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Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems. Classically it studies zeros of multivariate polynomials, the modern approach generalizes this in a few different aspects.
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
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In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. It was first published in 1928 by Dirac.
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In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint operator, endowed with supplemental structures. It was conceived by Alain Connes who was motivated by the Atiyah-Singer index theorem and sought its extension to 'noncommutative' spaces. Some authors refer to this notion as unbounded K-cycles or as unbounded Fredholm modules.
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In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language of G-structures on a manifold. Specifically, a quaternionic n-manifold can be defined as a smooth manifold of real dimension 4n equipped with a torsion-free -structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds.
Mariusz Wodzicki is a Polish mathematician, whose works primarily focus on analysis, algebraic k-theory, noncommutative geometry, and algebraic geometry.
Anne Schilling is an American mathematician specializing in algebraic combinatorics, representation theory, and mathematical physics. She is a professor of mathematics at the University of California, Davis.
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References
- ↑ "Irene Sabadini", People, Polytechnic University of Milan, Department of Mathematics, retrieved 2023-11-21
- ↑ Irene Sabadini at the Mathematics Genealogy Project
- ↑ Review of Analysis of Dirac systems and computational algebra: Michael Shapiro, MR 2089988, Zbl 1064.30049
- ↑ Review of Noncommutative functional calculus: Vladimir V. Kisil, Zbl 1228.47001
- ↑ Reviews of Entire slice regular functions: Alessandro Perotti, MR 3585395; Michael Shapiro, Zbl 1372.30001
- ↑ Reviews of Slice hyperholomorphic Schur analysis: Michael Shapiro, MR 3585855; Florian-Horia Vasilescu, Zbl 1366.30001
- ↑ Reviews of The mathematics of superoscillations: Raymond Johnson, Zbl 1383.42002; Gaetano Siciliano, MR 3633292
- ↑ Reviews of Quaternionic approximation: Amedeo Altavilla, Zbl 1432.30034; M. Elena Luna-Elizarrarás, MR 3930620
