The title is a talk given by Sir M. Atiyah in a conference with the following abstract:

**I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory) and algebraic number theory. In this analogy the group of exotic 4-spheres is the counterpart of the Tate-Shafarevich group**

I'm extremely curious about the analogy and I can't find a document on the subject. Would someone enlighten us about this analogy ?

**Edit** I found a link of the conference with many videos, and Atiyah's is now uploaded too!

he has set about tackling one of the greatest challenges in maths: to produce a simple proof of the Feit-Thompson theorem, ...The theorem runs to 255 pages of densely argued text. Sir Michael ... has reduced it to 12 pages, simply explained. This week he sent it to 15 experts in the field and is waiting for their reaction.He knows he faces an army of sceptics. Even his own family think he would be better off enjoying a quiet retirement. But he cannot accept that age is an insuperable barrier.$\endgroup$3more comments