São Paulo is a municipality in the Southeast Region of Brazil. The metropolis is an alpha global city and the most populous city in Brazil, the Western Hemisphere and the Southern Hemisphere, besides being the largest Portuguese-speaking city in the world; the municipality is the Earth's 11th largest city proper by population. The city is the capital of the surrounding state of São Paulo, the most populous and wealthiest state in Brazil, it exerts strong international influences in commerce, finance and entertainment. The name of the city honors Saint Paul of Tarsus; the city's metropolitan area, the Greater São Paulo, ranks as the most populous in Brazil and the 12th most populous on Earth. The process of conurbation between the metropolitan areas located around the Greater São Paulo created the São Paulo Macrometropolis, a megalopolis with more than 30 million inhabitants, one of the most populous urban agglomerations in the world. Having the largest economy by GDP in Latin America and the Southern Hemisphere, the city is home to the São Paulo Stock Exchange.
Paulista Avenue is the economic core of São Paulo. The city has the 11th largest GDP in the world, representing alone 10.7% of all Brazilian GDP and 36% of the production of goods and services in the state of São Paulo, being home to 63% of established multinationals in Brazil, has been responsible for 28% of the national scientific production in 2005. With a GDP of US$477 billion, the São Paulo city alone would have ranked 26th globally compared with countries by 2017 estimates; the metropolis is home to several of the tallest skyscrapers in Brazil, including the Mirante do Vale, Edifício Itália, North Tower and many others. The city has cultural and political influence both nationally and internationally, it is home to monuments and museums such as the Latin American Memorial, the Ibirapuera Park, Museum of Ipiranga, São Paulo Museum of Art, the Museum of the Portuguese Language. The city holds events like the São Paulo Jazz Festival, São Paulo Art Biennial, the Brazilian Grand Prix, São Paulo Fashion Week, the ATP Brasil Open, the Brasil Game Show and the Comic Con Experience.
The São Paulo Gay Pride Parade rivals the New York City Pride March as the largest gay pride parade in the world. São Paulo is a cosmopolitan, melting pot city, home to the largest Arab and Japanese diasporas, with examples including ethnic neighborhoods of Mercado and Liberdade respectively. São Paulo is home to the largest Jewish population in Brazil, with about 75,000 Jews. In 2016, inhabitants of the city were native to over 200 different countries. People from the city are known as paulistanos, while paulistas designates anyone from the state, including the paulistanos; the city's Latin motto, which it has shared with the battleship and the aircraft carrier named after it, is Non ducor, which translates as "I am not led, I lead." The city, colloquially known as Sampa or Terra da Garoa, is known for its unreliable weather, the size of its helicopter fleet, its architecture, severe traffic congestion and skyscrapers. São Paulo was one of the host cities of the 2014 FIFA World Cup. Additionally, the city hosted the IV Pan American Games and the São Paulo Indy 300.
The region of modern-day São Paulo known as Piratininga plains around the Tietê River, was inhabited by the Tupi people, such as the Tupiniquim and Guarani. Other tribes lived in areas that today form the metropolitan region; the region was divided in Caciquedoms at the time of encounter with the Europeans. The most notable Cacique was Tibiriça, known for his support for the Portuguese and other European colonists. Among the many indigenous names that survive today are Tietê, Tamanduateí, Anhangabaú, Diadema, Itapevi, Embu-Guaçu etc... The Portuguese village of São Paulo dos Campos de Piratininga was marked by the founding of the Colégio de São Paulo de Piratininga on January 25, 1554; the Jesuit college of twelve priests included Spanish priest José de Anchieta. They built a mission on top of a steep hill between the Tamanduateí rivers, they first had a small structure built of rammed earth, made by American Indian workers in their traditional style. The priests wanted to evangelize – teach the Indians who lived in the Plateau region of Piratininga and convert them to Christianity.
