Forms in many variables: algebra, geometry, logic

I shall discuss the following question: given a homogeneous polynomial in "much more" variables than its degree, does it have a nontrivial zero? The answer depends, of course, on the field of coefficients of the polynomial. I shall present several results in the area, from a classical theorem of Chevalley to a very recent result of Denef. I also intend to explain how methods of algebraic geometry and mathematical logic are useful in studying this kind of problem.

You can watch the lecture here: 

http://www.youtube.com/watch?v=8aniG9bu5OQ&list=PL_0phSnA7tyQJDn-5hacAuELRteKNuyl-