entries generates a free semigroup. Determining whether two finitely generated subsemigroups of integer matrices have a common element. Problems in combinatorial group theory The word problem for groups. The conjugacy problem. The group isomorphism problem. Problems in topology Determining whether two finite simplicial complexes are homeomorphic. Determining whether a finite simplicial complex is (homeomorphic to) a manifold. Determining whether the fundamental group of a finite simplicial complex is trivial. Determining whether two non-simply connected 5-manifolds are homeomorphic, or if a 5-manifold is homeomorphic to S5. Problems in analysis For functions in certain classes, the problem of determining: whether two functions are equal, known as the zero-equivalence problem (see Richardson's theorem); the zeroes of a function; whether the indefinite integral of a function is also in the class. Of course, some subclasses of these problems are decidable. For example, there is an effective decision procedure for the elementary integration of any function which belongs to a field of transcendental elementary functions, the Risch algorithm. "The problem of deciding whether the definite contour multiple integral of an elementary meromorphic function is zero over an everywhere real analytic manifold on which it is analytic", a consequence of the MRDP theorem resolving Hilbert's tenth problem. Determining the domain of a solution to an ordinary differential equation of the form where x is a vector in Rn, p(t, x) is a vector of polynomials in t and x, and (t0, x0) belongs to Rn+1. Problems about formal languages and grammars The Post correspondence problem. Determining if a context-free grammar generates all possible strings, or if it is ambiguous. Given two context-free grammars, determining whether they generate the same set of strings, or whether one generates a subset of the strings generated by the other, or whether there is any string at all that both generate. Other problems The problem of determining if a given set of Wang tiles can tile the plane. The problem of determining the Kolmogorov complexity of a string. Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers. Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions, or in a piecewise-linear flow in three dimensions. Determining whether a λ-calculus formula has a normal form. Conway's Game of Life