The site was separated from the coast by the Serra do Mar, called by the Indians Serra Paranapiacaba. The college was named for a Christian saint and its founding on the feast day of the celebration of the conversion of the Apostle Paul of Tarsus. Father José de Anchieta wrote this account in a letter to the Society of Jesus: The settlement of the region's Courtyard of the College began in 1560. During the visit of Mem de Sá, Governor-General of Brazil, the Captaincy of São Vicente, he ordered the transfer of the population of the Village of Santo André da Borda do Campo to the vicinity of the college, it was named "College of St. Paul Piratininga"; the new location was on a steep hill adjacent to a large wetland, the lowland do Carmo. It offered better protection from attacks by local Indian groups, it was renamed belonging to the Captaincy of São Vicente. For the next two centuries, São Paulo developed as a poor and isolated village that survived through the cultivation of subsistence crops by the labor of natives.
For a long time, São Paulo was the only village in Brazil's interior, as travel was too difficult for many to reach the area. Mem de Sá forbade colonists to use the "Path Pir
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, for research into vision and pattern theory. He was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science, he is a University Professor Emeritus in the Division of Applied Mathematics at Brown University. Mumford was born in West Sussex in England, of an English father and American mother, his father William started an experimental school in Tanzania and worked for the newly created United Nations. In high school, he was a finalist in the prestigious Westinghouse Science Talent Search. After attending the Phillips Exeter Academy, Mumford went to Harvard, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956, he completed his Ph. D. in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques, he published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety, on algebraic surfaces.
His books Abelian Varieties and Curves on an Algebraic Surface combined the new theories. His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique, the only accessible introduction, they are now available as The Red Book of Schemes. Other work, less written up were lectures on varieties defined by quadrics, a study of Goro Shimura's papers from the 1960s. Mumford's research did much to revive the classical theory of theta functions, by showing that its algebraic content was large, enough to support the main parts of the theory by reference to finite analogues of the Heisenberg group; this work on the equations defining abelian varieties appeared in 1966–7. He published some further books of lectures on the theory, he was one of the founders of the toroidal embedding theory. In a sequence of four papers published in the American Journal of Mathematics between 1961 and 1975, Mumford explored pathological behavior in algebraic geometry, that is, phenomena that would not arise if the world of algebraic geometry were as well-behaved as one might expect from looking at the simplest examples.
These pathologies fall into two types: bad behavior in characteristic p and bad behavior in moduli spaces. Mumford's philosophy in characteristic p was as follows: A nonsingular characteristic p variety is analogous to a general non-Kähler complex manifold. In the first Pathologies paper, Mumford finds an everywhere regular differential form on a smooth projective surface, not closed, shows that Hodge symmetry fails for classical Enriques surfaces in characteristic two; this second example is developed further in Mumford's third paper on classification of surfaces in characteristic p. This pathology can now be explained in terms of the Picard scheme of the surface, in particular, its failure to be a reduced scheme, a theme developed in Mumford's book "Lectures on Curves on an Algebraic Surface". Worse pathologies related to p-torsion in crystalline cohomology were explored by Luc Illusie. In the second Pathologies paper, Mumford gives a simple example of a surface in characteristic p where the geometric genus is non-zero, but the second Betti number is equal to the rank of the Néron–Severi group.
Further such examples arise in Zariski surface theory. He conjectures that the Kodaira vanishing theorem is false for surfaces in characteristic p. In the third paper, he gives an example of a normal surface; the first example of a smooth surface for which Kodaira vanishing fails was given by Michel Raynaud in 1978. In the second Pathologies paper, Mumford finds that the Hilbert scheme parametrizing space curves of degree 14 and genus 24 has a multiple component. In the fourth Pathologies paper, he finds reduced and irreducible complete curves which are not specializations of non-singular curves; these sorts of pathologies were considered to be scarce when they first appeared. But Ravi Vakil in a paper called "Murphy's law in algebraic geometry" has shown that Hilbert schemes of nice geometric objects can be arbitrarily "bad", with unlimited numbers of components and with arbitrarily large multiplicities. In three papers written between 1969 and 1976, Mumford extended the Enriques–Kodaira classification of smooth projective surfaces from the case of the complex ground field to the case of an algebraically closed ground field of characteristic p.
