Hilbert's problems
Hilbert's Problems
Hilbert's problems are a set of 23 unsolved problems in mathematics, presented by the German mathematician David Hilbert in 1900. These problems were intended to serve as a guiding framework for mathematical research in the 20th century. They cover a wide range of topics, from number theory to algebraic geometry, and have significantly influenced the development of modern mathematics.
Background
David Hilbert presented his list of problems at the International Congress of Mathematicians in Paris on August 8, 1900. His address, titled "Mathematical Problems," was a call to action for mathematicians to tackle these challenging questions. Hilbert's problems were not only a reflection of the mathematical landscape of his time but also a vision for future research directions.
The 23 Problems
The 23 problems Hilbert proposed are diverse in nature and complexity. Some have been solved, some partially resolved, and others remain open to this day. Below is a detailed examination of each problem:
Problem 1: The Continuum Hypothesis
The Continuum Hypothesis (CH) posits that there is no set whose cardinality is strictly between that of the integers and the real numbers. This problem was the first on Hilbert's list and has profound implications for set theory and the foundations of mathematics. In 1963, Paul Cohen proved that CH is independent of the standard axioms of set theory, meaning it can neither be proved nor disproved from these axioms.
Problem 2: The Consistency of Arithmetic
Hilbert's second problem asks for a proof that the axioms of arithmetic are consistent, meaning they do not lead to any contradictions. This problem is closely related to the work of Kurt Gödel, who, in 1931, demonstrated through his incompleteness theorems that no consistent system of axioms whose theorems can be listed by an algorithm is capable of proving its own consistency.
Problem 3: The Equality of Volumes of Tetrahedra
This problem involves proving that two tetrahedra of equal base area and height have equal volume, which was resolved by Max Dehn in 1902. Dehn's solution introduced the concept of the Dehn invariant, a crucial tool in the study of polyhedral volumes.
Problem 4: The Straight Line as the Shortest Distance Between Two Points
Hilbert's fourth problem is concerned with the characterization of geometries in which the straight line is the shortest path between two points. This problem has connections to the field of metric geometry and has seen various partial solutions over the years.
Problem 5: Lie Groups and Lie Algebras
This problem asks whether every locally Euclidean topological group is a Lie group. It was resolved affirmatively by Andrew Gleason, Deane Montgomery, and Leo Zippin in the 1950s, establishing a deep connection between topology and algebra.
Problem 6: Axiomatization of Physics
Hilbert's sixth problem calls for the development of a rigorous axiomatic framework for all of physics, akin to the axiomatization of geometry. This ambitious problem remains largely open, though significant progress has been made in areas such as quantum mechanics and statistical mechanics.
Problem 7: Irrationality and Transcendence of Certain Numbers
This problem involves proving the irrationality and transcendence of certain numbers, such as \( e^{a} \) where \( a \) is a non-zero algebraic number. Aleksandr Gelfond and Theodor Schneider independently solved this problem in 1934, leading to the Gelfond-Schneider theorem.
Problem 8: The Riemann Hypothesis
The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics. It posits that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2. This hypothesis has profound implications for number theory and the distribution of prime numbers.
Problem 9: Proof of the Most General Reciprocity Law in Any Number Field
This problem seeks a proof of the most general form of the reciprocity law in number fields. It was resolved by Emil Artin in 1927, who generalized the law of quadratic reciprocity to a broader class of number fields.
Problem 10: Determination of the Solvability of a Diophantine Equation
Hilbert's tenth problem asks for an algorithm to determine whether a given Diophantine equation has an integer solution. This problem was resolved negatively by Yuri Matiyasevich in 1970, who built on earlier work by Martin Davis, Hilary Putnam, and Julia Robinson.
Problem 11: Quadratic Forms with Any Algebraic Numerical Coefficients
This problem involves the representation of integers by quadratic forms with algebraic coefficients. Significant progress has been made, particularly through the work of Hermann Minkowski and others in the theory of quadratic forms.
Problem 12: Extension of Kronecker's Theorem on Abelian Fields to Any Algebraic Realm of Rationality
Hilbert's twelfth problem seeks to extend Leopold Kronecker's theorem on abelian fields to more general algebraic settings. This problem remains partially open, with significant contributions from the theory of complex multiplication and class field theory.
Problem 13: Impossibility of the Solution of the General Equation of the Seventh Degree by Means of Functions of Only Two Arguments
This problem asks whether a general equation of the seventh degree can be solved using functions of only two variables. Vladimir Arnold and Andrey Kolmogorov showed that such functions exist, providing a solution to this problem.
Problem 14: Finiteness of Certain Systems of Functions
Hilbert's fourteenth problem is concerned with the finiteness of certain systems of algebraic functions. This problem was resolved negatively by Masayoshi Nagata in 1959, who provided a counterexample.
Problem 15: Rigorous Foundation of Schubert's Enumerative Calculus
This problem seeks a rigorous foundation for Schubert calculus, a method in enumerative geometry. Significant progress has been made through the development of intersection theory and algebraic geometry.
Problem 16: Topology of Algebraic Curves and Surfaces
Hilbert's sixteenth problem involves the study of the topology of algebraic curves and surfaces, particularly the arrangement of ovals of a real algebraic curve. This problem remains partially open, with ongoing research in real algebraic geometry.
Problem 17: Representation of Definite Forms by Squares
This problem asks whether a positive definite rational function can be represented as a sum of squares of rational functions. Emil Artin solved this problem in 1927, showing that such a representation is always possible.
Problem 18: Building Space with Congruent Polyhedra
Hilbert's eighteenth problem involves the tiling of space with congruent polyhedra and the classification of such tilings. This problem includes the famous Kepler conjecture, which was resolved by Thomas Hales in 1998.
Problem 19: Analyticity of Solutions to Variational Problems
This problem asks whether the solutions to regular variational problems are always analytic. Ennio De Giorgi and John Nash independently resolved this problem in the 1950s, showing that solutions are indeed analytic under certain conditions.
Problem 20: General Boundary Value Problems
Hilbert's twentieth problem involves the existence of solutions to general boundary value problems. Significant progress has been made through the development of partial differential equations and functional analysis.
Problem 21: Linear Differential Equations with Prescribed Monodromy
This problem asks for the existence of linear differential equations with a given monodromy group. Josip Plemelj provided a solution in 1908, though further generalizations and refinements have been made since.
Problem 22: Uniformization of Analytic Relations by Means of Automorphic Functions
Hilbert's twenty-second problem involves the uniformization of analytic relations using automorphic functions. This problem has been largely resolved through the development of Teichmüller theory and moduli spaces.
Problem 23: Further Development of the Calculus of Variations
The final problem on Hilbert's list calls for the further development of the calculus of variations. This field has seen significant advancements, particularly in the study of minimal surfaces, geometric measure theory, and optimal control theory.
Impact and Legacy
Hilbert's problems have had a profound impact on the development of mathematics. They have inspired generations of mathematicians to tackle some of the most challenging questions in the field. The problems have also led to the creation of new areas of research and the development of new mathematical techniques and theories.