I’m going to be fairly busy in the coming days, so instead of the usual long post, I’m going to post here some links to interesting stuff I’ve found online (related to the subjects stated on the title of this post).
In the previous post, An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, we discussed Calabi-Yau manifolds (which are special cases of Kahler manifolds) and how their interesting properties, namely their Riemannian, symplectic, and complex aspects figure into the branch of mathematics called mirror symmetry, which is inspired by the famous, and sometimes controversial, proposal for a theory of quantum gravity (and more ambitiously a candidate for the so-called “Theory of Everything”), string theory.
We also mentioned briefly a famous open problem concerning Kahler manifolds called the Hodge conjecture (which was also mentioned in Algebraic Cycles and Intersection Theory). The links I’m going to provide in this post will be related to this conjecture.
As with the post An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, aside from introducing the subject itself, another of the primary intentions will be to motivate and explore aspects of algebraic geometry such as complex algebraic geometry, and their relation to other branches of mathematics.
Here is the page on the Hodge conjecture, found on the website of the Clay Mathematics Institute:
Hodge Conjecture on Clay Mathematics Institute
We have mentioned before that the Hodge conjecture is one of seven “Millenium Problems” for which the Clay Mathematics Institute is offering a million dollar prize. The page linked to above contains the official problem statement as stated by Pierre Deligne, and a link to a lecture by Dan Freed, which is intended for a general audience and quite understandable. The lecture by Freed is also available on Youtube:
Dan Freed on the Hodge Conjecture at the Clay Mathematics Institute on Youtube
Unfortunately the video of that lecture has messed up audio (although the lecture remains understandable – it’s just that the audio comes out of only one side of the speakers or headphones). Here is another set of videos by David Metzler on Youtube, which explains the Hodge conjecture (along with the other Millennium Problems) to a general audience:
Millennium Problem Talks on Youtube
The Hodge conjecture is also related to certain aspects of number theory. In particular, we have the Tate conjecture, which is another conjecture similar to the Hodge conjecture, but more related to Galois groups (see Galois Groups). Alex Youcis discusses it on the following post on his blog Hard Arithmetic:
The Tate Conjecture over Finite Fields on Hard Arithmetic
On the same blog there is also a discussion of a version of the Hodge conjecture called the -adic Hodge conjecture on the following post:
An Invitation to p-adic Hodge Theory; or How I Learned to Stop Worrying and Love Fontaine on Hard Arithmetic
The first part of the post linked to above discusses the Hodge conjecture in its classical form, while the second part introduces -adic numbers and related concepts, some aspects of which were discussed on this blog in Valuations and Completions.
A more technical discussion of the Hodge conjecture, Kahler manifolds, and complex algebraic geometry can be found in the following lecture of Claire Voisin, which is part of the Proceedings of the 2010 International Congress of Mathematicians in Hyderabad, India:
On the Cohomology of Algebraic Varieties by Claire Voisin
More about these subjects will hopefully be discussed on this blog at sometime in the future.