The final answer turns out to be as the answer in the complex case, once two important adjustments are made. The first is that one may get "non-classical" surfaces, which come about when p-torsion in the Picard scheme degenerates to a non-reduced group scheme; the second is the possibility of obtaining quasi-elliptic surfaces in characteristics three. These are surfaces fibred over a curve where the general fibre is a curve of arithmetic genus one with a cusp. Once these adjustments ar
André Weil was an influential French mathematician of the 20th century, known for his foundational work in number theory and algebraic geometry. He was the de facto early leader of the mathematical Bourbaki group; the philosopher Simone Weil was his sister. André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71; the famous philosopher Simone Weil was Weil's only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Muslim University. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he taught himself Sanskrit in 1920. After teaching for one year in Aix-Marseille University, he taught for six years in Strasbourg, he married Éveline in 1937. Weil was in Finland, his wife Éveline returned to France without him.
Weil was mistakenly arrested in Finland at the outbreak of the Winter War on suspicion of spying. Weil returned to France via Sweden and the United Kingdom, was detained at Le Havre in January 1940, he was charged with failure to report for duty, was imprisoned in Le Havre and Rouen. It was in the military prison in Bonne-Nouvelle, a district of Rouen, from February to May, that Weil completed the work that made his reputation, he was tried on 3 May 1940. Sentenced to five years, he requested to be attached to a military unit instead, was given the chance to join a regiment in Cherbourg. After the fall of France, he met up with his family in Marseille, he went to Clermont-Ferrand, where he managed to join his wife Éveline, living in German-occupied France. In January 1941, Weil and his family sailed from Marseille to New York, he spent the remainder of the war in the United States, where he was supported by the Rockefeller Foundation and the Guggenheim Foundation. For two years, he taught undergraduate mathematics at Lehigh University, where he was unappreciated and poorly paid, although he didn't have to worry about being drafted, unlike his American students.
But, he hated Lehigh much for their heavy teaching workload and he swore that he would never talk about "Lehigh" any more. He quit the job at Lehigh, he moved to Brazil and taught at the Universidade de São Paulo from 1945 to 1947, where he worked with Oscar Zariski, he returned to the United States and taught at the University of Chicago from 1947 to 1958, before moving to the Institute for Advanced Study, where he would spend the remainder of his career. He was a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts, in 1954 in Amsterdam, in 1978 in Helsinki. In 1979, Weil shared the second Wolf Prize in Mathematics with Jean Leray. Weil made substantial contributions in a number of areas, the most important being his discovery of profound connections between algebraic geometry and number theory; this began in his doctoral work leading to the Mordell–Weil theorem. Mordell's theorem had an ad hoc proof. Both aspects of Weil's work have developed into substantial theories. Among his major accomplishments were the 1940s proof of the Riemann hypothesis for zeta-functions of curves over finite fields, his subsequent laying of proper foundations for algebraic geometry to support that result.
The so-called Weil conjectures were hugely influential from around 1950. Weil introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, gave a proof of the Riemann–Roch theorem with them. His'matrix divisor' Riemann–Roch theorem from 1938 was a early anticipation of ideas such as moduli spaces of bundles; the Weil conjecture on Tamagawa numbers proved resistant for many years. The adelic approach became basic in automorphic representation theory, he picked up another credited Weil conjecture, around 1967, which under pressure from Serge Lang became known as the Taniyama–Shimura conjecture based on a formulated question of Taniyama at the 1955 Nikkō conference. His attitude towards conjectures was that one should not dignify a guess as a conjecture and in the Taniyama case, the evidence was only there after extensive computational work carried out from the late 1960s. Other significant results were on Pontryagin differential geometry, he introduced the concept of a uniform space in general topology, as a by-product of his collaboration with Nicolas Bourbaki.
His work on sheaf theory hardly appears in his published papers, but correspondence with Henri Cartan in the late 1940s, reprinted in his collected papers, proved most influential. He created the ∅, he discovered that the so-called Weil representation introduced in quantum mechanics by Irving Segal an
Oscar Zariski was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century. Zariski was born Oscher Zaritsky in 1918 studied at the University of Kiev, he left Kiev in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi. Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski. Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz, he had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school; the book was published in 1935 and reissued 36 years with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed.
