My research focusses on rational points on varieties. Of special interest are curves and Abelian varieties. For curves, p-adic analytic Chabauty methods in combination with algebraic covering techniques have recently yielded substantial results. The rational points of Abelian varieties, which form finitely generated groups called Mordell-Weil groups, form an important ingredient of those methods.
Mordell-Weil groups, and most importantly their free rank, are most commonly analysed using descent methods to compute Selmer groups. These provide a bound on the free rank but, unfortunately, not always a sharp one. The difference is measured by the Shafarevich-Tate group. Recently visualisation methods have been used to provide lower bounds on Shafarevich-Tate groups.