73 reads as


title: Succession to Muhammad

text: In the Twelver Shia belief, since the time of the first prophet, Adam, the earth has never remained without an Imam, in the form of prophets and their divinely-appointed successors. After Ali, imamate was passed down to his son, Hasan, through divinely-inspired designation ("nass"). In Shia theology, at any time, there is only one Imam and his successor, if alive, is called the silent Imam. After Hasan's death, his brother, Husayn, and nine of his descendants are regarded as Imams, the last of whom, Mahdi, went into occultation in 260 AH (874 CE), compelled by the hostility of his enemies. His advent is awaited by all Muslims, though different sects hold different views about Mahdi. In his absence, the vacuum in the Shia leadership is partly filled by "marjaiyya" and, more recently, "wilayat al-faqqih", i.e., guardianship of the Islamic jurist.


title: Succession to Muhammad

text: According to Jafri, it is widely reported that the fourth Shia Imam, Zayn al-Abidin, designated his son, Muhammad al-Baqir, as the next Imam before his death. Zayd, a half-brother of Muhammad al-Baqir, also asserted a claim to "imamate" on the basis that the title can belong to any descendant of Hasan or Husayn who is learned, pious, and revolts against the tyrants of his time. On this basis, his followers, known as Zaydis, consider Zayd as the rightful successor of the fourth Shia Imam, though the fourth Imam himself did not revolt against the Umayyads and instead adopted a policy of quiescence.


title: Succession to Muhammad

text: Initially, Zayd's activist approach gained him a large following. However, as he increasingly compromised with the traditionalists, some of Zayd's supporters are said to have returned to Muhammad al-Baqir. According to Jafri, a related incident is when two Kufan Shias asked Zayd if the first Shia Imam, Ali, was an Imam before he resorted to the sword. When Zayd refused to answer this question, the two broke their allegiance with him and went back to Muhammad al-Baqir. Eventually, Zayd took up arms against the Umayyads in 122 AH and was killed in Kufa by the forces of Caliph Hisham.


title: Succession to Muhammad

text: One faction of the Zaidiyyah, called the Batriyya, attempted a compromise between Sunni and Shia by accepting the legitimacy of the Sunni caliphs while maintaining that they were inferior to Ali. Imamat al-Mafdul (literally, ""imamate" of the inferior") is the belief that, while Ali was better suited to succeed Muhammad, the reigns of Abu Bakr and Umar must be acknowledged since Ali did not revolt against them.


title: Succession to Muhammad

text: The Ibadi, an Islamic school distinct from Sunni and Shia, believe that leadership of the Muslim community is not something which should be decided by lineage, tribal affiliations or divine selection, but rather through election by leading Muslims. They do not view their leaders as infallible. In particular, if a leader fails to maintain a legitimate government in accordance with the Islamic law, it is the duty of the population to remove him from power. The Rashidun Caliphs are seen as rulers who were elected in a legitimate fashion and, in particular, Abu Bakr and Umar are viewed as righteous leaders. However, Uthman is viewed as having committed grave sins during the latter half of his rule and was deserving of death. Ali is also similarly understood to have lost his mandate.


title: Succession to Muhammad

text: The Ibadi have been labeled by some scholars as the "moderate Kharijites." Their first Imam was Abd Allah ibn Wahb al-Rasibi, who led the Kharijites after their withdrawal from Ali's camp. Other Imams include Abu Ubaidah Muslim, Abdallah ibn Yahya al-Kindi, and Umar ibn Abdul Aziz.


title: Slivovitz

text: Slivovitz is a fruit spirit (or fruit brandy) made from damson plums, often referred to as plum spirit (or plum brandy). Slivovitz is produced in Central, Eastern and Southern Europe, both commercially and privately. Primary producers include Bosnia and Herzegovina, Bulgaria, Croatia, Czech Republic, Greece, Hungary, Italy, North Macedonia, Poland, Romania, Serbia, Slovakia, Slovenia, and Ukraine. In the Balkans, slivovitz is considered a kind of "rakia". In Central Europe it is considered a kind of pálinka (Austria, Czech Republic, Hungary, Poland, Slovakia, Ukraine—pálenka, or Greece, Romania and Italy-pălincă), and similar to Romanian țuică, corresponding to the distilled spirits category.


title: Slivovitz

text: UNESCO put it in a UNESCO Intangible Cultural Heritage Lists in 2022 on request of the county of geographic origin Serbia.


title: Slivovitz

text: The word "slivovitz" derives from Proto-Slavic "plum" (, or "slíva", , /шљива, or ) with the diminutive suffix (; "-ice" in Czech). Distilled spirits from different fruits are named similarly. For example, Czech apricot → apricot spirit; peach → peach spirit. Other names include slivovitza, slivovitsa, șliboviță, šljivovica, śliwowica, Schlivowitz, slivovice, slivovica or slivovka.


title: Slivovitz

text: The primary producers are Bosnia, Bulgaria, Croatia, Czech Republic, Hungary, Poland, Romania, Serbia, and Slovakia.


title: Slivovitz

text: Following the claims of several nations to the protected designation of origin, in October 2007 the European Union went for a compromise solution, leaving "slivovitz" as a generic name, and granting individual nations the right to protect the origin with their own adjective.


title: Slivovitz

text: Identical or similar spirits are also produced in Austria, Canada, France, Germany, Switzerland, the United States, and regional names include generic "plum spirit", "Pflümli", or "eau-de-vie de quetsche".


title: Slivovitz

text: Only Damson plums may be used as the base fruit for distillation. Before the production process, the plums may be slightly pressed or otherwise disrupted to speed up the fermentation process, but without damaging their ground kernels; yeast, starch, and sugar may be added to the juice. The mixture is then allowed to ferment. There may be one or more distillation stages, depending on the desired final product or region of production, and aging is common to enhance the distillate's finer flavours.


title: Slivovitz

text: Some producers have obtained a Hechsher certifying that it is kosher for Passover, and thus suitable for consumption during the festival when grain-based liquors are forbidden.


title: Slivovitz

text: Some modern production techniques, such as those used by Clear Creek in Oregon, omit the use of the pits in fermentation to create a less acerbic or bitter taste.


title: Slivovitz

text: Imitation slivovitz is made by flavouring spirits with prune juice and artificial oil of bitter almonds.


title: Slivovitz

text: In Bulgaria, the "Troyan plum spirit" (Troyanska Slivova) has been distilled in the Troyan Monastery by the monks since the founding of the monastery in the 14th century. The original recipe included 40 herbs, and was passed through the centuries from abbot to abbot. In 1894, the Monastery's spirit was presented at a spirits competition in Antwerp, Belgium, where it won a bronze medal. Some of its distinguishing characteristics include that it is made of a plum variety that only grows in the Troyan region, an old variety the locals call "Madzharkini plums", very juicy and aromatic whose stone, unlike the Teteven plums, is easily removed from the fruit; it is distilled in a vessel with a capacity of 80 to 120 litres; and only the best of the harvest is distilled. It is considered best distilled to an alcohol level between 39 and 41 degrees. Some celebrities who have tasted Troyan plum spirit are Pope John Paul II and former US President Bill Clinton. Patriarch Maxim of Bulgaria celebrated his 95th birthday in 2009 in the Sofia Metropolitanate with Troyan plum spirit.


title: Slivovitz

text: In 2007, in cooperation with the Bulgarian government, the Czech distillery "Rudolf Jelinek" protected the brands "Troyanska slivova" and "Tetevenska slivova" in the EU. That same year, this distillery, the largest European manufacturer of fruit distillates, bought a majority share in the largest Bulgarian slivovitsa vinery "Vinprom-Troyan", after having bought, half a year earlier, the second largest "Destila Teteven". However, the Czechs reduced the alcohol content to pay less duty. The production of "Vinprom-Troyan" is mainly for export. For the past 18 years, Troyan has had a special holiday: "The Festival of Plum". This holiday is celebrated at the end of September in Troyan and in the village of Oreshak, where the Troyan Monastery is located. The plum has always been an essential produce in this region. Since the beginning of the 20th century, plums have been made into marmalades, pesto, dried prunes, and pulps, all of which were exported into Western Europe.


