The Hodge conjecture is a major unsolved problem in algebraic geometry that posits that for certain classes of non-singular projective algebraic varieties, the de Rham cohomology classes that are of type (p, p) are precisely those that can be represented by algebraic cycles. It is one of the seven Millennium Prize Problems, reflecting its profound implications for understanding the relationship between algebraic geometry and topology.

Hodge Conjecture

Hodge conjecture

unsolved problem

Algebraic geometry is a branch of mathematics that studies the solutions of systems of polynomial equations using abstract algebraic techniques, primarily focusing on the properties and structures of algebraic varieties. It serves as a bridge between algebra and geometry, providing a deep understanding of both geometric shapes and algebraic equations through the lens of modern mathematics.

algebraic geometry

non-singular projective algebraic varieties

de Rham cohomology classes

Type (p, p)

algebraic cycles

The Millennium Prize Problems are a set of seven unsolved mathematical problems, each with a prize of one million dollars for a correct solution, established by the Clay Mathematics Institute in 2000 to celebrate the turn of the millennium and to stimulate interest in important mathematical questions. As of now, only one of the problems, the Poincaré Conjecture, has been solved, highlighting the complexity and significance of these challenges in the field of mathematics.

Millennium Prize Problems

profound implications

understanding the relationship

Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations such as stretching and bending, but not tearing or gluing. It provides a foundational framework for understanding concepts of convergence, continuity, and compactness in various mathematical contexts.

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