Michael Atiyah

“[T]he 19th century... started... with Gauss and went on with the development of the notion of curvature in 2 dimensions, by one function.”

“[A]lthough the sphere... is curved intrinsically because you can't draw a precise map of the atlas of the world on a piece of paper without distortion, if you have something on a scroll.. you... unroll that and there's no distortion. A cylinder looks curved, but it's... flat from the point of view of Gaussian curvature. It curves in one direction, but not in the other, and it's important that... both change.”

“Geometry starts in dimension 2, and it was Gauss... who defined... Gaussian... or scaler curvature of any surface... which varies from point to point. ...[T]his is a notion of ...intrinsic curvature to distinguish it from extrinsic curvature.”

“In dimension one, along a straight line or on the edge of a circle there is no real geometry. All you can do is to measure how far along the path you are.”

“[B]y the 19th century, end of the 18th century people were studying curved surfaces, and the {{w|history of geometry}} since then, has to a great extent, been the study of the notion of curvature. What is curvature? How do you define it precisely? What does it mean, and how does it change as you increase the number of dimensions..?”

“[J]ust as the past is important, so the future is important, although the future is a little harder to predict.”

“[M]athematics is not unconnected with the real world... [I]t has a very strong, intimate and long relationship with physics, which is pertinent in the context of geometry and of the analysis that often goes with it, and that... is true of the Poincaré conjecture.”

“As individuals, our personality resides entirely in our memory. ...The same is true of civilization. ...Without a knowledge of our past, we... are nothing, but the present... [T]hat applies... to mathematics. You must... understand the historical background of any part of mathematics, any problem, where it comes from, how it progresses and where it might lead to in the future.”

“This 'Hodge conjecture' has by now achieved a considerable status, almost on a par with the Riemann hypothesis or the Poincaré conjecture.”

“My own supervisor, William Hodge, the creator of the fertile theory of harmonic forms, was not a genius like Ramanujan but resembled Lefschetz.”

“In the broad light of day mathematicians check their equations and proofs,leaving no stone unturned in their search for rigor.But at night, under the full moon, they dream.They float among the stars and wonder at the miracle of the heavens.They are inspired.Without dreams there is no art, no mathematics, no life.”

“Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.'”