Abstract: What do the 'picture hanging problem' and 'Borromean rings' have in common? Their solution can be described by Milnor invariants in link theory, or equivalently by higher cohomological Massey products.

As noticed by B. Mazur, M. Morishita et al, there is a remarkable analogy between the theory of links and pro-$p$-extensions of number fields with ramification restricted to a finite set of primes. We discuss this analogy and give an arithmetic interpretation of Massey products in low degrees. It turns out that certain symmetry relations in the topological world carry over to number theory in special cases only.

We report on the work on applications of higher Massey products in order to construct so-called mild pro-$p$-groups and investigate recent progress in the theory of tamely ramified pro-$p$-extensions by J. Labute and A. Schmidt.

Abstract: In this talk, we will show how to obtain mild pro-$p$-groups in the arithmetic context.

Abstract: This talk is dedicated to the proof of Bott periodicity. First we generalize the clutching function to vector bundles over $X \times S^2$, then we can simplify the clutching function, first to a Laurent polynomial, then to a polynomial, finally to a linear function. And by the discussion on the linear clutching function, we can finally decompose it into a vector bundle with trivial clutching function and another one with clutching function $z$. Finally, we can construct a inverse by the simplified clutching function for the external product to prove that it is an isomorphism.