Michael Atiyah

“[[w:Elliptic function|[E]lliptic functions]]... have something to do with an ellipse... [I]f you go... to elementary calculus... you'll find formulas for the length of the arc of an ellipse. ...[E]lliptic functions come from that term.”

“Abel... died of tuberculosis before he could get his first job... These two figures are the pioneers of modern mathematics. All that we do... rests on their pioneering work. ...These ...are ...the two most romantic figures in the history of mathematics. We all dote on them. ...We all admire both of them... [T]hey showed... that an equation of degree 5 cannot be solved by... roots, but... you have to solve it by using more complicated... {{w|elliptic function}}s.”

“Galois... was... imprisoned... for radical views. ...[H]e was to take part in a duel, in which he was killed, but that... was an assassination plot. They covered it up. He was... murdered by the {{w|secret police}} because of his radical opinions.., but the last night.., he stayed up... scribbling his notes which formed the foundations of modern mathematical theory, the theory of groups.”

“Nobody succeeded until two of the most famous figures in mathematics appeared... Niels Abel and Évariste Galois.”

“[T]he quintic equation.., equations of degree 5 or more, cannot be solved using only... roots... [T]hat took mathematicians 100s of years.., painfully. ...[E]verybody knows how to solve quadratic equations.., and with a lot more work... over a century or two, people worked out how to solve {{w|cubic equation}}s... [W]ith even... more hard work, finally they worked out how to solve equations of degree 4.., a famous landmark in mathematics, and then they got stuck. They couldn't do 5.”

“There is no... rule for choosing 1 out of 5. If I give you 5 things and ask you to choose one, you can't do it... by the usual rules of either using ruler and compass, or equivalently, using only... roots... That comes under one of the most famous theorems in the history of mathematics..”

“The most important things in life.., whether you're a football player or a virus, is to deter attackers and run away when necessary.”

“Most viruses are of the form of an {{w|icosahedron}}... because it's the simplest thing you can do by doubling up. ...We wouldn't be here without viruses because the virus is also related to DNA. ...[V]iruses come in 2 forms, one more like the icosahedron. ...There are lots of nice [sharp] vertices..: what the virus uses to deter attackers. ...The round one is more like... the {{w|dodecahedron}}.., the one that is used to get away.”

“You start with the {{w|tetrahedron}} [in 3 dimensions], which is the simplest thing after the {{w|triangle}}... with the wave of a hand, you can construct two beautiful objects: the {{w|icosahedron}} and its dual, the {{w|dodecahedron}}. ...[This] was first discovered by the ...virus.”

“So spinor analysis has to be found as a substitute for complex analysis. That's the first stage in going from Cauchy's theorem in this idea of the square root of geometry.”

“[S]imilarly, you expect the deep part of {{w|spinor}}s, not to lie in the formal properties or {{w|representation theory}} algebra, but in their geometrical meaning.., and geometry here I identify, like Jean-Pierre, with analysis... You do geometry on {{w|curved space}}s. You have to use analysis, differential equations.”

“When people started using i in formulas they were thinking only in terms of solving polynomial equations... [T]hen in the hands of the analysts, Cauchy and others, it became an indispensible tool. So that's where the deep part of complex analysis is.”

“But {{w|spinor}}s exist without the need of complex structures. So what is a spinor when there's no complex structure? That's really the question.”

“If you have a {{w|complex manifold}}, if the {{w|manifold}} you are studying is complex, then... [by] Hodge... [etc.,] the differential forms on the manifold can be broken up into the forms of type (p,q) where p involves the dz's and q involves the d\bar{z}'s, the mixture of the two [is] the {{w|tensor product}}. So when you have a complex structure, you see the square root, and the square root is complex geometry. So when you have complex geometry, you've found the square root inside the real geometry. Complex structure has given you the square root.”

“So I don't know the answer to my question, and neither does Jean-Pierre here, and... maybe Dirac knows, but he's up there with God, so we are left on our own.”

“So it's a very deep notion, and it will take us... at least as long as it took to understand \sqrt{-1}, to understand what {{w|spinor}}s are, and they've only been around for perhaps 100 years, so we are still in the very early days, the equivalent perhaps, of the 15th century...”

“So what is a {{w|square root}}? Why do I say it's a square root? Because... the {{w|tensor product}} of the {{w|spinor}}s... is the {{w|exterior algebra}}. So the spinors are, in some sense, the square root, not of one form of a given degree, but [of] the sum of them all.”

“In geometry the fundamental elements... [have] to do with measurement: lengths, areas, volumes, and those are described mathematically by the {{w|exterior algebra}}, and it was used by {{w|Élie Cartan}} in the differential geometry, as a fundamental tool to describe... {{w|integral}}s.”

“[[w:Spinor|[S]pinor]]s are analogously... a {{w|square root}} of geometry.”

“I want to focus on the comparison between {{w|spinor}}s and the {{w|complex number}}s. Complex numbers... took mathematicians several hundred years to really feel they understood what i was. i is the {{w|square root}} of {{w|-1}}, what is called an {{w|imaginary number}}. It did not exist, but they found it very useful. They used it in formulas, and gradually, step by step, it became accepted... [F]inally the formal definitions were given so i... became and accepted part of mathematics, but it took a long, long time.”