Michael Atiyah

Sir Michael Atiyah (22 April 1929 – 11 January 2019) was a British mathematician, who specialised in geometry.

Quotes

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.'
- Michael Atiyah (2004). Collected works. Vol. 6. Oxford Science Publications. The Clarendon Press Oxford University Press. ISBN 978-0-19-853099-2.

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.
- "Dreams" from "Sir Michael Atiyah - From Algebraic Geometry to Physics - a Personal Perspective [2010]" 1:00:41, a YouTube video (Jun 15, 2018) from the Gaduate Mathematics channel.
  See also: 1) Jean-François Dars, Annick Lesne, Anne Papillault Les Déchiffreurs: Voyage en Mathématiques (2008) 2) Jean-François Dars, Annick Lesne, Anne Papillault, The Unravelers: Mathematical Snapshots (2008) the English translation.

My own supervisor, William Hodge, the creator of the fertile theory of harmonic forms, was not a genius like Ramanujan but resembled Lefschetz.
- Michael Atiyah (3 April 2014). Michael Atiyah Collected Works: Volume 7: 2002-2013. Oxford University Press. p. 286. ISBN 978-0-19-968926-2.

This 'Hodge conjecture' has by now achieved a considerable status, almost on a par with the Riemann hypothesis or the Poincaré conjecture.
- Michael Atiyah (28 April 1988). Collected Works: Michael Atiyah Collected Works: Volume 1: Early Papers; General Papers. Clarendon Press. pp. 250. ISBN 978-0-19-853275-0.

I always want to try to understand why things work. I'm not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it's there. And understanding is a very difficult notion. People think mathematics begins when you write down a theorem followed by a proof. That's not the beginning, that's the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You're trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You've got to feel it.
- Siobhan Roberts, Michael Atiyah’s Imaginative State of Mind (March 3, 2016) @QuantaMagazine.org
  Atiyah's response to: Is there one big question that has always guided you?

Geometry in 2, 3 and 4 Dimensions (Nov 24, 2011)

- Michael Atiyah, (Jun 8, 2010) Clay Research Conference in association with Institute Henri Poincaré (IHP) Paris. A YouTube video from the PoincareDuality channel. Also @Clay Mathematics Institute lecture video, Geometry in 2, 3 and 4 dimensions.

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.

[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.

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

[B]y the 19th century, end of the 18th century people were studying curved surfaces, and the 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..?

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.

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.

[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.

[T]he 19th century... started... with Gauss and went on with the development of the notion of curvature in 2 dimensions, by one function.
- $K=\kappa _{1}\kappa _{2}.$

In the 20th century we moved... from 2 dimensions to 3...
- Note: History does not... follow centuries exactly, so give or take 20 or 30 years. Allow me a little latitude.

The 19th century finished with a very very deep theory of 2 dimensional surfaces, in all aspects: geometrical, analytical, complex, real.

The 20th century... was devoted to ideas of 3 dimensional geometry, and this is the culmination... today...

In 3 dimensions the curvature is more complicated... It's determined not by by a single number r, but by a number R that depends on 2 directions i and j, called the Ricci curvature...
- Also see: Gregorio Ricci-Curbastro, Ricci calculus & Tensor field

4 dimensions is... very important, even if you don't believe in it. There are lots of ways in which 4 dimensions affects our world, especially if you include time. That is now... the center of attraction for mathematicians in the 21st century, and it may be not until the end of the 21st century that it is... better understood... [H]ere the curvature first introduced by Riemann... depends on 4 indices, or 4 directions. ...[A]fter that.., it doesn't get more complicated with 5 or more dimensions. It becomes simpler. These are the dimensions of interest.
- Also see: Bernhard Riemann.

[M]athematics is created by individual people... with real names.., faces and... personalities.

Let me go back to the history of 2 dimensional surfaces. ...The great Gauss ...laid the foundations for modern differential geometry. ...Niels Henrik Abel ...studied algebraic equations and showed for the first time that equations of degree 5 and more cannot be solved in terms of radicals, square roots... [etc.] [H]e brought into the theory of geometry, the complex variables... arising in polynomial equations. ...Bernhard Riemann ...whose collected works occupy one slim volume, but each chapter ...covers whole regions of mathematics.., brought the modern notion of topology into the game, to add to the differential geometry and the algebraic geometry of Gauss and Abel. These are... the three great figures in the [19th century] history of geometry...

