best, D

]]>Thanks for promoting the Taubes paper (and also for writing the other posts on this blog site). I look forward to investigating further!

Best wishes,

Paul.

]]>Once again many thanks!

Paul.

]]>I don’t think I’d ever claim anything like that.

]]>I note that at the end of Scorpan, the Fintushel-Stern conjecture states that K3(k’) and K3(k”) are diffeomorphic iff k’ and k” are equivalent knots (modulo reflections?) and that possibly all of the complexity of knot theory is contained in 4-manifold topology. Maybe this relates a little to the paper you recommend (?). I agree with you in that I like the style and broad sweep of the book. In it Scorpan also recommends a nice AT book called Algebraic Topology by Hatcher that clarifies a lot that I had previously found explained too cryptically in say the (original) book by Greenberg – and which is also much broader in scope than the latter.

Thanks for your recommendation of the Taubes 2016 paper and your perspective on this subject.

Paul.

]]>The most recent interesting paper on 4-manifolds that I read (well, skimmed) was http://arxiv.org/abs/1602.01687 by Cliff Taubes, where he gives examples of (families of) 4-manifolds for which certain families of anti-self dual metrics have “limits” of the form (S^3-K)xS^1 with a product metric, where K is a knot in the 3-sphere, and where S^3-K has its hyperbolic metric (so that the product is therefore conformally flat, and so trivially anti-self dual). This is a kind of connection between the 4 and 3 dimensional worlds that looks extremely important to me, but I have nothing nontrivial to add beyond that.

]]>Thanks,

Paul. ]]>

Over my career as an applied mathematician I have acquired many pure maths books (my hobby). One that I have bought recently is called the Wild World of 4-manifolds by Scorpan. It is relatively modern and seems to be packed with interconnections between many branches of pure maths. I had some postgraduate experience of these things before embarking on my (other) career and in the light of your apparent view of 3-manifolds and the way you say you moved away from studying them to some extent after the Poincare conjecture was proved, and also it seems your slight disinterest in algebraic topology that I read elsewhere, I wondered what you think of the study of 4-manifolds? – or is pure maths so vast that chance now plays a big role in what one ends up specialising in?

I hope this question is not too broad or vague. – I have been looking for some entry point that is not too specialised but gives a good overview of where ‘important’ current research areas are and where they may be going…..

My post – grad background is differential geometry and algebraic topology – I have to admit that in the 70’s when I studied it, AG made more of a profound impression on me…(I also very much like number theory and have been trying to learn some basic algebraic geometry – Atiyah Macdonald etc.

Best wishes,

Paul.

]]>Best,

D

]]>Many thanks for this very illuminating blog. Although I was aware that a differential geometric definition of grad, div and curl existed from my undergraduate days 40 years or so ago when Tom Willmore gave a course on DG at Durham University UK, I had never quite assimilated this until now. In the mean time I worked as an industrial mathematician until I retired a few years ago. I have since taken a keen interest again in pure maths.

Just a minor point, at the end of the first paragraph above under your description of div where you say ‘(by taking sharp and wedging)’- did you mean ‘flat’ (and not ‘sharp’)? (otherwise I am confused!).

Thanks again.

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