It is still an important reference. It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry, he addressed the question of rigour by recourse to commutative algebra. The Zariski topology, as it was known, is adequate for biregular geometry, where varieties are mapped by polynomial functions; that theory is too limited for algebraic surfaces, for curves with singular points. A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety; the description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use valuation theory to describe the phenomena such as blowing up. After spending a year 1946–1947 at the University of Illinois at Urbana–Champaign, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969.
In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests; the two sets of foundations weren't reconciled at that point. At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman—thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation. Zariski himself worked on equisingularity theory; some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that unified algebraic geometry. Zariski proposed the first example of a Zariski surface in 1958. Zariski was a Jewish atheist. Zariski was awarded the Steele Prize in 1981, in the same year the Wolf Prize in Mathematics with Lars Ahlfors.
He wrote Commutative Algebra in two volumes, with Pierre Samuel. His papers have been published in four volumes. Zariski, Abhyankar, Shreeram S.. Algebraic surfaces, Classics in mathematics, New York: Springer-Verlag, ISBN 978-3-540-58658-6, MR 0469915 Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, 4, The Mathematical Society of Japan, Tokyo, MR 0097403 Zariski, Cohn, James, ed. An introduction to the theory of algebraic surfaces, Lecture notes in mathematics, 83, New York: Springer-Verlag, doi:10.1007/BFb0082246, ISBN 978-3-540-04602-8, MR 0263819 Zariski, Oscar. Vol. II, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876 Zariski, Kmety, François; the moduli problem for plane branches, University Lecture Series, 39, Providence, R. I.: American Mathematical Society, ISBN 978-0-8218-2983-7, MR 0414561: Le problème des modules pour les branches planes Zariski, Collected papers. Vol. I: Foundations of algebraic geometry and resolution of singularities, Massachusetts-London: MIT Press, ISBN 978-0-262-08049-1, MR 0505100 Zariski, Collected papers.
Vol. II: Holomorphic functions and linear systems, Mathematicians of Our Time, Massachusetts-London: MIT Press, ISBN 978-0-262-01038-2, MR 0505100 Zariski, Artin, Michael. Collected papers. Volume III. Topology of curves and surfaces, special topics in the theory of algebraic varieties, Mathematicians of Our Time, Massachusetts-London: MIT Press, ISBN 978-0-262-24021-5, MR 0505104 Zariski, Lipman, Joseph. Collected papers. Vol. IV. Equisingularity on algebraic varieties, Mathematicians of Our Time, 16, MIT Press, ISBN 978-0-262-08049-1, MR 0545653 Zariski ring Zariski tangent space Zariski surface Zariski topology Zariski–Riemann surface Zariski space Zariski's lemma Zariski's main theorem
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 from commutative algebra, for solving geometrical problems about these sets of zeros; 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, parabolas, hyperbolas, cubic curves like elliptic curves, 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.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. 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. 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, 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 subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. 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 denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. 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
World War II
World War II known as the Second World War, was a global war that lasted from 1939 to 1945. The vast majority of the world's countries—including all the great powers—eventually formed two opposing military alliances: the Allies and the Axis. A state of total war emerged, directly involving more than 100 million people from over 30 countries; the major participants threw their entire economic and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. World War II was the deadliest conflict in human history, marked by 50 to 85 million fatalities, most of whom were civilians in the Soviet Union and China, it included massacres, the genocide of the Holocaust, strategic bombing, premeditated death from starvation and disease, the only use of nuclear weapons in war. Japan, which aimed to dominate Asia and the Pacific, was at war with China by 1937, though neither side had declared war on the other. World War II is said to have begun on 1 September 1939, with the invasion of Poland by Germany and subsequent declarations of war on Germany by France and the United Kingdom.
From late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, formed the Axis alliance with Italy and Japan. Under the Molotov–Ribbentrop Pact of August 1939, Germany and the Soviet Union partitioned and annexed territories of their European neighbours, Finland and the Baltic states. Following the onset of campaigns in North Africa and East Africa, the fall of France in mid 1940, the war continued between the European Axis powers and the British Empire. War in the Balkans, the aerial Battle of Britain, the Blitz, the long Battle of the Atlantic followed. On 22 June 1941, the European Axis powers launched an invasion of the Soviet Union, opening the largest land theatre of war in history; this Eastern Front trapped most crucially the German Wehrmacht, into a war of attrition. In December 1941, Japan launched a surprise attack on the United States as well as European colonies in the Pacific. Following an immediate U. S. declaration of war against Japan, supported by one from Great Britain, the European Axis powers declared war on the U.
