We recall the Zariski closure of an image, show that the image may not be the same as the Zariski closure, and ask what can be said about image of an affine variety under a polynomial map in general. We answer this question for the case V ⊂ Rn, and then prove that if K = C then the Zariski closure of the image coincides with the closure in the standard topology. Finally restricting our focus to regular maps, we show that the image of a projective variety in an algebraically closed field is Zariski closed and demonstrate some consequences of this result

Galois cohomology is a topic that should interest a wide range of audiences: number theorists, algebraic geometers, homotopy theorists, etc. It is a wonderful example of an application of ideas from algebraic topology to study algebra and number theory. In this talk, we will discuss the basics of Galois cohomology, and we demonstrate its power by using it to prove the Mordell-Weil theorem for elliptic curves. If time permits, we might also discuss a little bit about its generalization: étale cohomology.