In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, المقابلة al-muqābala "balancing" referred to adding equal terms to both sides. The word algebra comes from the Arabic: الجبر, romanized: al-jabr, lit.'reunion of broken parts, bonesetting ' from the title of the early 9th century book ʿIlm al-jabr wa l-muqābala "The Science of Restoring and Balancing" by the Persian mathematician and astronomer al-Khwarizmi. The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi. A mathematician specialized in algebra is called an algebraist. Sometimes, the same phrase is used for a subarea and its main algebraic structures for example, Boolean algebra and a Boolean algebra. The word algebra is not only used for naming an area of mathematics and some subareas it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an algebra. There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields. Įlementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Comprehensive text including the geometric point of view.The quadratic formula expresses the solution of the equation ax 2 + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c.Īlgebra (from Arabic الجبر ( al-jabr) 'reunion of broken parts, bonesetting' ) is the study of variables and the rules for manipulating these variables in formulas it is a unifying thread of almost all of mathematics. Useful reference.Ĭommutative algebra with a view toward algebraic geometry, D. Classic text (very concise).Ĭommutative ring theory, H. Thorough development of the analytic approach (more careful than Griffiths-Harris, but fewer examples). Hodge theory and complex algebraic geometry I, C. Describes the analytic approach to algebraic geometry. Standard text covering modern techniques in algebraic geometry. Recent book with lots of examples.Īlgebraic geometry, R. Fairly extensive introduction with few prerequisites.Īlgebraic geometry: a first course, J. Very accessible.īasic algebraic geometry 1, I. Reid, googlebooks.Įlementary introduction to algebraic geometry. The red book of varieties and schemes, D. Examples will include projective space, the Grassmannian, the group law on an elliptic curve, blow-ups and resolutions of singularities, algebraic curves of low genus, and hypersurfaces in projective 3-space. Topics will include projective varieties, singularities, differential forms, line bundles, and sheaf cohomology, including the Riemann-Roch theorem and Serre duality for algebraic curves. Passing from local to global data is delicate (as in complex analysis) and is either accomplished by working in projective space (corresponding to a graded polynomial ring) or by using sheaves and their cohomology. In the algebraic approach to the subject, local data is studied via the commutative algebra of quotients of polynomial rings in several variables. This course will be a fast-paced introduction to the subject with a strong emphasis on examples. It is a central subject in mathematics with strong connections to differential geometry, number theory, and representation theory. Homeworks will be due every 1-2 weeks at the beginning of Thursday's class.Īlgebraic geometry is the study of geometric spaces locally defined by polynomial equations. Some prior experience of manifolds would be useful (but not essential). Office hours: Tuesdays 3:00PM-4:00PM and Wednesdays 2:30PM-3:30PM in my office LGRT 1235H.Ĭommutative algebra (rings and modules) as covered in 611-612. Instructor: Paul Hacking, LGRT 1235H, Tuesdays and Thursdays, 11:30AM-12:45PM in LGRT 1114. Math 797W: Algebraic geometry Math 797W: Algebraic geometry
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