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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".
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.
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds.The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.
There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ∈ X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and ...
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.
The Hodge conjecture generalises this statement to higher dimensions. In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
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 ...
Witten, Edward (1993), "Algebraic geometry associated with matrix models of two-dimensional gravity", in Goldberg, Lisa R.; Phillips, Anthony V. (eds.), Topological methods in modern mathematics (Stony Brook, NY, 1991), Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook ...
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