Imperial mathematician wins Satter Prize for groundbreaking number theory work

Professor Ana Caraiani / Image credit: Louise Rose Photography

Professor Ana Caraiani has won the Ruth Lyttle Satter Prize by the American Mathematical Society for her work on the Langlands Program.

The award honours her contributions to arithmetic geometry and number theory, which builds upon the proof employed in Fermat's Last Theorem.

The Satter Prize, awarded every two years to a woman for outstanding research in mathematics, is a significant recognition for Professor Caraiani, whose work leading up to this prize spans more than a decade.

Professor Caraiani's research primarily focuses on the Langlands program, one of the most ambitious projects in modern mathematics that seeks to bridge disparate areas of mathematics, particularly number theory and analysis.

Building bridges in mathematics

Professor Caraiani’s work, characterised by its depth and technical difficulty, tackles some of the most challenging concepts in mathematics. One of her key achievements relates to a concept called ‘modularity,’ which was famously used by Sir Andrew Wiles from the University of Oxford to prove Fermat’s Last Theorem.

Sir Andrew proved that one could translate between two seemingly unrelated concepts in mathematics – modular forms and elliptic curves.

Elliptic curves are smooth curves that can be described using specific polynomial equations. Modular forms are highly symmetrical objects in 'harmonic analysis,' a branch of mathematics developed to study problems in physics such as planetary orbits and vibrating strings.

To prove mathematics’ most infamous theorem, Sir Andrew showed that every elliptic curve had a matching modular form. This fact is also a crucial instance of Langlands’ conjectures, which seek to form a ‘grand unified theory’ by bridging distant areas in mathematics.

One problem that seems very difficult in one area can be translated through this connection to a different area, where the problem might become easier. Professor Ana Caraiani Department of Mathematics

Building such bridges not only reveals profound relationships between seemingly unrelated concepts, but can also provide tools to solve challenging problems. Professor Caraiani said: “One problem that seems very difficult in one area can be translated through this connection to a different area, where the problem might become easier.”

Professor Caraiani extended the idea of modularity to show this relationship holds over more generally defined number systems, known as imaginary quadratic fields. The Taniyama-Shimura-Weil conjecture was known to hold true over all rational numbers, which are numbers that can be defined in terms of a fraction.

The number system Professor Caraiani tackled contains not only these rational numbers, but also the square roots of certain negative numbers (hence the term ‘imaginary’).

These more complex number systems make the math significantly more challenging, but by applying new techniques, Professor Caraiani and her collaborators were able to show that modularity still holds in these settings.

A role model in mathematics

Professor Caraiani’s achievements are not limited to theoretical breakthroughs. Her work also serves as an inspiration for future mathematicians.

“Pure mathematics was quite male-dominated back then,” she said, “When I was a student, I didn't have so many role models to look up to, and I guess one way to find role models is to look at the winners of a prize like this.”

Reflecting on the personal significance of receiving the Satter Prize, Professor Caraiani mentioned how the women who won it before inspired her. “The people who had got this prize back then were heroes to me,” she said.

Professor Caraiani has been at Imperial since 2017 where she continues her influential work on the Langlands program.