Selected problems in applied and computational topology

Applied and computational topology is a new mathematical field where deep achievements of algebraic and geometric topology, which made a history of 20th century mathematics (index theorems, K‐theory, Poincare conjecture, knot theory…) are combined with powerful computational techniques, which allow explicit determination of basic topological quantities that were previously beyond reach. Although sporadic applications of topology to robotics, complexity theory and image analysis were known before, the new field emerged around the turn of the century with almost contemporary appearance of persistent homology of point clouds (Carlsson, Edelsbrunner), topological robotics (Farber) and discrete Morse theory (Forman). New methods were soon applied to a wide range of problems in data analysis, sensor networks, robot motion planning, pattern recognition, computer graphics, machine learning, numerical PDE, statistical ranking and many others. Our proposal is built around two main research themes: (1) Explicit estimates of the size of triangulations of manifolds and polyhedra. In most applications geometric objects are triangulated, i.e., decomposed into segments, triangles, tetrahedra, and their higher-dimensional analogues called simplexes. Efficiency of algorithms that analyze and manipulate geometric objects is crucially affected by the size (number of simplexes) of a specific triangulation. To exemplify, in Blue Brain Project, a major international multidisciplinary scientific project (https://www.epfl.ch/research/domains/bluebrain/), geometry of human brain is given as millions of measured points and its shape is described by data analysis based on statistical and topological methods. To a topologist, brain is like a gigantic network with dots (neurons) and pathways (synapses). From these one can infer the existence of higher dimensional geometric structures, which arise when a group of neurons forms a clique, with each neuron connecting to every other neuron in the group (see https://actu.epfl.ch/news/scientists‐discover‐hidden‐patterns‐of‐brain‐activ/). The minimal size of a triangulation is a very elusive quantity, due to many ways a given space can be triangulated. We have recently discovered that covering type, a new invariant introduced by Karoubi and Weibel in 2016, allows estimating the number of vertices in a triangulation. Combined with the Generalized Lower Bound Theorem (proved by Adiprasito in 2018, crowning almost 50 years of research), we can obtain very good estimates of the overall size of triangulations. Covering type of X is the minimal number of sets in a good cover of any space homotopy equivalent to X. Being homotopy invariant by definition, it is related to other invariants, in particular Lusternik‐Schnirelmann category and cup‐length. We propose several strategies how to determine covering type of geometric objects that arise in applications, including computational methods for spaces given as point clouds. (2) Study of optimal paths in Riemannian manifolds. A central problem in robotics is motion planning. For us a robot is any mechanical device (cleaning machine, drone, industrial arm, etc.) that is self‐guided according to some algorithm. Often a robot must remain operative in a changing environment like rearranged workstations in a factory, presence of obstacles and other robots, etc. Usually small changes do not affect the robot, what counts are changes in the topology of its workspace. This led Farber to introduce topological complexity as a measure for the minimal number of instructions that enable a robot to operate autonomously. We will study complexity of robot motion planning under several constraints regarding efficiency (optimality with respect to trajectory length or energy consumption) and ambient variations (parametrized complexity). We will use computational methods to study motion planning in workspaces that are only partly known or given in a discretized form.