Some thoughts on mathematics

We are now deep into hiring season, and lots of interviews and offers are happening.  This will be short, but I want to point out a few things to keep in mind when negotiating.  (This will be mostly for postdocs/tenure-track positions at R1 universities.)  Also, remember that this is my opinion, and there are lots of other legitimate views on negotiating. - My view is it is unlikely you will be viewed as a "primadonna" or come in with enemies if you negotiate!  Many people do it and it does not mean you do not appreciate the offer/opportunity that you might be getting.  - Reach out to lots of people to ask for advice negotiating.  Ask your postdoc advisor, PhD advisor, recent hires you know in your field, senior faculty in your department in other fields, etc.  Someone will give you bad advice, someone will have a clever idea you haven't thought of, etc.  It's likely people had different job market experiences than you, and so your advisor's negotiating might have

Here's a quick comment which is orthogonal/complementary to a recent paper , which is also related to some of the results in this paper.  What I'm going to say below has an analogue for Donaldson invariants, but I thought I'd talk about Seiberg-Witten theory today to switch things up.  I want to talk about how certain five-dimensional homology cobordisms govern the Seiberg-Witten invariants of four-manifolds.  The Seiberg-Witten invariants are a powerful gauge theoretic invariant of smooth four-manifolds (and three-manifolds) that gives lots of exciting topological and geometric information.   At a first pass, the Seiberg-Witten invariants take in a four-manifold (closed, oriented, $b^+ > 1$) and can be thought of as vanishing (such as for symplectic manifolds) or non-vanishing (manifolds with positive scalar curvature).  For the second pass, think of it as a function $SW: Spin^c(X) \to \mathbb{Z}/2$.  Per usual, if you don't like spin$^c$ structures, then just