title: Slivovitz

text: In the Czech Republic, Hungary and Slovakia "slivovice", or " slivovica", respectively (as well as other fruit-based distilled alcoholic drinks) due to the somewhat symbolic status of the Moravian "national" drink is strongly presented in local traditions, culture and pop culture like in proverbs, songs, TV shows and movies.


title: Slivovitz

text: "Slivovice" is primarily produced in the southern and eastern provinces of Moravia and in Vysočina, where the country retains its rural character. Although not legal, traditional moonshine home distilleries still exist, nonetheless, the majority of private production moved to certified local community-owned distilleries to prevent errors during the distillation process (leading to concentration of toxic methanol). Certified production also allows state authorities to collect respective taxes (based on the proof of the alcohol in the final product), however tax-reliefs for limited private and non-commercial production of the drink exist. The usual proof of private-produced "slivovice" is over 50% of alcohol in the final product, commercially available mass-produced slivovice is proofed less.


title: Slivovitz

text: "Slivovice" is mostly served in small shot glass known locally as ""panák"" (literally: a dummy), ""kalíšek" (colloquial for a small cup) or ""štamprle" (from German ""das Stamperl", a little glass). Keeping slivovice cold helps to reduce the effects of high proof; however, the drink is almost never served on the rocks. To enjoy the aroma and taste of the original fruit it is better and more common to drink "slivovice" served at room temperature.


title: Slivovitz

text: The mountainous region of southern Poland has a long tradition of making slivovitz. Historically it has been distilled by local Goral highlanders and not necessarily given any particular brand names. One of the more recognised of such products is Śliwowica łącka associated with the town of Łącko. However, due to the fact that it is made without any specific control and brand ownership it has declined in quality, especially in recent years.


title: Slivovitz

text: Slivovitz was also distilled in large quantities by the Jewish community of Poland before the Second World War. As a popular Passover alcohol, slivovitz had a strong standing among the traditional Jewish-Orthodox communities. State-owned distillers tried to reintroduce the slivovitz based on Jewish traditions in the form of "Pejsachówka". After 1989, Pejsachówka had disappeared from the market. Polish distillers continue to produce Śliwowica Paschalna and Śliwowica Strykowska (Strykover Slivovitz), which are both certified as kosher.


title: Slivovitz

text: Šljivovica (, ) is the national drink of Serbia in domestic production for centuries, and the plum is the national fruit. Šljivovica has a Protected Designation of Origin (PDO).


title: Slivovitz

text: Plum and its products are of great importance to Serbs and are a part of numerous traditional customs. A Serbian meal sometimes starts or ends with plum products and šljivovica is served as an apéritif. A saying goes that the best place to build a house is where a plum tree grows the best. Traditionally, šljivovica (commonly referred to as "rakija") is connected to a Serbian culture as a drink used at all important rites of passage (birth, baptism, military service, marriage, death, etc.). It is used in the Serbian Orthodox patron saint celebration, Slava. It is used in numerous folk remedies, and is given certain degree of respect above all other alcoholic drinks. The fertile region of Šumadija in central Serbia is particularly known for its plums and šljivovica. In 2004, over 400 000 litres of šljivovica was produced in Serbia. In 2017, Serbia was the 5th largest producer of plums and sloes in the world.


title: Slivovitz

text: After legal disputes, following the compromise of 2007 "Serbian Slivovitz" ( / "Srpska šljivovica") became Serbia's first certified national brand.


title: Slivovitz

text: Šljivovica is consumed either directly from a leather-wrapped round bottle, or chilled in a shot glass called "čokanjčić" (plural: "čokanjčići"). There is also a town in Zlatibor, called Šljivovica. A popular print in Serbia depicts a moustached peasant wearing the "šubara" (fur hat), drinking šljivovica from a leather-wrapped bottle, with the motto: ""Fuck the Coca, fuck the pizza, all we need is šljivovica"".


title: Slivovitz

text: In 2021, Serbia's sljivovica was added to the United Nations Intangible Cultural Heritage List as a "cherished tradition to be preserved by humanity".


title: The Phantom Tollbooth

text: The Phantom Tollbooth is a children's fantasy adventure novel written by Norton Juster, with illustrations by Jules Feiffer, first published in 1961. The story follows a bored young boy named Milo who unexpectedly receives a magic tollbooth that transports him to the once prosperous, but now troubled, Kingdom of Wisdom. Along with a dog named Tock and the Humbug, Milo goes on a quest to the Castle in the Air seeking the kingdom's two exiled princesses, named Rhyme and Reason. As Milo learns valuable lessons, he finds a love of learning in a story full of puns and wordplay, such as exploring the literal meanings of idioms.


title: The Phantom Tollbooth

text: In 1958, Juster had received a Ford Foundation grant for a children's book about cities. Unable to make progress on that project, he turned to writing what became "The Phantom Tollbooth", his first book. His housemate, Feiffer, a cartoonist, interested himself in the project. Jason Epstein, an editor at Random House, bought the book and published it. "The Phantom Tollbooth" received rave reviews and has sold in excess of three million copies, far more than expected. It has been adapted into a film, opera, and play, and translated into many languages.


title: The Phantom Tollbooth

text: Though the book is on its face an adventure story, a major theme is the need for a love of education; through this, Milo applies what he has learned in school, advances in his personal development, and learns to love the life that previously bored him. Critics have compared its appeal to that of Lewis Carroll's "Alice's Adventures in Wonderland" and to L. Frank Baum's "The Wonderful Wizard of Oz". Additionally Maurice Sendak, in his introductory "An Appreciation" included in editions of the book since 1996, quotes a critic as comparing "The Phantom Tollbooth" to Bunyan's "Pilgrim's Progress": "As "Pilgrim’s Progress" is concerned with the awakening of the sluggardly spirit, "The Phantom Tollbooth" is concerned with the awakening of the lazy mind."


title: The Phantom Tollbooth

text: Milo is a boy bored by the world around him; every activity seems a waste of time. He arrives home from another boring day at school to find a mysterious package. Among its contents are a small tollbooth and a map of "the Lands Beyond," illustrating the Kingdom of Wisdom (which will also guide the reader from its place on the endpapers of the book). Attached to the package is a note "For Milo, who has plenty of time." Warned by an included sign to have his destination in mind, he decides without much thought to go to Dictionopolis, assuming this is a pretend game to be played on the floor of his room. He maneuvers through the tollbooth in his electric toy car, and instantly finds himself driving on a road that is clearly not in his city apartment.


title: The Phantom Tollbooth

text: Milo begins with Expectations, a pleasant place where he starts on Wisdom's road. In Expectations, he seeks directions from the Whether Man, who is full of endless talk. As Milo drives on, he daydreams and gets lost in the Doldrums, a colorless place where nothing ever happens. Milo soon joins the inhabitants, the Lethargarians, in killing time there, a pastime angrily interrupted by the arrival of Tock, a talking, oversized dog with an alarm clock on each side (a "watchdog"), who tells Milo that only by thinking can he get out of the Doldrums. Head abuzz with unaccustomed thoughts, Milo is soon back on his road, and Tock joins him on his journey through Wisdom.


title: The Phantom Tollbooth

text: Milo and Tock travel to Dictionopolis, one of two capital cities of the divided Kingdom of Wisdom, and home to King Azaz the Unabridged. They meet King Azaz's cabinet officials and visit the Word Market, where the words and letters are sold that empower the world. A fight between the Spelling Bee and the blustering Humbug breaks up the market, and Milo and Tock are arrested by the very short Officer Shrift. In prison, Milo meets Faintly Macabre, the Not so Wicked Which (not to be confused with Witch), long in charge of which words should be used in Wisdom. She tells him how the two rulers, King Azaz and his brother, the Mathemagician, had two adopted younger sisters, Rhyme and Reason, to whom everyone came to settle disputes. All lived in harmony until the rulers disagreed with the princesses' decision that letters (championed by Azaz) and numbers (by the Mathemagician) were equally important. They banished the princesses to the Castle in the Air, and since then, the land has had neither Rhyme nor Reason.


