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Open Problems

First, I want to give a quick advertisement for a Floer homology open problem website run by John Baldwin ( https://floerhomologyproblems.blogspot.com/ ).  This has a mix of problems, some of which seem very hard, and some of which might be a bit more manageable.  If I was more thoughtful, I would collect a bunch of different open problem lists here for reference, but I'm not that thoughtful.  Speaking of which, if you're a younger grad student or someone adjacent but not in low-dimensional topology, you may not be familiar with the famous Kirby problem list from the 1990s.  (That is an update of an initial list made by Kirby many years earlier.)  Many of these problems have been solved (e.g. the 3D Poincare Conjecture), but many of the outstanding ones are the biggest problems in the field (e.g. the 4D Smooth Poincare Conjecture, additivity of unknotting number under connected sum, etc) and have guided a lot of research directions.  If you solve a Kirby problem, it definitely
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Negotiating job offers

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
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Surgery, Seiberg-Witten invariants, and cobordisms

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
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Smith theory (part 2)

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In Smith theory (part 1) we discussed the Smith Conjecture which stated that the fixed point set of a smooth cyclic action on the 3-sphere is always unknotted.   Today, the goal is to give a hands on disproof of the 4D Smith Conjecture.  We'll cook up $\mathbb{Z}/2$-actions on $S^4$ with knotted fixed point set.  (The action we build will be smooth and the fixed point set will be knotted in the topological and smooth categories.) To do this, we need a more interesting way to decompose $S^4$ into two pieces.  First, an observation.  Consider a Mazur manifold $W$ - a four-manifold built by attaching a single 1-handle to $B^4$ and a 2-handle that algebraically links it once.  (The framing can be anything, the 2-handle can be knotted in whatever way.)  Here's an example: These manifolds will be contractible, but might still not be homeomorphic to $B^4$; in fact, if your 2-handle is attached along anything other than $pt \times S^1$, the resulting boundary will *not*
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