Explore the fascinating world of conical sums in this one-hour lecture by Federico Zerbini at the Hausdorff Center for Mathematics. Delve into the concept of periods defined by infinite sums over lattice points in cones of $\mathbb R^n$, including special cases like multiple zeta values and Matsumoto-Witten zeta values associated with semisimple Lie algebras. Discover the relevance of conical sums in string theory amplitude calculations and learn about Terasoma's proof regarding the $\mathbb Q^{\mathrm{ab}}$-algebra generated by conical sums. Examine the conjecture that all relations in this algebra stem from cone decompositions and investigate Dupont's conjecture on the motivic nature of conical sums. Gain insights into the current state of research on these open questions and their potential implications for Matsumoto-Witten zeta values.