title: The Phantom Tollbooth

text: Milo and Tock leave the dungeon. King Azaz hosts them at a banquet where the guests literally eat their words, served to them on plates. After the meal, King Azaz lets Milo and Tock talk themselves into a dangerous quest to rescue the princesses. Azaz flatters the Humbug into being their guide, and boy, dog and insect set off for the Mathemagician's capital of Digitopolis as they must gain his approval before they can begin their quest.


title: The Phantom Tollbooth

text: Along the way, they meet such characters as Alec Bings, a little boy suspended in the air who sees through things and who will grow down until he reaches the ground. He then meets the world's smallest giant, the world's biggest dwarf, the world's thinnest fat man, and the world's fattest thin man, who turn out to just be one regular man. Milo then loses time in substituting for Chroma the Great, a conductor whose orchestra creates the colors of the world.


title: The Phantom Tollbooth

text: They meet a twelve-sided creature called the Dodecahedron, who leads them to Digitopolis. There, they meet the Mathemagician, who is still angry at Azaz and who will not give his blessing to anything that his brother has approved. Milo maneuvers him into saying he will permit the quest if the boy can show the two have concurred on anything since they banished the princesses. To the number wizard's shock, Milo proves that the two have agreed to disagree, and the Mathemagician gives his reluctant consent.


title: The Phantom Tollbooth

text: In the Mountains of Ignorance, the journeyers contend with demons like the Terrible Trivium and the Gelatinous Giant. After overcoming obstacles and their own fears, they reach the Castle in the Air. Princesses Rhyme and Reason welcome Milo and agree to return to Wisdom. Unable to enter the castle, the demons cut it loose, letting it drift away, but Milo realizes Tock can carry them down because 'time flies'. The demons pursue, but the armies of Wisdom arrive and repel them. Rhyme and Reason heal the divisions in the old Kingdom of Wisdom, Azaz and the Mathemagician are reconciled, and all enjoy a three-day celebration.


title: The Phantom Tollbooth

text: Milo says goodbye and drives back through the tollbooth. Suddenly he is back in his own room, and discovers he has been gone only an hour, though his journey seemed to take weeks. He awakens the next day with plans to return to the kingdom, but finds the tollbooth gone when he gets home from school. A note instead is there, "For Milo, who now knows the way." The note states that the tollbooth is being sent to another child who needs help finding direction in life. Milo is somewhat disappointed but agrees and looks at a now-interesting world around him, concluding that even if he found a way back, he might not have time to go, for there is so much to do right where he is.


title: The Phantom Tollbooth

text: Architect Norton Juster was living in his hometown of Brooklyn, after three years in the Navy. In June 1960, he gained a $5,000 grant from the Ford Foundation to write a children's book about cities. Juster argued that the young baby boomers would soon have responsibility for the cities, and many lived in the suburbs and did not know them. In his proposal, he said he wanted "to stimulate and heighten perception – to help children notice and appreciate the visual world around them – to help excite them and shape their interest in an environment they will eventually reshape." Beginning with great enthusiasm, he ground to a halt with too many notes and too little progress. He took a weekend break with friends at Fire Island, and came back determined to put aside the cities book and seek inspiration in another writing project.


title: The Phantom Tollbooth

text: Juster's guilt over his lack of progress on the cities book had led him to write pieces of stories about a little boy named Milo, which he began to develop into a book.


title: The Phantom Tollbooth

text: Juster quit his job so that he could work on the book. His imagination fired by a boy who approached him on the street and with whom he discussed the nature of infinity, Juster wanted to finish the story about "a boy who asked too many questions" before returning to the book on cities. Juster shared his house in Brooklyn Heights with cartoonist Jules Feiffer whose bedroom was immediately below, and who could hear him pacing in the night. Feiffer was surprised to learn that his friend's insomnia was not caused by the cities book, but by a book about a boy. Juster showed Feiffer the draft to date and, unbidden, the artist began sketching illustrations. Feiffer knew Judy Sheftel, who put deals together in the publishing trade and was his future bride. Sheftel got Jason Epstein, an innovative editor at Random House with a deep appreciation for children's literature, to agree to review the manuscript. Some at Random House considered the book's vocabulary too difficult: at the time, educators advised against children's literature containing words the target audience did not already know, fearing the unfamiliar would discourage young learners. Based on seven chapters of manuscript, plus a three-page outline of the rest of the story, Epstein bought the book.


title: The Phantom Tollbooth

text: Since Juster did the cooking for the housemates, if Feiffer wanted to eat, he had to do the drawings. Feiffer quickly realized the book would require illustrations of the type and quality that John Tenniel had created for Lewis Carroll's "Alice's Adventures in Wonderland" and although a nationally known artist, doubted his competence to do the text justice. Feiffer considers the double-spread illustration of demons late in the book to be a success and one of his favorites. It differs from his usual style (which would involve a white background), and instead uses Gustave Doré's drawings as an inspiration.


title: The Phantom Tollbooth

text: It became a game, with Feiffer trying to draw things the way he wanted, and Juster trying to describe things that were impossible to sketch. These included the Triple Demons of Compromise—one short and fat, one tall and thin, and the third exactly like the first two. Feiffer got his revenge by depicting the author as the Whether Man, clad in a toga.


title: The Phantom Tollbooth

text: Repeated edits altered the protagonist's name (originally Tony), removed his parents entirely from the book, and deleted text attempting to describe how the tollbooth package had been delivered. Milo's age was removed from the text—early drafts have him aged eight or nine—as Juster decided not to state it, lest potential readers decide they were too old to care.


title: The Phantom Tollbooth

text: Since no one has ever bothered to explain the importance of learning to Milo, he regards school as the biggest waste of time in his life. Juster intended that the book speak to the importance of learning to love learning. Teaching methods that might bore children, by memorization for example, are mocked in the book, as in the case of the Spelling Bee. Like the Bee, the Humbug's insult to his fellow insect goes over Milo's head, but possibly not the reader's: "A slavish concern for the composition of words is the sign of a bankrupt intellect." According to Mary Liston in her journal article on law in fantasy realms, ""The Phantom Tollbooth" concerns the difference between education and wisdom and what processes are conducive to synthesizing both, so as to encourage an attitude of engagement, alertness, and responsibility within an increasingly autonomous individual."


title: The Phantom Tollbooth

text: Another theme is the need for common sense to back up rules. Milo journeys through a land where, without Rhyme or Reason, the art of governance has been lost, leading to bizarre results. Milo repeatedly meets characters to whom words are more important than their meaning. The Whether Man, for all his talk, is unable to provide Milo with the information or guidance the boy wants, while Officer Shrift's investigation of the overturning of the Word Market contains the forms of law, without justice. The denizens around Digitopolis are little better; the twelve-faced Dodecahedron, named for what he is, turns the logic of his naming on its head when he asks if everyone with one face is called Milo. The attitudes now displayed by the adherents of both brothers are summed up by the Dodecahedron, "as long as the answer is right, who cares if the question is wrong?"


title: The Phantom Tollbooth

text: As Milo struggles with words and begins the process of making himself their master, he also has difficulty with numbers, especially when he speaks with .58 of a child, who with parents and two siblings (whom Milo does not meet) makes up an average family. Milo has had difficulties in school with mathematics and problem solving; his reaction to this encounter is to protest that averages are not real. The partial child enlightens Milo that there is beauty in math beyond the tedium of learning an endless set of rules, "one of the nicest things about mathematics, or anything else you might care to learn, is that many of the things which can never be, often are". Late in the book, Princess Reason counsels Milo, who has much learning ahead of him, not to be discouraged by its complexity, "You must never feel badly about making mistakes ... as long as you take the trouble to learn from them. For you often learn more by being wrong for the right reasons than you do by being right for the wrong reasons". An index card in the Juster papers sets forth the germ of the princess' "memorable" counsel to Milo, "Quite often the road to Rhyme + Reason is through the right mistakes."


