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Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. [1] Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure.
Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V. The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x.
Among the properties of Hodge numbers are Hodge symmetry h p,q = h q,p (because H p,q (X) is the complex conjugate of H q,p (X)) and h p,q = h n−p,n−q (by Serre duality). The Hodge numbers of a smooth complex projective variety (or compact Kähler manifold) can be listed in the Hodge diamond (shown in the case of complex dimension 2):
Gross and Bernd Siebert jointly developed a program (known as the Gross–Siebert Program) for studying mirror symmetry within algebraic geometry. [ 1 ] [ 9 ] The Gross–Siebert program builds on an earlier, differential-geometric, proposal of Strominger , Yau , and Zaslow , in which the Calabi–Yau manifold is fibred by special Lagrangian ...
Along with the homological mirror symmetry conjecture, it is one of the most explored tools applied to understand mirror symmetry in mathematical terms. While the homological mirror symmetry is based on homological algebra, the SYZ conjecture is a geometrical realization of mirror symmetry.
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1 ⁄ 3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations.
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, that were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry.
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