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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.
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.
Phase portrait of the Van der Pol oscillator, a one-dimensional system. Phase space was the original object of study in symplectic geometry. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.
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.
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.
In algebraic geometry, a Fano variety, introduced by Gino Fano (Fano 1934, 1942), is an algebraic variety that generalizes certain aspects of complete intersections of algebraic hypersurfaces whose sum of degrees is at most the total dimension of the ambient projective space.
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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