S. in solidarity with their Japanese ally. Rapid Japanese conquests over much of the Western Pacific ensued, perceived by many in Asia as liberation from Western dominance and resulting in the support of several armies from defeated territories; the Axis advance in the Pacific halted in 1942. Key setbacks in 1943, which included a series of German defeats on the Eastern Front, the Allied invasions of Sicily and Italy, Allied victories in the Pacific, cost the Axis its initiative and forced it into strategic retreat on all fronts. In 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained its territorial losses and turned toward Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in Central China, South China and Burma, while the Allies crippled the Japanese Navy and captured key Western Pacific islands; the war in Europe concluded with an invasion of Germany by the Western Allies and the Soviet Union, culminating in the capture of Berlin by Soviet troops, the suicide of Adolf Hitler and the German unconditional surrender on 8 May 1945.
Following the Potsdam Declaration by the Allies on 26 July 1945 and the refusal of Japan to surrender under its terms, the United States dropped atomic bombs on the Japanese cities of Hiroshima and Nagasaki on 6 and 9 August respectively. With an invasion of the Japanese archipelago imminent, the possibility of additional atomic bombings, the Soviet entry into the war against Japan and its invasion of Manchuria, Japan announced its intention to surrender on 15 August 1945, cementing total victory in Asia for the Allies. Tribunals were set up by fiat by the Allies and war crimes trials were conducted in the wake of the war both against the Germans and the Japanese. World War II changed the political social structure of the globe; the United Nations was established to foster international co-operation and prevent future conflicts. The Soviet Union and United States emerged as rival superpowers, setting the stage for the nearly half-century long Cold War. In the wake of European devastation, the influence of its great powers waned, triggering the decolonisation of Africa and Asia.
Most countries whose industries had been damaged moved towards economic expansion. Political integration in Europe, emerged as an effort to end pre-war enmities and create a common identity; the start of the war in Europe is held to be 1 September 1939, beginning with the German invasion of Poland. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or the Japanese invasion of Manchuria on 19 September 1931. Others follow the British historian A. J. P. Taylor, who held that the Sino-Japanese War and war in Europe and its colonies occurred and the two wars merged in 1941; this article uses the conventional dating. Other starting dates sometimes used for World War II include the Italian invasion of Abyssinia on 3 October 1935; the British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the fo
Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.:58Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility; the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot; the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex number coefficients is determined by the set of its roots in the complex plane.
Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory; this correspondence is a defining feature of algebraic geometry. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define quasi-projective varieties in a similar way; the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s. For an algebraically closed field K and a natural number n, let An be affine n-space over K; the polynomials f in the ring K can be viewed as K-valued functions on An by evaluating f at the points in An, i.e. by choosing values in K for each xi.
For each set S of polynomials in K, define the zero-locus Z to be the set of points in An on which the functions in S vanish, to say Z =. A subset V of An is called an affine algebraic set if V = Z for some S.:2 A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets.:3 An irreducible affine algebraic set is called an affine variety.:3 Affine varieties can be given a natural topology by declaring the closed sets to be the affine algebraic sets. This topology is called the Zariski topology.:2Given a subset V of An, we define I to be the ideal of all polynomial functions vanishing on V: I =. For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.:4 Let k be an algebraically closed field and let Pn be the projective n-space over k. Let f in k be a homogeneous polynomial of degree d, it is not well-defined to evaluate f on points in Pn in homogeneous coordinates.
However, because f is homogeneous, meaning that f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish: Z =. A subset V of Pn is called a projective algebraic set if V = Z for some S.:9 An irreducible projective algebraic set is called a projective variety.:10Projective varieties are equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.:10A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice that the complement of an algebraic set in an affine variety is a quasi-projective variety. In classical algebraic geometry, a