When it comes to postdoc positions in mathematics in the US, does it make any difference for the hiring committees if your thesis adviser is a big name (say, someone like Gromov or Milnor)?
I'm surely not the most qualified person on this site to answer, but I have been on a postdoc hiring committee in pure mathematics and I quite disagree with the existing answers. The single most important factor is going to be your letters of recommendation (with second on the list being your publications and preprints). I think having a famous advisor is going to be of minor importance in comparison; most people understand that also great mathematicians sometimes have students that are just mediocre, so that it's no great predictor of future success. In fact I heard in a sense the opposite argument being made, that if an applicant from a smaller university with a less famous advisor has strong letters from big names, then this is more impressive than if a student at a "top" university had the exact same letters.
In spite of the obsession with objective assessment and quantification that supposedly governs science, personal recommendations often still matter the most during the hiring process. Hence, if you have a big shot PhD advisor, you'll probably have an easier time finding a good postdoc in a group that your advisor has good contacts with.
If you don't have the big shot backing, you can (partly) make up for it by approaching potential postdoc advisors in person (e.g. at a conference) and convincing them that you're not only a scientist with high potential but also a nice person.
The point is, when you apply for a postdoc in a good group, your application is likely to go to the "meh" pile if neither your nor your advisor's name ring a bell with the group leader.