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...

unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem...

particular to an idea of mixed Hodge structure on singular varieties, and to deep analogies with étale cohomology. The Hodge conjecture on the 'middle' spaces...

theory). This conjecture implies the Lefschetz conjecture. If the Hodge standard conjecture holds, then the Lefschetz conjecture and Conjecture D are equivalent...

higher dimensions; this duality is now known as the Hodge star operator. He further conjectured that each cohomology class should have a distinguished...

Hodge conjecture, for some applications. The theory of mixed Hodge structures, a powerful tool in algebraic geometry that generalizes classical Hodge...

quintic threefold is in the study of the infinitesimal generalized Hodge conjecture where this difficult problem can be solved in this case. In fact, all...

mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to...

In the mathematical field of geometric topology, the Poincaré conjecture (UK: /ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about...

cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture. Let V be...

is normal usage. The importance of Hodge cycles lies primarily in the Hodge conjecture, to the effect that Hodge cycles should always be algebraic cycles...

including Shing-Tung Yau's proof of the Calabi conjecture, the Hitchin–Kobayashi correspondence, the nonabelian Hodge correspondence, and existence results for...

problems in mathematics In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even...

in a related paper they showed that the Hodge conjecture for integral cohomology is false. The Hodge conjecture for rational cohomology is, as of 2008...

proved that the generalization of the Hodge conjecture for compact Kähler varieties is false. The Hodge conjecture is one of the seven Clay Mathematics Institute...

Game complexity List of unsolved problems in mathematics Unique games conjecture Unsolved problems in computer science A nondeterministic Turing machine...

deal with in terms of the conjectural algebraic geometry (Hodge conjecture and Tate conjecture). In those problems the special situation is more demanding...

theory) is used. Its purpose is to shed light on both the Hodge conjecture and the Tate conjecture, the outstanding questions in algebraic cycle theory. Fix...

number conjecture is a major research problem. Tate conjecture The Tate conjecture (John Tate, 1963) provided an analogue to the Hodge conjecture, also...

to classes in its integral cohomology. It is the only case of the Hodge conjecture which has been proved for all Kähler manifolds. Let X be a compact...

Hodge modules similar to the ones for sheaves. Mixed Hodge structure Hodge conjecture Jacobian ideal Hodge–Tate structure, a p-adic analogue of Hodge...

for Vojta's conjecture, ABC conjecture and so on; in 2012, he published his Inter-universal Teichmuller theory, in which he didn't use Hodge-Arakelov theory...

(coming from the Hodge structure on H1(A)), to the extent that knowledge of G determines the Lie algebra of the Galois image. This conjecture is known only...

conjecture Kelvin's conjecture Kouchnirenko's conjecture Mertens conjecture Pólya conjecture, 1919 (1958) Ragsdale conjecture Schoenflies conjecture (disproved...

will be measured. The seven problems are: P versus NP The Hodge conjecture The Poincaré conjecture – solved, by Grigori Perelman The Riemann hypothesis Yang–Mills...

among the most important open questions in modern mathematics. The Hodge conjecture, one of the Clay Mathematics Institute's Millennium Prize Problems...

Birch and Swinnerton-Dyer conjecture Hodge conjecture Navier–Stokes existence and smoothness P versus NP problem Poincaré conjecture (solved) Riemann hypothesis...

important unsolved problems in mathematics. The first conjecture, which is known as the "smoothness" conjecture, states that there should always exist smooth...

two quintic threefolds in this paper have the following Hodge diamonds. Mirror symmetry conjecture - more mathematically based article Topological quantum...

also follow from the Bass conjecture in algebraic K-theory. For a smooth complex projective variety X, the Hodge conjecture predicts the image (tensored...