Riemann surfaces, [in] our 2 dimensional study, is a... simple story. The surfaces are all characterized topologically by... the genus. When the genus is 0 the surface is the surface of a sphere, when it's 1, it's the surface of a torus.., when it's greater than or equal to 2, that's everything else... [T]hese surfaces... can have... metrics describing geometry, and they are characterized by having 0 curvature in the... torus (the torus is flat, in the sense like a cylinder); the sphere, which has positive curvature which curves one way; and the surfaces of negative... [curvature], which curve the other way; where Gauss curvature takes values either zero, plus or minus.

The genus is the number of holes. The sphere [surface] has no hole... The torus is a sphere with a hole bored through...

"The Youthful Spirit of Mathematics" (Nov 14, 2012)

Michael Atiyah, Distinguished Lecture [2012], co-organized by Department of Mathematics, The Hong Kong University of Science and Technology. A YouTube video from the Graduate Mathematics channel.

Mathematics is a very ancient subject... mainly created by the young, and... it's cumulative. ...How doews this happen?

Mathematics was well known to the Babylonians. ...[A]bout 2,000 BC ...this tablet shows how sophisticated the mathematics was ...An approximation to ${\sqrt {2}}$ ${\sqrt {2}}$ ...accurate to 4 or 5 decimal places ...written in decimal notation ...[P]eople think the decimal system was created by the Arabs, Indians, Chinese, but ...the Babylonians had a system ...based on 60. So when they wanted to describe a number, they did it with 1/60, 1/60² ...[etc.]
- Overhead: Babylonian clay tablet YBC 7289. Note: YBC 7289 (ca 1800–1600 BCE) is from the Yale Babylonian Collection

Given the antiquity.., the cumulative nature of mathematical knowledge, and the fact that the young people... produce the original ideas, how does it keep going? ...How do young mathematicians cope with with this vast amount of knowledge from the past and still turn out... new ideas? ...[T]here's an external answer and an internal answer.

Externally there's... diversification of mathematics. New ideas keep coming in... from the outside world, and they generate new problems, new theories. So you advance by expanding your frontiers as you interact with the outside world, and the outside world keeps changing.

Internally.., the mathematics... at any given time, it's a vast amount of information.., you have to organize it... in such a way that it can be packaged and passed on... [A] lot of facts in a lot of books aren't much help. ...You organize it by unification. You bring together many different parts ...into smaller, simpler packages which can ...be learned.

You expand the content and... unify the subject.

Diversification [External]

[A]stronomy was the key driver of mathematics in the early days. Physics came along... closely with it... Then you go into chemistry... biology, economics, finance... [etc.] A... range of... fields... have contributed to the development of mathematics by posing problems.., giving ideas and concepts.

As civilization developed, it got new tools. ...Archimedes was supposed to have drawn in the sand ...then they had discovered chalk... [etc.] ...Then we had telescopes.., microscopes, computers ...All these advances in technology, which come indirectly out of the advances in science.., provide... new opportunities. You can look further.., or at things of a very small scale... [W]ith computers we can handle all this information and data at incredible speeds... [W]e have powerful [technological] tools which assist, (I don't say they replace traditional ways of thinking and doing.., but) they... expand your scope. You get new material for each generation... to work on.

Unification [Internal]

All mathematics builds upon abstraction. When you count... people, I don't look at individual people... I just count... all people... abstracted as one...

[C]oncepts are very important and abstraction develops new concepts, new principles, how you organize them, structure, all... enable you to climb higher... up the heirarchy. Arithmetic... is an abstraction... then... algebra... gives abstract symbols for numbers... [etc.]

Abstraction enables you to put lots of things in the past [together] in one package, which you label "algebra" or whatever it may be.

Part of that process is... intellectual technology. Not only do we have hardware.., we... have the software... [e.g.,] the development of calculus, you can think of as an intellectual technology.., a machine of ideas and techniques in the brain... which you can... apply to... problems. The development of intellectual technology is... one of the secrets of advancing. Each generation inherits the intellectual technology... [with] more powerful tools it can push the ideas further.

[A]t the same time you have to do pruning.., you encourage growth and you remove unnecessary clutter so you can see through the jungle, and it simplifies... and... removes unnecessary dead wood.

[P]runing is a way of slimming down the past, to make it more productive... and... abstraction helps to use new principles, which enables you to climb higher in the process of moving forward.

