Explore computational topology in this conference talk that presents two new results. Delve into the algorithmic embeddability problem and its undecidability in higher dimensions. Discover the r-Tverberg problem, a generalization of embeddability, and learn about its polynomial-time decidability in the metastable range of dimensions. Gain insights into piecewise-linear embeddings, simplicial complexes, and Euclidean space as the speaker discusses joint work on these complex topological challenges.