Some thoughts on mathematics

As we're getting back into job and grantwriting season, maybe I'll make another quick post with some unsolicited advice: Sometimes, you need to have a little *less* faith in your knowledge of the system.   What do I mean by this?  I talk to a lot of folks who seem to tell me things like "I'm not qualified for that job", "I won't apply to School X because they don't want someone in my area", "I got no traction from School Y last year, so won't try again this year", etc.  Almost everyone who has said that to me has received some version of "you're wrong".  There's a good chance if you're saying that, you're probably wrong too.  I know not everyone can get all the jobs or win all the awards, but when someone says that to me often they're usually better than they think they are.  I should write a whole post on impostor syndrome, etc but actually what I want to emphasize in this post is that there's just

Here's a cool fact.  If you have a fibered three-manifold, i.e. your three-manifold is a fiber bundle with fiber a surface and base $S^1$, then the fiber surface minimizes genus among all surfaces in its homology class.  (If the fibered three-manifold has boundary, necessarily a union of tori, then we think about the homology class of the surface rel boundary.)   This says that for fibered knots in $S^3$, the fiber surface is a minimal genus Seifert surface!  Just for funsies I wanted to put a proof of this here, because I think it's a nice idea that shows up in various guises in low-dimensional topology.   If you haven't thought much about fibered three-manifolds, some good examples to keep in mind are: $\Sigma^2 \times S^1$, the complement of a torus knot ( here is an awesome visualization), the complement of the figure eight knot, or 0-surgery on any fibered knot in the three-sphere.       Let $Y$ be a fibered three-manifold with a fiber $\Sigma$.  Form the infinite