The Hodge Conjecture
Su-Jeong Kang
In 2000, the Clay Mathematics Institute published a list of seven important unsolved mathematics problems. Because of their importance the problems on the Clay Institute’s list were titled the Millennium Prize Problems and the Hodge conjecture was one of the seven problems singled out for recognition. As of today, six of the problems, including the Hodge conjecture, remain unsolved with the Poincaré conjecture being the only one of the Millennium Prize Problem to have been solved.
The Hodge conjecture, originally formulated by the Scottish mathematician William V.D. Hodge and later modified by several other mathematicians, including Alexander Grothendieck, the central figure in modern algebraic geometry, postulates the existence of a surprisingly intimate relationship between the geometry of complex algebraic varieties and the topology of complex algebraic manifolds. The conjecture has been shown to be true in several special instances but a general proof has eluded mathematicians for the last 60 years.
In this talk, I will introduce the historical development of this challenging, but beautiful problem. Several elementary examples will be discussed and we will prove the Hodge conjecture in these instances. If time permits, I will also discuss the progress of my project to generalize the Hodge conjecture to bad objects.
This talk is intended for a general audience, especially for those people who find great beauty in great mathematics!