title: The Phantom Tollbooth

text: Although Milo is bored with learning, that does not mean he knew nothing before his journey. He exhibits characteristics of a well-schooled child of his time; his speech is polite and peppered with "please" and "thank you", and when he unexpectedly encounters the partial child, he requests pardon for staring. He can count to a thousand, though his tact in bringing up the subject in the presence of the numbers-hating King Azaz is not the best. Mindful of his mother's admonition to eat lightly when a guest, he initially orders a light meal at the banquet, only to find the waiters bringing in insubstantial light beams. Not realizing he will be asked to eat his words, he then makes a plausible start at a speech before being interrupted by the king.


title: The Phantom Tollbooth

text: "The Phantom Tollbooth" displays Milo's growth; Leonard S. Marcus in his notes to its annotated edition writes that the boy learns to think in the abstract, pledging after his unintentional jump to Conclusions that he will not make up his mind again without a good reason. Milo does not accept the word of the demon of insincerity that he is a threat and is rewarded by learning he is not. Just for a moment, Milo is able to float in the air beside Alec Bings and see things from the perspective he will have as an adult, allowing the young reader to contemplate what it will be like to do the same. According to Liston, Milo "transforms himself from an unthinking and compliant Lethargarian to a young adult with greater consciousness, a firmer sense of self, and a newly found set of responsibilities".


title: The Phantom Tollbooth

text: Even though the day is won by Milo and his fellow questers, it is a great but not a permanent victory, as he hears the kingly brothers begin to argue again as he departs. Juster has written that it was his intent to get Milo out of there as quickly as possible, and that "the fight would have to be won again and again".


title: The Phantom Tollbooth

text: Milo's trip through the lands beyond the tollbooth teaches the previously bored child to appreciate both journey and destination. This is a lesson that had been unlearned by the citizens of Wisdom, as exemplified by the described fate of the twin cities of Reality and Illusions. Although the city of Illusions never actually existed, Reality was lost as its residents concentrated on getting to their destination as quickly as possible, and, unappreciated, the city withered away, unnoticed by the busy people who still hasten along its former streets. Milo meets his trials by defining himself as different from the kingdom's inhabitants, who either demand or accept conformity, as enforced by the kingdom's laws, which discourage (and even outlaw) thought. Milo cannot accept such laws, beginning when, in the Doldrums, he thinks, thus violating a local ordinance and separating himself from the thoughtless inhabitants. Liston opined that because the Kingdom of Wisdom's "laws require the impossible, they contradict what it means to be fully human".


title: The Phantom Tollbooth

text: "The Phantom Tollbooth" contains allusions to many works, including those loved by Juster in his own upbringing. Some of Juster's favorite books as a child, including "The Wind in the Willows", had endpaper maps; Juster insisted on one, over Feiffer's opposition, going so far as to sketch one and require that his collaborator reproduce it in his own style. Juster was also inspired by his father Samuel's love of puns, with which the book is more than sprinkled. In his childhood, Juster spent much time listening to the radio. According to Juster, the need to envision the action when listening to radio serials helped inspire "The Phantom Tollbooth", as well as yielding the character of Tock, based on sidekick Jim Fairfield from "Jack Armstrong, the All-American Boy". Jim gave Tock his wisdom, courage, and adventurous spirit. As a child, Juster had synesthesia, and could only do arithmetic by making associations between numbers and colors. He remembered that the condition affected word associations. "One of the things I always did was think literally when I heard words. On the "Lone Ranger" [radio serial] they would say, 'Here come the Injuns!' and I always had an image of engines, of train engines."


title: The Phantom Tollbooth

text: Some of the incidents in the book stem from Juster's own past. In Digitopolis, the Numbers Mine, where gemlike numerals are dug for, recalled one of Juster's architecture professors at the University of Pennsylvania, who compared numbers and equations to jewels. The Marx Brothers films were a staple for Juster as a child and his father would quote lengthy passages from the movies; this inspired the unending series of straight-faced puns that fills the book.


title: The Phantom Tollbooth

text: Growing up in a Jewish-American household where the parents demanded high achievement, Juster was intimately familiar with expectations, though in his case many of his parents' hopes were centered on his older brother, an academic star. The Terrible Trivium, the well-dressed, polite demon who sets the questers to mindless tasks, was Juster's way of representing his own tendency to avoid what he should be doing in favor of a more congenial occupation, such as his evasion of the grant project to write "The Phantom Tollbooth". Juster drew on Feiffer's life experiences as well; the Whether Man's adage "Expect everything, I always say, and the unexpected never happens" was a favorite of the cartoonist's mother.


title: The Phantom Tollbooth

text: Juster had not read "Alice's Adventures in Wonderland" when he wrote "The Phantom Tollbooth", but the two books, each about a bored child plunged into a world of absurd logic, have repeatedly been compared. According to Daniel Hahn in his 2012 article on the Juster book, "Alice is clearly Milo's closest literary kin". Milo's conversation with the Whether Man, which leaves him no more comprehending than when he came, recalls that of Alice with the Cheshire Cat. The questions of authority (something omnipresent for a child) and of justice run through both books; the Queen of Hearts' arbitrary justice is echoed, though with less violence, by Officer Shrift. Alice's sovereigns, representing the authority figures of Victorian childhood life, rule absolutely (though not necessarily effectively); a child of the post-World War II world, Milo journeys through a more bureaucratic realm. His quest is far more purposeful than the frustrating journey Alice experiences, and the outcome differs as well—Milo restores his kingdom while Alice overturns hers. Carroll leaves us uncertain if Alice has learned anything from her adventures, but Juster makes it clear that Milo has acquired tools he will need to find his way through life.


title: The Phantom Tollbooth

text: "The Phantom Tollbooth" was published in September 1961. Its competition among new books for the minds and hearts of children included Roald Dahl's "James and the Giant Peach". "The Bronze Bow", set in the Biblical times, was newly available, and would bring Elizabeth George Speare her second Newbery Award in three years. Neither publisher nor first-time author expected many sales for "The Phantom Tollbooth", but Juster was nevertheless disappointed not to find his work on store shelves. His mother, Minnie, did her part, as her son put it, "terrorizing" bookstore owners into displaying it.


title: The Phantom Tollbooth

text: Juster says the book was rescued from the remainders table when Emily Maxwell wrote a strong review of it in "The New Yorker". Maxwell wrote, "As "Pilgrim's Progress" is concerned with the awakening of the sluggardly spirit, "The Phantom Tollbooth" is concerned with the awakening of the lazy mind." Hers was far from the only positive piece; children's author Ann McGovern reviewed it for "The New York Times", writing "Norton Juster's amazing fantasy has something wonderful for anyone old enough to relish the allegorical wisdom of "Alice in Wonderland" and the pointed whimsy of "The Wizard of Oz" ". John Crosby wrote for the "New York Herald Tribune", "In a world which sometimes seems to have gone mad, it is refreshing to pause and consider for a moment a book for children which contains a character called 'Faintly Macabre, the not-so-wicked Which.' The name of the book is "The Phantom Tollbooth" and it was written by a bearded elf named Norton Juster and illustrated by Jules Feiffer, who is the cleverest of the young neurotics". Dissenting was the "Bulletin of the Center for Children's Books", which in March 1962 deemed the book an "intensive and extensive fantasy, heavily burdened with contrivance and whimsy".


title: The Phantom Tollbooth

text: In 1962, the book was published in Britain. Siriol Hugh-Jones wrote for "The Times Literary Supplement", ""The Phantom Tollbooth" is something every adult seems sure will turn into a modern "Alice" ... The obvious guess is that the appeal of this sort of writing is directed towards just the sort of adults who derive a perfectly grown-up pleasure from regularly rereading the "Alices". As one might expect, it is illustrated by every grown-up's favourite child-like pictures with the built-in sad sophistication, the work of Jules Feiffer." Jennifer Bourdillon reviewed it for "The Listener", "This is the story of an imaginary journey, a sort of "Pilgrim's Progress" of a little boy in his car ... One would hardly have thought from the sound of this that it would have so magnetic an appeal, but the brilliant verbal humour and the weird and wonderful characters (the Dodecahedron. the Watchdog, Faintly Macabre) make it that rare delight, a book which parents and children can share." It reached Australia in 1963; "The Canberra Times"' reviewer, J.E.B., deemed it memorable, causing readers to quote from it and leaf through its pages again.


