"Title: Discrete and convex geometry: challenges, methods, applicationsAbstract: Research in discrete and convex geometry, using tools from combinatorics, algebraictopology, probability theory, number theory, and algebra, with applications in theoreticalcomputer science, integer programming, and operations research. Algorithmic aspects areemphasized and often serve as motivation or simply dictate the questions. The proposedproblems can be grouped into three main areas: (1) Geometric transversal, selection, andincidence problems, including algorithmic complexity of Tverberg's theorem, weakepsilon-nets, the k-set problem, and algebraic approaches to the Erdos unit distance problem.(2) Topological methods and questions, in particular topological Tverberg-type theorems,algorithmic complexity of the existence of equivariant maps, mass partition problems, and thegeneralized HeX lemma for the k-coloured d-dimensional grid. (3) Lattice polytopes and randompolytopes, including Arnold's question on the number of convex lattice polytopes, limitshapes of lattice polytopes in dimension 3 and higher, comparison of random polytopes andlattice polytopes, the integer convex hull and its randomized version."