10 breakthroughs in mathematics and theoretical computer science, explained simply

Mathematics and theoretical computer science rarely make headlines the way a new smartphone or a court ruling does, but a handful of results in these fields have quietly reshaped what computers can do, what encryption can promise, and what counts as a solved problem. Here are ten of the most significant, chosen for how much they changed the field around them rather than for pure technical difficulty alone.

Fermat's Last Theorem, which states that no three positive whole numbers can satisfy a certain simple equation once the exponent is greater than two, went unproved for roughly 350 years after the 17th-century mathematician Pierre de Fermat claimed to have a proof he never wrote down. The British mathematician Andrew Wiles finally proved it in 1994, in a proof running hundreds of pages that drew on techniques far outside what Fermat could have known, closing one of mathematics' most famous open questions.

The Poincaré Conjecture, a statement about the shape of three-dimensional spaces first posed in 1904, remained unresolved until the Russian mathematician Grigori Perelman published a proof in 2002 and 2003 using a technique called Ricci flow, which smooths out irregularities in geometric shapes over time. It remains the only one of the seven Millennium Prize Problems, a list of major unsolved questions with a million-dollar reward each, to have been solved to date; Perelman turned down both the prize money and the Fields Medal.

In 1976, the Four Color Theorem, which holds that any map can be coloured using only four colours so that no two adjacent regions share a colour, became the first major mathematical theorem proved with substantial help from a computer, when Kenneth Appel and Wolfgang Haken used a program to check thousands of individual map configurations that would have been impractical to verify by hand. The result sparked a long-running debate over whether a proof too large for a human to check line by line still counts as a proof.

Progress on the twin prime conjecture, the still-unproved idea that infinitely many pairs of primes differ by exactly two, took a major step forward in 2013 when Yitang Zhang, then a relatively unknown lecturer, proved that infinitely many pairs of primes differ by less than 70 million. A rapid collaborative effort involving mathematician James Maynard and others brought that gap down to 246, though closing it to the conjectured value of two remains unsolved.

Peter Shor's 1994 algorithm showed that a sufficiently powerful quantum computer could factor large numbers exponentially faster than any known method on a classical computer, a result that mattered enormously because much of modern encryption, including the systems that secure online banking and communications, relies on factoring large numbers being computationally infeasible. Shor's algorithm remains a central reason governments and companies are racing to develop both quantum computers and quantum-resistant encryption.

The PCP theorem, developed in the early 1990s by a group of theoretical computer scientists, showed that every mathematical proof can be rewritten in a form that a verifier can check for correctness by examining only a small, randomly chosen fraction of it, while still catching any error with high probability. The result transformed the study of approximation algorithms, providing a rigorous way to show that certain optimization problems cannot be approximated efficiently, not just that nobody had yet found a way to do so.

Deciding whether two large, complex graphs are structurally identical, known as the graph isomorphism problem, had resisted efficient general algorithms for decades. In 2015, the mathematician László Babai announced a quasi-polynomial time algorithm for the problem, a significant improvement over previous approaches and a notable advance in computational complexity theory, an area concerned with classifying how difficult different computational problems are.

The question of how to pack spheres as densely as possible in a given number of dimensions had been solved only for a handful of low dimensions until the mathematician Maryna Viazovska proved the optimal packing for eight-dimensional space in 2016, and shortly after, working with several collaborators, extended the technique to solve the same problem in 24 dimensions, results that also had implications for information theory and error-correcting codes.

The Erdős discrepancy problem, posed by the prolific mathematician Paul Erdős, concerned how unevenly a sequence of plus-one and minus-one values could be distributed. A special case was verified in 2014 using a computer-based SAT solver in a proof so large it required roughly 13 gigabytes to store, at the time one of the largest mathematical proofs ever produced. The following year, Terence Tao proved the full, general conjecture using more traditional analytic methods, resolving a problem that had stood open since the 1930s.

What unites these results, mathematicians say, is less their subject matter than what they reveal about how mathematical progress happens: sometimes through decades of incremental refinement by many researchers, as with prime gaps, and sometimes through a single insight that reframes an entire problem, as with Ricci flow or the PCP theorem, with computers increasingly playing a role either as a verification tool or, more recently, as a research collaborator in its own right.