We will present some very recent results on CMC surfaces in spheres, investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any \(h\in (0,1)\) there exist CMC planes and cylinders in \(S^5\), where \(h\) is the mean curvature of the immersion, while a necessary and sufficient condition on \(h\) is found for the existence of CMC tori in \(S^5\).

Let \(b\geq 2\) be an integer and \(v\) a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers \(\xi\) with the property that, for every sufficiently large integer \(N\), there exists an integer \(n\) such that \(1 \leq n \leq N\) and the distance between \(b^n \xi\) and its nearest integer is at most equal to \(b^{-v N}\). We further solve the same question when replacing \(b^n\xi\) by \(T^n_\beta \xi\), where \(T_\beta\) denotes the classical \(\beta\)-transformation. This is a joint work with Yann Bugeaud.

In my talk I report about a recent joint work with Alexei Myasnikov and Armin Weiss which was presented at LATIN 2014 in Montevideo. The motivation stems from algorithmic group theory. It concerns the conjugacy problem for two prominent groups: the Baumslag-Solitar group and the Baumslag's group \(BG(1,2)\). The word problem and the conjugacy problem in the Baumslag-Solitar group is easy, but this does not transfer to \(BG(1,2)\).

Our main result shows that conjugacy in \(BG(1,2)\) can be solved in polynomial time in a strongly generic setting. The result is surprising because our algorithm has non-elementary average case complexity; and we conjecture that this is the best we can expect. This is interesting in a broader sense since it relates a natural conjugacy problem in algorithmic group theory to integer division in power circuits. A power circuit is a data structure which allows to represent huge numbers involving tower functions by small graphs. Actually, the complexity of the division problem in power circuits is an open and interesting problem in arithmetic.