title: The Phantom Tollbooth

text: After publication, Juster sent a copy of the book to the Ford Foundation, with an explanation of how the projected book on cities had transformed into "The Phantom Tollbooth". He never heard back from them, and learned years later that they were delighted by the turn of events. With the book having become an unexpected hit, Juster found himself answering letters from young readers, and a few parents. He found that children understood the wordplay at different ages, and heard from the occasional college student as well. Some students wrote a second time after a gap of years "and they'll talk to me about a whole different book, normally. But now they've got a lot more of the words right. A lot more of the fun kind of crazy references". He learned too that readers were capable of more than he had intended, as in the case of the letter sent by the Mathemagician to King Azaz. Composed entirely of numbers, some readers assumed it was a code and set about breaking it, only to appeal to Juster for help when they were not successful. The numbers were not intended to have any meaning, and were meant to convey that the Mathemagician's letter could not have been understood by Azaz or his advisers.


title: The Phantom Tollbooth

text: As the book became acclaimed as a modern classic, it began to be used in the classroom, and Juster corresponded with some teachers. After the book's readers attained adulthood, they wrote of its influence on them. Novelist Cathleen Schine recalled, "it was as if someone had turned on the lights. The concepts of irony, of double entendre, of words as play, of the pleasure and inevitability of intellectual absurdity, were suddenly accessible to me. They made sense to me in an extremely personal way." British fantasy writer Diana Wynne Jones read her copy so often it fell apart: "it didn't occur to us that it might be about something. It struck us as a little like "The Wizard of Oz", only better." One reader, signing himself "Milo", wrote to "Rolling Stone" in 1970, "If you want to get freaked out of your undernourished head, pick up "The Phantom Tollbooth", by Norton Juster. They tell you it's a kids' book, but take my word for it, no one who reads it is ever the same. No hype."


title: The Phantom Tollbooth

text: The book continued to garner positive reviews and comments. In 1998, Amanda Foreman wrote for "The Sunday Times" of London, "I want to shout about "The Phantom" from the rooftops. I want to stand in Waterloo and press copies into people's hands. This is a book that should be in every home. ... Whether you are 8 or 88 Juster's mixture of allegorical wisdom and logical whimsy will take you on a journey of the spirit. "The Phantom" is a "mappa mundi" of our hearts, proving once again that in laughter and simplicity lies the truth of life". In a 2011 article written for the book's fiftieth anniversary, Adam Gopnik wrote, "The book is made magical by Juster's and Feiffer's gift for transforming abstract philosophical ideas into unforgettable images."


title: The Phantom Tollbooth

text: The book has been translated into many foreign languages, including Chinese, Croatian, German, Hebrew, Japanese, Korean, Russian, Thai, Turkish and three different Spanish editions: one for Spain, one for Latin America, and one for Spanish speakers in the United States. Juster stated that he did not know if the wordplay of the original carries through to the translated works. In 1969, Chuck Jones made "The Phantom Tollbooth" into a musical film of the same name, produced and released by Metro-Goldwyn-Mayer in 1970. Milo's town and room were depicted in live-action, and with the film changing to animation beyond the tollbooth. Juster disliked the film, describing it as "drivel". In February 2010, director Gary Ross began development of a remake of "The Phantom Tollbooth", with the first draft of the script written by Alex Tse. As of August 2016, the remake has moved to TriStar Pictures, with Michael Vukadinovich writing the adaption. The classical composer Robert Xavier Rodriguez in 1987 composed "A Colorful Symphony" based on the work.


title: The Phantom Tollbooth

text: In 2011, "The Annotated Phantom Tollbooth" was published, which includes sketches and copies of Juster's handwritten drafts and word lists, Feiffer's early drawings, and an introduction and annotations by Leonard S. Marcus. A fiftieth anniversary edition was also published, with appreciations by Maurice Sendak, Michael Chabon and Philip Pullman. More than three million copies have been sold of the original book in the U.S. alone. It has been adapted into a small-scale opera with music by Arnold Black, and a book by Juster and Sheldon Harnick, produced by Opera Delaware in 1995. It was then revamped into a musical that had its debut to strong reviews at the Kennedy Center in Washington, D.C., and then made a national tour.


title: The Phantom Tollbooth

text: "The Phantom Tollbooth" remains acknowledged as a classic of children's literature. Based on a 2007 online poll, the U.S. National Education Association listed it as one of the "Teachers' Top 100 Books for Children". In 2012, it was ranked number 21 among all-time children's novels in a survey published by "School Library Journal".


title: The Phantom Tollbooth

text: In 2017, TriStar Pictures announced that Matt Shakman would direct its upcoming "live-action/hybrid" film adaptation of "The Phantom Tollbooth". In 2018, Carlos Saldanha replaced Shakman due to scheduling conflicts. Theodore Melfi is drafting the screenplay after Michael Vukadinovich and Phil Johnston.


title: The Phantom Tollbooth

text: In 2011, Juster stated that he felt that his book still had relevance, even though children's lives had changed since 1961:


title: Agonist

text: An agonist is a chemical that activates a receptor to produce a biological response. Receptors are cellular proteins whose activation causes the cell to modify what it is currently doing. In contrast, an antagonist blocks the action of the agonist, while an inverse agonist causes an action opposite to that of the agonist.


title: Agonist

text: From the Greek αγωνιστής (agōnistēs), contestant; champion; rival < αγων (agōn), contest, combat; exertion, struggle < αγω (agō), I lead, lead towards, conduct; drive


title: Agonist

text: Receptors can be activated by either endogenous agonists (such as hormones and neurotransmitters) or exogenous agonists (such as drugs), resulting in a biological response. A physiological agonist is a substance that creates the same bodily responses but does not bind to the same receptor.


title: Agonist

text: New findings that broaden the conventional definition of pharmacology demonstrate that ligands can concurrently behave as agonist "and" antagonists at the same receptor, depending on effector pathways or tissue type. Terms that describe this phenomenon are "functional selectivity", "protean agonism", or selective receptor modulators.


title: Agonist

text: As mentioned above, agonists have the potential to bind in different locations and in different ways depending on the type of agonist and the type of receptor. The process of binding is unique to the receptor-agonist relationship, but binding induces a conformational change and activates the receptor. This conformational change is often the result of small changes in charge or changes in protein folding when the agonist is bound. Two examples that demonstrate this process are the muscarinic acetylcholine receptor and NMDA receptor and their respective agonists.


title: Agonist

text: For the muscarinic acetylcholine receptor, which is a G protein-coupled receptor(GPCR), the endogenous agonist is acetylcholine. The binding of this neurotransmitter causes the conformational changes that propagate a signal into the cell. The conformational changes are the primary effect of the agonist, and are related to the agonist's binding affinity and agonist efficacy. Other agonists that bind to this receptor will fall under one of the different categories of agonist mentioned above based on their specific binding affinity and efficacy.


title: Agonist

text: The NMDA receptor is an example of an alternate mechanism of action, as the NMDA receptor requires co-agonists for activation. Rather than simply requiring a single specific agonist, the NMDA receptor requires both the endogenous agonists, N-methyl-D-aspartate (NMDA) and glycine. These co-agonists are both required to induce the conformational change needed for the NMDA receptor to allow flow through the ion channel, in this case calcium. An aspect demonstrated by the NMDA receptor is that the mechanism or response of agonists can be blocked by a variety of chemical and biological factors. NMDA receptors specifically are blocked by a magnesium ion unless the cell is also experiencing depolarization.


title: Agonist

text: These differences show that agonists have unique mechanisms of action depending on the receptor activated and the response needed. The goal and process remains generally consistent however, with the primary mechanism of action requiring the binding of the agonist and the subsequent changes in conformation to cause the desired response at the receptor. This response as discussed above can vary from allowing flow of ions to activating a GPCR and transmitting a signal into the cell.


