Topology and symmetry are two fundamental and intertwined concepts driving the behavior of fermionic systems in both condensed-matter and high-energy physics. The goal of the TOPSIM project is to address open problems concerning topological states of fermionic matter from an experimental point of view, by taking advantage of novel possibilities of quantum control on synthetic systems formed by ultracold neutral atoms. We will investigate the behavior of fermionic matter under strong gauge fields in order to study quantum Hall physics and the emergence of topological order in a fully tunable experimental geometry. We will also synthesize fermionic systems exhibiting enlarged interaction symmetries beyond the SU(2) symmetry of electrons, which will allow us to experimentally realize, for the first time, SU(N) models that have no other experimental counterpart in physics, and to use them to study the emergence of long-sought topological states of matter. With these ambitious goals, the TOPSIM project will considerably advance our understanding of topological fermionic matter, paving the way to new methods of investigation of open questions in both high- and low-energy physics, by approaching many-body problems with metrological quantum control.