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"Higher Dimensional Group Theory". groupoids.org.uk. Bangor University. A web article with many references explaining how the groupoid concept has led to notions of higher-dimensional groupoids, not available in group theory, with applications in homotopy theory and in group cohomology. Brown, R.; Higgins, P.J. (1981). "On the algebra of cubes".
Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any smooth projective surface.
In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higher-dimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as the Siegel modular variety. This is the problem underlying Siegel modular form theory.
An arithmetic cycle of codimension p is a pair (Z, g) where Z ∈ Z p (X) is a p-cycle on X and g is a Green current for Z, a higher-dimensional generalization of a Green function. The arithmetic Chow group C H ^ p ( X ) {\displaystyle {\widehat {\mathrm {CH} }}_{p}(X)} of codimension p is the quotient of this group by the subgroup generated by ...
The two-dimensional complex tori include the abelian surfaces. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of ...
Algebraic geometry is the place where the algebra involved in solving systems of simultaneous multivariable polynomial equations meets the geometry of curves, surfaces, and higher dimensional algebraic varieties.
It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves. The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory.
Category theory was originally introduced for the need of homological algebra, and widely extended for the need of modern algebraic geometry (scheme theory). Category theory may be viewed as an extension of universal algebra , as the latter studies algebraic structures , and the former applies to any kind of mathematical structure and studies ...
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