"I love Phillip Griffiths' books. Every time I read one I want togo and hug the guy." That's rare praise for maths textbooks, bestowedon Griffiths by a mathematician I met during coffee at theInternational Congress of Mathematicians in Seoul. Griffiths came upin our chat because he had just been awarded the Chernmedal which bestows the "highest level of recognition foroutstanding achievements in the field of mathematics". These includeGriffiths' mathematical work, but also his books, his work for mathseducation and for the mathematical community as a whole.

Beauty and truth

Listening to Griffiths talk about maths you get a clear sense ofwhat draws him to it. He quotes the mathematician

It's a side of mathematics that is often hidden from the outsideview. What we see in school books are stern formulas andrigid logic — no room for choice, let alone personal preference. Butthose textbooks only present the final results, pulledlike a rabbit out of a hat. The course of mathematics,whether historical or in the work of a single person, is way moremeandering and haphazard than that rabbitsuggests. When you are wandering the wilderness of the unknown,discovering (or inventing) new maths, youneed something to guide you, and often that something is a sense of aesthetics. "You find out what the most harmonious properties of a[mathematical] structure are, the most aesthetically pleasing way tosee it, and then you let that guide you," says Griffiths.

Griffiths takes delight in thosetwists and turns of mathematical history. When hestarted out in maths there were no textbooks covering the area heworked in, so he had to go back to the old masters. Reformulatingtheir work and bringing it up to modern standards of rigour is inpart what he is being honoured for. He told me how his own area of maths took one startingpoint around 1812 in a Russian jail. Here JeanVictor Poncelet, a French engineer and mathematician captured duringNapoleon's campaign against the Russian Empire, gave some seriousthought to geometry.

Poncelet was thinking about problems such as this one. Suppose youhave two ellipses, one sitting within the other. Pick a pointP on theouter ellipse and draw a straight line that just touches (istangent to) the inner ellipse. Extend that line until it hitsthe outer ellipse again and repeat the process. It's conceivable thatafter drawing a few of these tangent lines, you end up at the point P you started at. Naively,you would think that this periodic behaviour is pretty rare; that itonly happens for some very special starting points P. ButPoncelet proved a surprising result: if such periodic behaviouroccurs for one starting point P, then it occurs forall starting points on the outer ellipse. So, depending onthe ellipses, thisbilliard-like game is periodic everywhere, or nowhere.

It's all connected

This type of problem seems worthy of an engineer, but also ratherisolated. Yet it turned out that Poncelet's result is closely related to aground breaking one, proved a few years later in1822, by the twenty-year-old prodigy NielsHenrik Abel. Building on work of many notable mathematicians Abelwas trying to evaluate objects called ellipticintegrals. These cropped up in a number of contexts. For example,you needed to evaluate elliptic integrals to figure out the length ofan arcof an ellipse, and also in mechanics, for example to describe the motion of apendulum. (Elliptic integrals are special cases of the more general integrals of algebraic functions.)

With these kind of problems you hope for a general formula that gives you the solution. For example, the length

of an arc of a circle is where is the radius of the circle and is the angle at the centre of the circle that defines the arc (see the figure below). For elliptic integrals (and also for the integrals of most algebraic functions more generally) no such formulasexisted. "What you got by integrating algebraic functions was a very mysteriousquantity," explains Griffiths. "You cannot evaluate it by anythingthat is in closed form," that is, by any neat formula. With his deep insight into the innersymmetries of such integrals, Abel realised that while you could noteasily evaluate them, you could decompose them into componentsthat are much easier to deal with: they can be expressed usingelementary functions mathematicians were familiar with.

Surprising though it may seem, Abel's result andPoncelet's result on bouncing around between ellipses were bothmanifestations of the same underlying mathematics. Neither Poncelet orAbel were aware of this link and they never had the pleasure ofmeeting. Abel's work found no immediate recognition and he died a fewyears later, in 1829, from tuberculosis and in poverty.

Geometry via algebra

Abel's result was a breakthrough, and it marked the birth ofHodge theory, Griffiths' favourite area of maths. Ellipses andcircles are geometric shapes, but they can be described using algebra(see the box on the right). In a similar vein, you can study othergeometric shapes that are defined by certain algebraic equations. These can becurves such as hyperbolas or parabolas, they can be surfaces such asspheres, and they caneven be higher dimensional shapes you cannot visualise. The good thingabout such algebraic varieties, as they are called, is that you canuse the poweful tools of algebra to understand them, even if you can'tpicture them.

Hodge theory is about understanding thegeometry of complicated algebraic varieties using information encoded ina much simpler algebraic structure, called the Hodge structure of an algebraic variety. It's on these Hodge structures Griffiths has worked on most of his life, making major contributions to the area. But there is still much work to be done. One open question, called the Hodge conjecture, was designated in 2000 as one of the seven most difficult problems in maths by the Clay Mathematics Institute. Very loosely speaking, the conjecture concerns the question whether every component in the Hodge structure of an algebraic variety also has an algebro-geometric interpretation.

The Clay Mathematics Institute is offering $1 million for a resolution of the Hodgeconjecture, but so far no one has been able to claim it. "It's one of these problems that probably requires some deeper levelof understanding," says Griffiths. "It's not a problem you can just work ondirectly at the moment. My guess is that it's going to be [solved using]some combination of ways of looking at it from different perspectives, puttogether in a way nobody has thought of before." But as Griffiths alsopoints out, there is always the chance that a lone engineer in jail,or perhaps atwenty-year-old genius, comes up with the magic ingredient thatclinches the prize.

When it comes to teaching such complexmathematics, Griffiths isn't a stickler for formalism. He tends towardsteaching students how to think about a problem, he says, rather than howto actually do it (though he notes the latter is important too). And he expresses admiration for the mathematician SolomonLefschetz, who reportedly never stated a false result nor wrote down acompletely correct proof (according to the prevailing current standards). It's about gaining a deep sense of aproblem, rather than getting all the formalities right straightaway. Griffiths' teaching technique seems to work. He has had 29 PhDstudents many of whom had glittering careers, producing a total of 460"doctoral descendents".

But he is also interested in how mathematics is taught to people who won't go on to become mathematicians, but will need mathematics in one form or another in their jobs. In this context Griffiths is down toEarth. That kind of teaching should be "demand driven" he says: don't teach the maths in isolation,but also teach people how they are going touse it in their jobs and lives.

Being practical

Griffiths' influence goes way beyond personal contacts. In various roles he has shaped science policy, from maths and science education back home in the US to forming the Science Initiative Group, which is dedicated to fostering science in developing nations. Between 1991 and 2003 Griffiths was director of the Institute forAdvanced Study in Princeton, one of the best research centres inworld. According to the International Mathematical Union, the body that awards the Chern medal, Griffiths has had"a substantial impact on the entire scientific enterprise the world over". (For a comprehensive review of all of Griffiths' achievements, which we haven't got space to list here, see thissummary produced by the International Mathematical Union.)

Now aged 76 Griffiths has left behind the "wilderness of administration"as he calls it, and is devoting most of his time to themaths. Receiving the Chern Medal is "the most movingexperience" of his professional life he says. It's got a very specialsignificance for him: Shiing-shenChern, after whom it is named, was Griffiths' mentor, collaboratorand friend. With all of Griffiths' achievements the medal couldn't bemore deserved.

---

About this article

Marianne Freiberger is Editor of Plus. She interviewed Phillip Griffiths at the International Congress of Mathematicians in Seoul in August 2014.