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Yoichi Miyaoka
is a mathematician who works in algebraic geometry and who proved (independently of Shing-Tung Yau's work) the Bogomolov–Miyaoka–Yau inequality in an Inventiones Mathematicae paper. In 1984, Miyaoka extended the Bogomolov–Miyaoka–Yau inequality to surfaces with quotient singularities, and in 2008 to orbifold surfaces. Doing so, he obtains sharp bound on the number of quotient singularities on surfaces of general type. Moreover, the inequality for orbifold surfaces gives explicit values for the coefficients of the so-called Vojta's conjecture, Lang-Vojta conjecture relating the degree of a curve on a surface with its geometric genus. References 20th-century Japanese mathematicians 21st-century Japanese mathematicians Living people Year of birth missing (living people) {{Asia-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
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. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University. Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bogomolov–Miyaoka–Yau Inequality
In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality : c_1^2 \le 3 c_2 between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topological types of the underlying real 4-manifold. It was proved independently by and , after and proved weaker versions with the constant 3 replaced by 8 and 4. Armand Borel and Friedrich Hirzebruch showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: and gave examples of surfaces in characteristic ''p'', such as generalized Raynaud surfaces, for which it fails. Formulation of the inequality The conventional formulation of the Bogomolov–Miyaoka–Yau inequality is as follows. Let ''X'' be a compact complex surface of general type, and let ''c''1 = ''c''1(''X'') and ''c''2 = ''c''2(''X'') be the first and second Chern class of the complex tangent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirô Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vojta's Conjecture
In mathematics, Vojta's conjecture is a conjecture introduced by about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic. Statement of the conjecture Let F be a number field, let X/F be a non-singular algebraic variety, let D be an effective divisor on X with at worst normal crossings, let H be an ample divisor on X, and let K_X be a canonical divisor on X. Choose Weil height functions h_H and h_ and, for each absolute value v on F, a local height function \lambda_. Fix a finite set of absolute values S of F, and let \epsilon>0. Then there is a constant C and a non-empty Zariski open set U\subseteq X, depending on all of the above choices, such that :: \sum_ \lambda_(P) + h_(P) \le \epsilon h_H(P) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, the dimension of the canonical linear system plus one. In other words for a variety of complex dimension it is the number of linearly independent holomorphic -forms to be found on .Danilov & Shokurov (1998), p. 53/ref> This definition, as the dimension of : then carries over to any base field, when is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle. The geometric genus is the first invariant of a sequence of invariants called the plurigenera. Case of curves In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century Japanese Mathematicians
The 1st century was the century spanning AD 1 ( I) through AD 100 ( C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or historical period. The 1st century also saw the appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius (AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and instability, which was finally brought to an end by Vespasian, ninth Roman emperor, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Living People
Related categories * :Year of birth missing (living people) / :Year of birth unknown * :Date of birth missing (living people) / :Date of birth unknown * :Place of birth missing (living people) / :Place of birth unknown * :Year of death missing / :Year of death unknown * :Date of death missing / :Date of death unknown * :Place of death missing / :Place of death unknown * :Missing middle or first names See also * :Dead people * :Template:L, which generates this category or death years, and birth year and sort keys. : {{DEFAULTSORT:Living people 21st-century people People by status ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
