Dr. Atiyah’s early work was in topology, a field of mathematics that studies shape, including that of mathematical objects with many more than three dimensions. Though such objects can’t be visualized, topology provides tools to figure out how many holes they have and how different parts of an object are connected to one another. Topology considers shapes to be elastic and malleable — able to be stretched or squished without their fundamental nature being changed, as long as no new holes are punched and no pieces are newly joined together. Working with the German mathematician Friedrich Hirzebruch, Dr. Atiyah developed a topological tool called K-theory.

Dr. Atiyah teamed up with Dr. Singer in the early 1960s. Dr. Singer is a specialist in mathematical analysis, the study of differential equations, which are used to describe physical phenomena in the language of calculus.

The equations are extremely useful for describing real-world situations, but they have a wicked problem: No one knows how to solve them precisely. Dr. Atiyah and Dr. Singer set out to see if Dr. Atiyah’s topological tools might help find the solutions.

Although they couldn’t find the exact solutions to any differential equation, they did manage to use topology to reveal the number of solutions such an equation has. This became their famous Atiyah-Singer Index Theorem, which they developed into an entire field, called index theory.

“It’s a bit of black magic,” Dr. Atiyah said in 2015, “to figure things out about differential equations even though you can’t solve them.”

But that was just the beginning of the connections that the index theory would make. In the mid-1970s, in the middle of this work, Dr. Atiyah learned something surprising: Physicists had been creating their own, less formal version of index theory in parallel with the mathematicians. They were using it to try to understand quantum field theory.

Dr. Atiyah and Dr. Singer teamed up with the mathematician Raoul Bott and Dr. Witten, who was then barely out of graduate school. The team (and soon many others) used index theory to see how discoveries in mathematics revealed truths about physics, and how physical facts revealed mathematical insights. In the process, they transformed both fields.