title: Agonist

text: Potency is the amount of agonist needed to elicit a desired response. The potency of an agonist is inversely related to its EC value. The EC can be measured for a given agonist by determining the concentration of agonist needed to elicit half of the maximum biological response of the agonist. The EC value is useful for comparing the potency of drugs with similar efficacies producing physiologically similar effects. The smaller the EC value, the greater the potency of the agonist, the lower the concentration of drug that is required to elicit the maximum biological response.


title: Agonist

text: When a drug is used therapeutically, it is important to understand the margin of safety that exists between the dose needed for the desired effect and the dose that produces unwanted and possibly dangerous side-effects (measured by the TD, the dose that produces toxicity in 50% of individuals). This relationship, termed the therapeutic index, is defined as the ratio TD:ED. In general, the narrower this margin, the more likely it is that the drug will produce unwanted effects. The therapeutic index emphasizes the importance of the margin of safety, as distinct from the potency, in determining the usefulness of a drug.


title: Algebraic geometry

text: Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.


title: Algebraic geometry

text: The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.


title: Algebraic geometry

text: Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.


title: Algebraic geometry

text: Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach.


title: Algebraic geometry

text: In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R could be defined as the set of all points ("x","y","z") with


title: Algebraic geometry

text: A "slanted" circle in R can be defined as the set of all points ("x","y","z") which satisfy the two polynomial equations


title: Algebraic geometry

text: First we start with a field "k". In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that "k" is algebraically closed. We consider the affine space of dimension "n" over "k", denoted A("k") (or more simply A, when "k" is clear from the context). When one fixes a coordinate system, one may identify A("k") with "k". The purpose of not working with "k" is to emphasize that one "forgets" the vector space structure that "k" carries.


title: Algebraic geometry

text: A function "f" : A → A is said to be "polynomial" (or "regular") if it can be written as a polynomial, that is, if there is a polynomial "p" in "k"["x"...,"x"] such that "f"("M") = "p"("t"...,"t") for every point "M" with coordinates ("t"...,"t") in A. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system in A.


title: Algebraic geometry

text: When a coordinate system is chosen, the regular functions on the affine "n"-space may be identified with the ring of polynomial functions in "n" variables over "k". Therefore, the set of the regular functions on A is a ring, which is denoted "k"[A].


title: Algebraic geometry

text: We say that a polynomial "vanishes" at a point if evaluating it at that point gives zero. Let "S" be a set of polynomials in "k"[A]. The "vanishing set of S" (or "vanishing locus" or "zero set") is the set "V"("S") of all points in A where every polynomial in "S" vanishes. Symbolically,


title: Algebraic geometry

text: A subset of A which is "V"("S"), for some "S", is called an "algebraic set". The "V" stands for "variety" (a specific type of algebraic set to be defined below).


title: Algebraic geometry

text: Given a subset "U" of A, can one recover the set of polynomials which generate it? If "U" is "any" subset of A, define "I"("U") to be the set of all polynomials whose vanishing set contains "U". The "I" stands for ideal: if two polynomials "f" and "g" both vanish on "U", then "f"+"g" vanishes on "U", and if "h" is any polynomial, then "hf" vanishes on "U", so "I"("U") is always an ideal of the polynomial ring "k"[A].


title: Algebraic geometry

text: The answer to the first question is provided by introducing the Zariski topology, a topology on A whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of "k"[A]. Then "U" = "V"("I"("U")) if and only if "U" is an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that "I"("V"("S")) is the radical of the ideal generated by "S". In more abstract language, there is a Galois connection, giving rise to two closure operators; they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection.


title: Algebraic geometry

text: For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set "U". Hilbert's basis theorem implies that ideals in "k"[A] are always finitely generated.


title: Algebraic geometry

text: An algebraic set is called "irreducible" if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the "irreducible components" of the algebraic set. An irreducible algebraic set is also called a "variety". It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring.


title: Algebraic geometry

text: Some authors do not make a clear distinction between algebraic sets and varieties and use "irreducible variety" to make the distinction when needed.


title: Algebraic geometry

text: Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called "regular functions" or "polynomial functions". A regular function on an algebraic set "V" contained in A is the restriction to "V" of a regular function on A. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even analytic.


title: Algebraic geometry

text: It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.


title: Algebraic geometry

text: Just as with the regular functions on affine space, the regular functions on "V" form a ring, which we denote by "k"["V"]. This ring is called the "coordinate ring of V".


title: Algebraic geometry

text: Since regular functions on V come from regular functions on A, there is a relationship between the coordinate rings. Specifically, if a regular function on "V" is the restriction of two functions "f" and "g" in "k"[A], then "f" − "g" is a polynomial function which is null on "V" and thus belongs to "I"("V"). Thus "k"["V"] may be identified with "k"[A]/"I"("V").


title: Algebraic geometry

text: Using regular functions from an affine variety to A, we can define regular maps from one affine variety to another. First we will define a regular map from a variety into affine space: Let "V" be a variety contained in A. Choose "m" regular functions on "V", and call them "f", ..., "f". We define a "regular map" "f" from "V" to A by letting . In other words, each "f" determines one coordinate of the range of "f".


title: Algebraic geometry

text: If "V"′ is a variety contained in A, we say that "f" is a "regular map" from "V" to "V"′ if the range of "f" is contained in "V"′.


title: Algebraic geometry

text: The regular maps are also called "morphisms", as they make the collection of all affine algebraic sets into a category, where the objects are the affine algebraic sets and the morphisms are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets.


title: Algebraic geometry

text: Given a regular map "g" from "V" to "V"′ and a regular function "f" of "k"["V"′], then . The map is a ring homomorphism from "k"["V"′] to "k"["V"]. Conversely, every ring homomorphism from "k"["V"′] to "k"["V"] defines a regular map from "V" to "V"′. This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced "k"-algebras. This equivalence is one of the starting points of scheme theory.


title: Algebraic geometry

text: In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions.


title: Algebraic geometry

text: If "V" is an affine variety, its coordinate ring is an integral domain and has thus a field of fractions which is denoted "k"("V") and called the "field of the rational functions" on "V" or, shortly, the "function field" of "V". Its elements are the restrictions to "V" of the rational functions over the affine space containing "V". The domain of a rational function "f" is not "V" but the complement of the subvariety (a hypersurface) where the denominator of "f" vanishes.


title: Algebraic geometry

text: As with regular maps, one may define a "rational map" from a variety "V" to a variety "V"<nowiki>'</nowiki>. As with the regular maps, the rational maps from "V" to "V"<nowiki>'</nowiki> may be identified to the field homomorphisms from "k"("V"<nowiki>'</nowiki>) to "k"("V").


title: Algebraic geometry

text: Two affine varieties are "birationally equivalent" if there are two rational functions between them which are inverse one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic.


title: Algebraic geometry

text: An affine variety is a "rational variety" if it is birationally equivalent to an affine space. This means that the variety admits a "rational parameterization", that is a parametrization with rational functions. For example, the circle of equation formula_5 is a rational curve, as it has the parametric equation


title: Algebraic geometry

text: The problem of resolution of singularities is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular (see also smooth completion). It was solved in the affirmative in characteristic 0 by Heisuke Hironaka in 1964 and is yet unsolved in finite characteristic.


title: Algebraic geometry

text: Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the number "i", a root of the polynomial , projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet.


title: Algebraic geometry

text: To see how this might come about, consider the variety . If we draw it, we get a parabola. As "x" goes to positive infinity, the slope of the line from the origin to the point ("x", "x") also goes to positive infinity. As "x" goes to negative infinity, the slope of the same line goes to negative infinity.


title: Algebraic geometry

text: Compare this to the variety "V"("y" − "x"). This is a cubic curve. As "x" goes to positive infinity, the slope of the line from the origin to the point ("x", "x") goes to positive infinity just as before. But unlike before, as "x" goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behavior "at infinity" of "V"("y" − "x") is different from the behavior "at infinity" of "V"("y" − "x").


title: Algebraic geometry

text: The consideration of the "projective completion" of the two curves, which is their prolongation "at infinity" in the projective plane, allows us to quantify this difference: the point at infinity of the parabola is a regular point, whose tangent is the line at infinity, while the point at infinity of the cubic curve is a cusp. Also, both curves are rational, as they are parameterized by "x", and the Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular.