Sir Michael Atiyah, What is a Spinor? (Sep 13, 2013)

Conférence en l'honneur de Jean-Pierre Bourguignon, IHÉS, Centre de conférences Marilyn et James Simons. A Creative Commons video from the Institut des Hautes Etudes Scientifiques (IHES) YouTube channel

My favorite story... is about the high school physics teacher who... noticed somebody... was not paying attention. So he turned to him and... said, "...[W]hat is an electron?" So the boy scratched his head and said, "Oh well, I used to know sir, but I've forgotten." So the teacher say, "Ah, what a pity... Only two people in the universe knew what an electron was, you and God, and you've forgotten!" So replace "electron" by "spinor" and you get the tone of this lecture. ...I've spent most of my life working on spinors ...and I don't know. Only God knows! Maybe Dirac, but he's no longer with us.

I want to focus on the comparison between spinors and the complex numbers. Complex numbers... took mathematicians several hundred years to really feel they understood what i was. i is the square root of -1, what is called an 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.

[S]pinors are analogously... a square root of geometry.

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

So what is a square root? Why do I say it's a square root? Because... the tensor product of the spinors... is the 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.

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 spinors 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 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.

If you have a complex manifold, if the 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 $(p,q)$ $(p,q)$ where $p$ $p$ involves the $dz$ $dz$ 's and $q$ $q$ involves the $d{\bar {z}}$ $d{\bar {z}}$ 's, the mixture of the two [is] the 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.
- Overhead: Complex geometry is a square root of real geometry.

But spinors exist without the need of complex structures. So what is a spinor when there's no complex structure? That's really the question.

[C]omplex numbers... are... not important only in algebra. They're important in geometry, and it was known... early.., that you could do the theory of Riemann surfaces.., fundamentally... formalized by Hermann Weyl... [T]he important thing in complex numbers is the analysis. The algebra, the ${\sqrt {-1}}$ is interesting. We know that the complex numbers are the algebraic closure of the real numbers... You don't need to keep going... higher up, just equations of degree 2. But the... deep part about complex numbers is their role in analysis: complex analysis, Cauchy theorem... [etc.] That's a very unexpected bonus.

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.

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

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.

Quotes about Atiyah

Sir Michael Atiyah, a Knight Mathematician (Oct 17, 2009)

A tribute to Michael Atiyah, an inspiration and a friend, by Alain Connes, Joseph Kouneiher, arXiv:1910.07851v1 Source

Sir Michael Atiyah was considered one of the world’s foremost mathematicians. ...[H]is work has helped theoretical physicists to advance their understanding of quantum field theory and general relativity.

Michael’s approach to mathematics was based... on the idea of finding new horizons and opening up new perspectives. ...For him an idea was justified by the new links between different problems which it illuminated.

[H]e spent the first half of his career connecting mathematics to mathematics, and the second half connecting mathematics to physics.

[H]e showed how the study of vector bundles on spaces could be regarded as the study of a cohomology theory called K-theory.

Today the index theorem plays an essential role in partial differential equations, stochastic processes, Riemannian geometry, algebraic geometry, algebraic topology and mathematical physics.
Since the appearance of the Atiyah-Singer index theorem generalizations have been obtained encompassing many new geometric situations.

Besides topological K-theory, Michael Atiyah made two discoveries that played a major role in the elaboration of the fundamental paradigm of noncommutative geometry...

Atiyah has been influential in stressing the role of topology in quantum field theory and in bringing the work of theoretical physicists to the attention of the mathematical community.

The term “gauge theory” refers to a quite general class of quantum field theories... These theories involve deep and interesting nonlinear differential equations, in particular, the Yang-Mills equations... Atiyah initiated much of the early work in this field, culminated by Simon Donaldson’s work in four-dimensional geometry.

External links

Wikipedia

Wikipedia has an article about:

Michael Atiyah

Michael Atiyah tells his life story @WebOfStories.com
Atiyah80+ Michael Atiyah's 80th birthday celebrations, Edinburgh (April 20-24, 2009)
"I'm a bit of a jack of all trades" @theguardian.com
"Atiyah and Singer receive 2004 Abel prize" @ams.org
"Details for Michael F. Atiyah" Oberwolfach Photo Collection @owpdb.mfo.de
Michael F. Atiyah complete bibliography @celebratio.org
"Sir Michael Atiyah, a Knight Mathematician:" A Tribute to Michael Atiyah, an Inspiration and a Friend by Alain Connes, Joseph Kouneiher @ams.org
Sir Michael Francis Atiyah @npg.org.uk