Notices of the AMS - CK Raju

additional 6 (or 7) dimensions are hidden fromour normal experience of reality.2. Whereas more classically, the basic objects suchas electrons, protons, and quarks, could bethought of as point particles, string theory interpretssuch objects as being very small“strings”. This allows for a “smoothing out” ofthe singularities that arise in the classical picturewhen such particles come close together.Even the singularities associated with mixingquantum mechanical and gravitational forcesare resolved in this manner.3. Very sophisticated geometry is used, involvinga vast amount of mathematics, both old andnew.4. No unique model or picture has emerged out ofstring theory, but rather several versions exist.These different theories are now known to bedifferent facets of the same theory. What hashappened to the “real world”?5. Quantum mechanics remains the basic framework.String theory has had a remarkable and mysteriousimpact on pure mathematics, leading to manynew concepts and results. In some cases, such resultshave been given proofs in the traditionalmathematical sense. In other situations, the “results”merely fit well with known mathematical resultsor accepted features of string theory. In particular,string theory has had an impact in1. algebraic geometry, by addressing enumerativequestions concerned with counting algebraiccurves satisfying certain conditions;2. knot theory, by construction of new topologicalinvariants of knots that can sometimes distinguisha knot from its mirror image;3. four-dimensional geometry, by giving new, unexpected,and very deep results that are uniqueto four dimensions; and4. various branches of algebra.A New Paradigm?If a “theory of everything” emerges from stringtheory, we will discover a universe built on fantasticallyintricate mathematics. In particular, theCalabi-Yau manifolds that make up the hidden dimensionsare extremely complicated. Atiyah suggestedthat it is not satisfying that the true theorywould be so complicated—even writing down theterms of the theory requires a vast amount of background.Perhaps, according to Atiyah, a new paradigmis needed; perhaps the complicated mathematicsappearing in string theory is merely “in the eye ofthe beholder”. That is, maybe we do not understandthe fundamental nature of reality well enough, andthis misunderstanding is leading to such exceptionallycomplicated mathematics. String theory,from this point of view, is only our method ofapproximating a simple reality. Perhaps, Atiyahsuggested, we should follow Einstein and questionquantum mechanics.In order to make progress, we might need to dispensewith some piece of accepted dogma. Relativity,quantum mechanics, and string theory havealready dispensed with many previously heldtenets, and so one might ponder whether there remainsany such dogma left to throw away. Atiyahnoted that all physical models since Newton, includingeven quantum mechanics, have assumedone basic premise—that we can predict the futurefrom full knowledge of the present. Atiyah suggestedan alternative to this paradigm: Perhaps weneed full knowledge of the present and the past inorder to predict the future. That is, maybe the universehas memory. As a simple example, the notionof the velocity of an object is viewed as beinga property of the present, but, in reality, to measurevelocity one needs to know not only where theobject is now but where it was a moment earlier.Atiyah’s hypothesis possibly leads to several interestingconsequences:1. The mathematics used in physical theory wouldbecome more difficult, since all previously usedmathematics in physics assumes that knowledgeof the present suffices. With the new paradigm,for example, retarded (or delay) differentialequations would become necessary.2. Since we do not have complete knowledge of thepast, uncertainty would arise. This might shedlight on the uncertainty inherent in quantum mechanics.3. Perhaps the complicated mathematics of stringtheory arises from our attempt to understandthe full implications of the theory of general relativitywithout incorporating the knowledge ofthe past.Atiyah does not promote discarding older, timetestedphysical theories. Rather, such a newJUNE/JULY 2006 NOTICES OF THE AMS 677

paradigm ought to build on the old theories, muchas relativity builds on Newtonian mechanics.Speculations and QuestionsThere are various attitudes among physicists towardstring theory. Some dismiss it as fancy mathematicsthat is unrelated to the real world, sincestring theory makes no testable predictions. Othersbelieve the mathematical applications of stringtheory give confidence in the physical insights andindicate that the theory is on the right track. Fromthis point of view, mathematical applications becomea kind of alternative to experimental evidence.A third point of view is that we should continueto push forward with string theory in the hopethat the new results and ideas that emerge will serveas a guide for finding a final unified theory.Atiyah concluded his talk by speculating on themeaning of all this—quantum field theory, stringtheory, and their mathematical applications. Whatwill the future physical theory look like? The aimis to unify quantum mechanics, the physics of thevery small, with general relativity, the physics ofthe very large. Supersymmetry is a symmetry inwhich physical laws are unchanged when bosonsand fermions are interchanged. Superstring theory,a supersymmetric string theory, is a perturbativeapproach, one that Atiyah compared with the theoryof epicycles developed by Ptolemy. But what isthe real theory; that is, what is being perturbed? Isit M-theory, a currently incomplete theory unifyingall five versions of string theory? Is the universereally built using all this sophisticated machineryor is this an example of mathematics imposed byus? Perhaps the real physics is simpler and oneshould adhere to the dictate of Occam’s razor—conceptsshould not be multiplied beyond necessity.Do we need to modify quantum mechanics? Atiyahclosed by saying “This is for young people: Goaway and explore it. If it works, don’t forget I suggestedit. If it doesn’t, don’t hold me responsible.”The second Einstein Public Lecture in Mathematicswas delivered on April 29, 2006, in conjunctionwith the AMS Spring Western SectionalMeeting at San Francisco State University. BenoîtMandelbrot of Yale University spoke on “The natureof roughness in mathematics, science, and art”.All photographs used in this article are courtesyof Gregg Johnson (Suitefreedom.com).678 NOTICES OF THE AMS VOLUME 53, NUMBER 6