title: Algebraic geometry

text: Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry.


title: Algebraic geometry

text: Nowadays, the "projective space" P of dimension "n" is usually defined as the set of the lines passing through a point, considered as the origin, in the affine space of dimension , or equivalently to the set of the vector lines in a vector space of dimension . When a coordinate system has been chosen in the space of dimension , all the points of a line have the same set of coordinates, up to the multiplication by an element of "k". This defines the homogeneous coordinates of a point of P as a sequence of elements of the base field "k", defined up to the multiplication by a nonzero element of "k" (the same for the whole sequence).


title: Algebraic geometry

text: A polynomial in variables vanishes at all points of a line passing through the origin if and only if it is homogeneous. In this case, one says that the polynomial "vanishes" at the corresponding point of P. This allows us to define a "projective algebraic set" in P as the set , where a finite set of homogeneous polynomials vanishes. Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced homogeneous ideals which define them. The "projective varieties" are the projective algebraic sets whose defining ideal is prime. In other words, a projective variety is a projective algebraic set, whose homogeneous coordinate ring is an integral domain, the "projective coordinates ring" being defined as the quotient of the graded ring or the polynomials in variables by the homogeneous (reduced) ideal defining the variety. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties.


title: Algebraic geometry

text: The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations. On the other hand, the "field of the rational functions" or "function field " is a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring.


title: Algebraic geometry

text: The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. For example, the curve of equation formula_8 is a circle if formula_9, but does not have any real point if formula_10. It follows that real algebraic geometry is not only the study of the real algebraic varieties, but has been generalized to the study of the "semi-algebraic sets", which are the solutions of systems of polynomial equations and polynomial inequalities. For example, a branch of the hyperbola of equation formula_11 is not an algebraic variety, but is a semi-algebraic set defined by formula_12 and formula_13 or by formula_12 and formula_15.


title: Algebraic geometry

text: One of the challenging problems of real algebraic geometry is the unsolved Hilbert's sixteenth problem: Decide which respective positions are possible for the ovals of a nonsingular plane curve of degree 8.


title: Algebraic geometry

text: One may date the origin of computational algebraic geometry to meeting EUROSAM'79 (International Symposium on Symbolic and Algebraic Manipulation) held at Marseille, France, in June 1979. At this meeting,


title: Algebraic geometry

text: Since then, most results in this area are related to one or several of these items either by using or improving one of these algorithms, or by finding algorithms whose complexity is simply exponential in the number of the variables.


title: Algebraic geometry

text: A body of mathematical theory complementary to symbolic methods called numerical algebraic geometry has been developed over the last several decades. The main computational method is homotopy continuation. This supports, for example, a model of floating point computation for solving problems of algebraic geometry.


title: Algebraic geometry

text: A Gröbner basis is a system of generators of a polynomial ideal whose computation allows the deduction of many properties of the affine algebraic variety defined by the ideal.


title: Algebraic geometry

text: Gröbner basis computations do not allow one to compute directly the primary decomposition of "I" nor the prime ideals defining the irreducible components of "V", but most algorithms for this involve Gröbner basis computation. The algorithms which are not based on Gröbner bases use regular chains but may need Gröbner bases in some exceptional situations.


title: Algebraic geometry

text: Gröbner bases are deemed to be difficult to compute. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential. However, this is only a worst case complexity, and the complexity bound of Lazard's algorithm of 1979 may frequently apply. Faugère F5 algorithm realizes this complexity, as it may be viewed as an improvement of Lazard's 1979 algorithm. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than 100. This means that, presently, the difficulty of computing a Gröbner basis is strongly related to the intrinsic difficulty of the problem.


title: Algebraic geometry

text: CAD is an algorithm which was introduced in 1973 by G. Collins to implement with an acceptable complexity the Tarski–Seidenberg theorem on quantifier elimination over the real numbers.


title: Algebraic geometry

text: This theorem concerns the formulas of the first-order logic whose atomic formulas are polynomial equalities or inequalities between polynomials with real coefficients. These formulas are thus the formulas which may be constructed from the atomic formulas by the logical operators "and" (∧), "or" (∨), "not" (¬), "for all" (∀) and "exists" (∃). Tarski's theorem asserts that, from such a formula, one may compute an equivalent formula without quantifier (∀, ∃).


title: Algebraic geometry

text: The complexity of CAD is doubly exponential in the number of variables. This means that CAD allows, in theory, to solve every problem of real algebraic geometry which may be expressed by such a formula, that is almost every problem concerning explicitly given varieties and semi-algebraic sets.


title: Algebraic geometry

text: While Gröbner basis computation has doubly exponential complexity only in rare cases, CAD has almost always this high complexity. This implies that, unless if most polynomials appearing in the input are linear, it may not solve problems with more than four variables.


title: Algebraic geometry

text: Since 1973, most of the research on this subject is devoted either to improving CAD or finding alternative algorithms in special cases of general interest.


title: Algebraic geometry

text: As an example of the state of art, there are efficient algorithms to find at least a point in every connected component of a semi-algebraic set, and thus to test if a semi-algebraic set is empty. On the other hand, CAD is yet, in practice, the best algorithm to count the number of connected components.


title: Algebraic geometry

text: The basic general algorithms of computational geometry have a double exponential worst case complexity. More precisely, if "d" is the maximal degree of the input polynomials and "n" the number of variables, their complexity is at most formula_16 for some constant "c", and, for some inputs, the complexity is at least formula_17 for another constant "c"′.


title: Algebraic geometry

text: During the last 20 years of the 20th century, various algorithms have been introduced to solve specific subproblems with a better complexity. Most of these algorithms have a complexity formula_18.


title: Algebraic geometry

text: Among these algorithms which solve a sub problem of the problems solved by Gröbner bases, one may cite "testing if an affine variety is empty" and "solving nonhomogeneous polynomial systems which have a finite number of solutions." Such algorithms are rarely implemented because, on most entries Faugère's F4 and F5 algorithms have a better practical efficiency and probably a similar or better complexity ("probably" because the evaluation of the complexity of Gröbner basis algorithms on a particular class of entries is a difficult task which has been done only in a few special cases).


title: Algebraic geometry

text: The main algorithms of real algebraic geometry which solve a problem solved by CAD are related to the topology of semi-algebraic sets. One may cite "counting the number of connected components", "testing if two points are in the same components" or "computing a Whitney stratification of a real algebraic set". They have a complexity of


title: Algebraic geometry

text: formula_18, but the constant involved by "O" notation is so high that using them to solve any nontrivial problem effectively solved by CAD, is impossible even if one could use all the existing computing power in the world. Therefore, these algorithms have never been implemented and this is an active research area to search for algorithms with have together a good asymptotic complexity and a good practical efficiency.


title: Algebraic geometry

text: The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes, ind-schemes, algebraic spaces, algebraic stacks and so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions and developing formal grounds for natural intersection theory and deformation theory lead to some of the further extensions.


title: Algebraic geometry

text: Most remarkably, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field "k", and the category of finitely generated reduced "k"-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent including Nisnevich topology. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks.


title: Algebraic geometry

text: Sometimes other algebraic sites replace the category of affine schemes. For example, Nikolai Durov has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field of one element and an algebraic analogue of Arakelov's geometry were realized in this setup.


title: Algebraic geometry

text: Another formal generalization is possible to universal algebraic geometry in which every variety of algebras has its own algebraic geometry. The term "variety of algebras" should not be confused with "algebraic variety".


title: Algebraic geometry

text: The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry.


title: Algebraic geometry

text: Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach. One can extend the Grothendieck site of affine schemes to a higher categorical site of derived affine schemes, by replacing the commutative rings with an infinity category of differential graded commutative algebras, or of simplicial commutative rings or a similar category with an appropriate variant of a Grothendieck topology. One can also replace presheaves of sets by presheaves of simplicial sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which is satisfying certain infinite categorical version of a sheaf axiom (and to be algebraic, inductively a sequence of representability conditions). Quillen model categories, Segal categories and quasicategories are some of the most often used tools to formalize this yielding the "derived algebraic geometry", introduced by the school of Carlos Simpson, including Andre Hirschowitz, Bertrand Toën, Gabrielle Vezzosi, Michel Vaquié and others; and developed further by Jacob Lurie, Bertrand Toën, and Gabriele Vezzosi. Another (noncommutative) version of derived algebraic geometry, using A-infinity categories has been developed from the early 1990s by Maxim Kontsevich and followers.


