List of undecidable problems

In computability theory, an undecidable problem is a decision problem for which an effective method to derive the correct answer does not exist. More formally, an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language. There are uncountably many undecidable problems, so the list below is necessarily incomplete. Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e., such undecidable languages may be recursively enumerable.
Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols represent the same object or not.
For undecidability in axiomatic mathematics, see List of statements undecidable in ZFC.

Problems about abstract machines

The halting problem and the mortality problem.
Determining whether a Turing machine is a busy beaver champion.
Rice's theorem states that for all nontrivial properties of partial functions, it is undecidable whether a given machine computes a partial function with that property.
The halting problem for a register machine: a finite-state automaton with no inputs and two counters that can be incremented, decremented, and tested for zero.
Universality of a nondeterministic pushdown automaton: determining whether all words are accepted.
Conway's Game of Life on whether, given an initial pattern and another pattern, the latter pattern can ever appear from the initial one.
Rule 110 - most questions involving "can property X appear later" are undecidable.

Hilbert's Entscheidungsproblem.
Type inference and type checking for the second-order lambda calculus.
Determining whether a first-order sentence in the logic of graphs can be realized by a finite undirected graph.
Trakhtenbrot's theorem - Finite satisfiability is undecidable.
Satisfiability of first order Horn clauses.
Determining whether a λ-calculus formula has a normal form.
The Post correspondence problem: whether a tag system halts. There are many variants thereof.
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.
Some other problems about CFG are also undecidable. See the page section for details.
The emptiness problem: determining whether a language is empty given some representation of it, such as a finite-state automaton. It is undecidable for context-sensitive grammars.

The mortal matrix problem.
Determining whether a finite set of upper triangular 3 × 3 matrices with nonnegative integer entries generates a free semigroup.
Determining whether two finitely generated subsemigroups of integer matrices have a common element.
Given a finite set of n×n matrices, their joint spectral radius is defined as. For a set of 2 matrices with rational entries, the problem of deciding whether their joint spectral radius is is undecidable.

Determining whether two finite simplicial complexes are homeomorphic.
Determining whether a finite simplicial complex is 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.

Hilbert's tenth problem: the problem of deciding whether a Diophantine equation has a solution in integers.

For functions in certain classes, the problem of determining: whether two functions are equal, known as the zero-equivalence problem ; 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

Determining whether a quantum mechanical system has a spectral gap.
In the ray tracing problem for a 3-dimensional system of reflective or refractive objects, determining if a ray beginning at a given position and direction eventually reaches a certain point.
Determining if a particle path of an ideal fluid on a three dimensional domain eventually reaches a certain region in space.