The Hodge conjecture: The complications of understanding the shape of geometric spaces

Authors

  • Vicente Muñoz Velázquez Complutense University of Madrid (Spain).

DOI:

https://doi.org/10.7203/metode.0.8253

Keywords:

complex geometry, topology, homology, Hodge theory, manifolds

Abstract

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

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Author Biography

Vicente Muñoz Velázquez, Complutense University of Madrid (Spain).

Full professor of Geometry and Topology at the Complutense University of Madrid (Spain). He is a researcher in differential geometry, algebraic geometry, and algebraic topology, specifically in gauge theory, moduli spaces, symplectic geometry, complex geometry, and rational homotopy.

References

Atiyah, M. F., & Hirzebruch, F. (1962). Analytic cycles on complex manifolds. Topology, 1, 25–45. doi: 10.1016/0040-9383(62)90094-0

Grothendieck, A. (1969). Hodge’s general conjecture is false for trivial reasons. Topology, 8, 299–303. doi: 10.1016/0040-9383(69)90016-0

Hodge, W. V. D. (1950). The topological invariants of algebraic varieties. In Proceedings of the International Congress of Mathematicians (pp. 181–192). Cambridge, MA: American Mathematical Society.

Poincaré, H. (1895). Analysis situs. Journal de l’École Polytechnique, 1, 1–123.

Voisin, C. (2002). A counterexample to the Hodge Conjecture extended to Kähler varieties. International Mathematics Research Notices, 20, 1057–1075. doi: 10.1155/S1073792802111135

Weil, A. (1980). Abelian varieties and the Hodge ring. In Oeuvres Scientifiques Collected Papers III (pp. 421–429). New York: Springer-Verlag.

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Published

2018-06-05

How to Cite

Muñoz Velázquez, V. (2018). The Hodge conjecture: The complications of understanding the shape of geometric spaces. Metode Science Studies Journal, (8), 51–57. https://doi.org/10.7203/metode.0.8253
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Issue

Making science: A multitude of perspectives (2018)

Section

The millennium problems. Challenges to further mathematics

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