When simulating fluids with moving boundaries, current methods avoid geometry and topology by converting them into numerically solving PDEs. To remove limitations of this approach, we tackle geometric and topological problems with tools in geometry and topology. First, we propose a topological space, called the Yin space, as a mathematical model of physically meaningful regions. Second, we equip the Yin space, both theoretically and algorithmically, with a Boolean algebra for fluids of arbitrarily complex topology. Third, we develop the MARS framework to unify current interface tracking methods, to analyze their accuracy and stability, and to foster fourth- and higher-order methods of interface tracking and curvature estimation. Traditional finite difference and finite volume methods have long been criticized for their inadequacy of dealing with irregular geometries. This weakness can be very much alleviated by MARS. We demonstrate this point by a fourth-order projection method for numerically solving the incompressible Navier-Stokes equations with structured rectangular grids on irregular domains.