We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

We exhibit large families of K3 surfaces with real multiplication, both abstractly, using lattice theory, the Torelli theorem and the surjectivity of the period map, as well as explicitly, using dihedral covers and isogenies.

We consider the extremes of the logarithm of the characteristic polynomial of matrices from the C$\beta $E ensemble. We prove convergence in distribution of the centered maxima (of the real and imaginary parts) toward the sum of a Gumbel variable and another independent variable, which we characterize as the total mass of a ‘derivative martingale’. We also provide a description of the landscape near extrema points.