The Clay Mathematics Institute is a global organisation dedicated to furthering the beauty, power and universality of mathematical thought.
The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.
Experiment and computer simulations suggest the existence of a “mass gap” in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.
The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper, it asserts that all the ‘non-obvious’ zeros of the zeta function are complex numbers with real part 1/2.
Mathematical Institute, University of Oxford
Mathematical Institute, University of Oxford
Mathematical Institute, University of Oxford
Mathematical Institute, University of Oxford