title: Algebraic geometry

text: Some of the roots of algebraic geometry date back to the work of the Hellenistic Greeks from the 5th century BC. The Delian problem, for instance, was to construct a length "x" so that the cube of side "x" contained the same volume as the rectangular box "a""b" for given sides "a" and "b". Menaechmus () considered the problem geometrically by intersecting the pair of plane conics "ay" = "x" and "xy" = "ab". In the 3rd century BC, Archimedes and Apollonius systematically studied additional problems on conic sections using coordinates. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years.His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding cordinates using geometric methods like using parabolas and curves. Medieval mathematicians, including Omar Khayyam, leonardo of Pisa, Gersonides and Nicole Oresme in the Medieval Period , solved certain cubic and quadratic equations by purely algebraic means and then interpreted the results geometrically. The Persian mathematician Omar Khayyám (born 1048 AD) believed that there was a relationship between arithmetic, algebra and geometry. This was criticized by Jeffrey Oaks, who claims that the study of curves by means of equations originated with Descartes in the seventeenth century.


title: Algebraic geometry

text: Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of Renaissance mathematicians such as Gerolamo Cardano and Niccolò Fontana "Tartaglia" on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry. The French mathematicians Franciscus Vieta and later René Descartes and Pierre de Fermat revolutionized the conventional way of thinking about construction problems through the introduction of coordinate geometry. They were interested primarily in the properties of "algebraic curves", such as those defined by Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes).


title: Algebraic geometry

text: During the same period, Blaise Pascal and Gérard Desargues approached geometry from a different perspective, developing the synthetic notions of projective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek "ruler and compass construction". Ultimately, the analytic geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Newton and Leibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the "calculus of infinitesimals" of Lagrange and Euler.


title: Algebraic geometry

text: It took the simultaneous 19th century developments of non-Euclidean geometry and Abelian integrals in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by Edmond Laguerre and Arthur Cayley, who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea of "homogeneous polynomial forms", and more specifically quadratic forms, on projective space. Subsequently, Felix Klein studied projective geometry (along with other types of geometry) from the viewpoint that the geometry on a space is encoded in a certain class of transformations on the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamental Kleinian geometry on projective space, they concerned themselves also with the higher degree birational transformations. This weaker notion of congruence would later lead members of the 20th century Italian school of algebraic geometry to classify algebraic surfaces up to birational isomorphism.


title: Algebraic geometry

text: The second early 19th century development, that of Abelian integrals, would lead Bernhard Riemann to the development of Riemann surfaces.


title: Algebraic geometry

text: In the same period began the algebraization of the algebraic geometry through commutative algebra. The prominent results in this direction are Hilbert's basis theorem and Hilbert's Nullstellensatz, which are the basis of the connexion between algebraic geometry and commutative algebra, and Macaulay's multivariate resultant, which is the basis of elimination theory. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed by singularity theory and computational algebraic geometry.


title: Algebraic geometry

text: B. L. van der Waerden, Oscar Zariski and André Weil developed a foundation for algebraic geometry based on contemporary commutative algebra, including valuation theory and the theory of ideals. One of the goals was to give a rigorous framework for proving the results of Italian school of algebraic geometry. In particular, this school used systematically the notion of generic point without any precise definition, which was first given by these authors during the 1930s.


title: Algebraic geometry

text: In the 1950s and 1960s, Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of sheaf theory. Later, from about 1960, and largely led by Grothendieck, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilized in the 1970s, and new applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities, moduli, and formal moduli.


title: Algebraic geometry

text: An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's Last Theorem and are also used in elliptic-curve cryptography.


title: Algebraic geometry

text: In parallel with the abstract trend of the algebraic geometry, which is concerned with general statements about varieties, methods for effective computation with concretely-given varieties have also been developed, which lead to the new area of computational algebraic geometry. One of the founding methods of this area is the theory of Gröbner bases, introduced by Bruno Buchberger in 1965. Another founding method, more specially devoted to real algebraic geometry, is the cylindrical algebraic decomposition, introduced by George E. Collins in 1973.


title: Algebraic geometry

text: An analytic variety is defined locally as the set of common solutions of several equations involving analytic functions. It is analogous to the included concept of real or complex algebraic variety. Any complex manifold is an analytic variety. Since analytic varieties may have singular points, not all analytic varieties are manifolds.


title: Algebraic geometry

text: Modern analytic geometry is essentially equivalent to real and complex algebraic geometry, as has been shown by Jean-Pierre Serre in his paper "GAGA", the name of which is French for "Algebraic geometry and analytic geometry". Nevertheless, the two fields remain distinct, as the methods of proof are quite different and algebraic geometry includes also geometry in finite characteristic.


title: Algebraic geometry

text: Algebraic geometry now finds applications in statistics, control theory, robotics, error-correcting codes, phylogenetics and geometric modelling. There are also connections to string theory, game theory, graph matchings, solitons and integer programming.


title: Awake (2007 film)

text: Awake is a 2007 American conspiracy thriller film written and directed by Joby Harold (in his directorial debut). It stars Hayden Christensen, Jessica Alba, Terrence Howard and Lena Olin. The film was released in the United States and Canada on November 30, 2007.


title: Awake (2007 film)

text: Young billionaire Clay Beresford, Jr. is in love with beautiful Samantha "Sam" Lockwood, his mother's personal assistant. Clay requires a heart transplant and Dr. Jack Harper is Clay's heart surgeon and friend. Clay asks Dr. Harper to arrange his elopement with Sam and they marry privately at midnight, then Clay goes to the hospital for the operation. While Clay's mother Lilith awaits completion of his surgery, Clay encounters anesthesia awareness. The surgical pain causes Clay to have a clairvoyant experience exposing Dr. Harper's plot to murder him, also revealing that Sam worked at the hospital under Dr. Harper and has conspired with him against Clay. Sam's plan was to poison the donor heart by injecting Adriamycin to cause its rejection, thus murdering Clay to collect insurance money to pay off Dr. Harper's malpractice lawsuits.


title: Awake (2007 film)

text: The scheme unravels and Lilith, realizing what has happened, sacrifices her own life so that Clay, who is close to death, can live: she dies by suicide so her heart can be switched for the poisoned one, and save Clay. While Sam tries to get away with what she did, Dr. Harper feels guilty and he holds onto proof so she can be arrested too. Another surgical team takes over the operation, as Clay barely clings to life and the conspirators are arrested. The new team takes Lilith's heart and transplants it into Clay's body, as Clay and Lilith have their final moments together in spirit (in an out-of-body experience).


title: Awake (2007 film)

text: The new head surgeon announces that Clay has come back to life, as the new team stitch Clay's wound. Clay, in spirit, is still in the afterlife with Lilith, tries to take his own life to stay with his mother. Clay makes his new heart stop beating and the surgeons have to use the defibrillator in attempt to revive Clay. As Clay resists being revived, Lilith forces Clay (in the "afterlife-world") to revisit a scene from his childhood, when Lilith accidentally killed Clay's abusive father. This scene reveals the truth for Clay and connects his childhood flashbacks. After seeing this scene, Clay gives away to revival, and before the surgeons could shock his body again, Clay allows his new heart to begin beating. Clay opens his eyes when the surgeons remove the eye tapes while Harper ends his narratings with "He is awake".


title: Awake (2007 film)

text: Portions of "Awake" were filmed on Fordham University's Lincoln Center campus (Lowenstein Hall is converted to look like a hospital; the statue of "St. Peter: Fisher of Men" is visible in the film). In addition, many scenes, including Dr. Jack Harper's office, the cafeteria where Lilith dies by suicide, and the elevator bank, were filmed in Bellevue Hospital.