VIEW FROM THE PENNINES: USING MATHS I, TOMOGRAPHY AND FRITZ JOHN PAUL GLENDINNING There is one man in the village who is instantly recognisable at al- most any distance, even when in his jeep. The reason is simple: he is never seen without his hat. This hat is a leather, wide-brimmed cow- boy hat, and it is the hat, not the man, which makes the identi cation possible. Some people seem born to wear hats. I nd it hard to picture Humphrey Bogart without a trilby, JR without a stetson or Churchill without a bowler. The mass wearing of hats on a daily basis is a thing of the past, but some descriptions still evoke an image with a hat as a necessary accessory. Take the following for example ... he had the air of a southern gentleman, still extremely good-looking, cigarillo-smoking, marvelously kind, po- lite and modest. [4] To me, this passage suggests that a Panama hat would be almost de rigeur for the man described. That man is Fritz John. Born in Berlin, John went to school in Danzig (now in Poland) and moved to Gottingenas a young man to join the group of mathematicians around Courant. His doctoral thesis was examined in haste as he ed Germany in the wake of the Nazi rise to power which remains so painful and so immediate to those who experienced it even at a this distance of time, as the following quote from Hildebrandt [4] shows. By the end of April [1933], Richard Courant and Max Born were kicked out of their professorships. Hoards of Nazi students, led by the assistant Weber and the student Teichmuller,terrorized the mathematical insti- tute. The "half-Jew" Fritz John lost his scholarship; only "Aryans" were now eligible for stipends. At the last minute, by good luck and the help of several colleagues, Fritz John was able to receive his doctoral degree; he was examined and promoted by Franz Rellich, who was only 27 years old but, as a Privatdozent, had the right to give Date: August 22, 2006. 1 2 PAUL GLENDINNING examinations. [...] Ten days later he married Charlotte Woellmer, equally without money, and their situation became even more precarious since by this time mar- riages between Aryans and non-Aryans were ocially discouraged, though not yet forbidden by law. John went to Cambridge in 1934, and ended up in Kentucky until the war, eventually rejoining Courant in New York in 1946 where he went on to become a founder member of the institute which was to be the Courant Institute of Mathematical Sciences. Whilst in Kentucky John published a paper which has become one of the main sources for three dimensional tomography (the non-invasive determination of density or other structure obtained by attenuation of waves as they pass through the body) [5]. This paper was from material in his thesis, and built on results of Johann Radon (1917) [9], which provides the mathematics for two dimensional tomography [3]. He considered the problem of relating the integrals along lines (as a function of the line), i.e. the ordered pairs Z ¨ © L; f(x(s); y(s); z(s))ds j lines L in R3 (1) L to the function f. In tomography, which hadn't been dreamed of in 1938, f represents the density of an object, and the key question is how much information about f can be obtained from measurements which amount to the evaluation of these line integrals over a subset of all possible lines. The rst remark (cf. [3]) is that by choosing an appropriate parametrization such as choosing lines to be of the form (a1t + b1; a2t + b2; t) the integrals take the form Z 1 J(a1; a2; b1; b2) = f(a1t + b1; a2t + b2; t)dt (2) 1 and so there are naturally four parameters in the integral description of the problem. However the function f itself is a function of three variables, and so there must be some redundancy in the integral for- mulation of (2). There are thus two natural questions. First, indepen- dent of the tomography applications, what is the connection between J(a1; a2; b1; b2) for di erent parameters? Second, if a point source is moved on a curve C around a body, what properties must C have in order for the information obtained about line integrals via attenuation to be enough to uniquely determine f from J along this path? TOMOGRAPHY AND FRITZ JOHN 3 John [5] showed that the answer to the rst question is that the redundancy can be described by a partial di erential equation, the ul- trahyperbolic equation, which is a generalization of the wave equation: @2J @2J = (3) @a1@b2 @a2@b1 and conversely that every solution of this equation with appropriate boundary conditions provides a solution to the inverse problem [7]. Equation (3) is easy to derive formally for suciently smooth functions f since by the chain rule Z @J 1 = tfi(a1t + b1; a2t + b2; t)dt @ai 1 th where fi is the partial derivative of f with respect to its i argument. Hence Z @2J 1 = tf12(a1t + b1; a2t + b2; t)dt (4) @a1@b2 1 and (3) follows from a similar calculation for @2J and the symmetry @a2@b1 of second derivatives, f12 = f21. There are di erent philosophies as to how this result can be used, but rst consider the second problem, which is more rooted in the methodology of tomography. To some extent this question is the to- mographic equivalent of asking which subsets of (1) are large enough to specify f uniquely. A point source is moved around the continuous, almost everywhere di erentiable, closed curve C and the attenuation of the x-rays is measured, thus providing the integrals along all lines which intersect C. Kirillov [6] shows that if the object to be measured lies in a bounded region then this provides enough information to reconstitute f provided every plane containing a point in intersects C, and if at least one of these intersections is transversal for each such plane then an explicit inversion formula is available [10]. This is a suf- cient, but not a necessary condition for unique inversion, and other arrangements are possible [2]. Explicit algorithms using John's ultrahyperbolic equation in tomog- raphy are still under development. There are currently two approaches. Patch [8] solves John's equations to augment the set of lines on which information is known, and hence simplify the inversion formula. On the other hand Defrise et al [1] use (3) as a consistency condition that can be used to obtain rst order correction terms to their rebinning algorithm. Whichever approach turns out to be most helpful, it is un- deniable that a problem which Fritz John chose because of its elegance and intrinsic interest has had a major in uence on the development of 4 PAUL GLENDINNING an area of practical importance which simply did not exist at the time at which the work was done. This provides another example of the diculties that directed research in mathematics imposes on the com- munity: applications and their theory do not necessarily follow in that order, and the identi cation of theory which is likely to have unforeseen practical applications is necessarily impractical. This is not to deny the need for direct applications in many contexts, but to emphasize the im- portance and complexity of the interaction between the fundamental or core areas of mathematics and the application of mathematics. As Klainerman decribes John [7]: ... Though the deepest motivation of his work came from the physical world, his modus operandi was that of the "pure mathematician." His starting point would be a simple-to-state and intriguing mathematical question from which he would proceed with utmost care, using rigorous and elegant arguments, simpli ed model prob- lems, examples, and counterexamples. Some of the most important practical applications of his work, such as nonlinear programming or the Radon-John transform, came as an afterthought, a bonus, rather than from an original applied intention. He didn't much care about devisive labels such as "pure" and "applied". He was simply a mathematician in the exemplary tradition of men like Fermat, Newton, Euler, Lagrange, Gauss, Rie- mann, Poincare,Hilbert, and Weyl. And the potatoes [3]? The earlies were sublime: creamy and smooth. The lates are still being eaten. They taste like potatoes. That's the comforting thing: you know where you are with potatoes. Acknowledgements: I am grateful to Bill Lionheart for useful guid- ance on both this View from the Pennines and [3]. Readers may be interested to learn that Bill is another hat wearer. He can frequently be seen in a grey tweed trilby sporting a green feather. c Paul Glendinning (2006) References [1] M. Defrise, F. Noo and H. Kudo (2003) Improved two-dimensional rebinning of helical cone-beam computerized tomography data using John's equation, In- verse Problems 19 S41-S54. [2] D.V. Finch (1985) Cone beam reconstruction with sources on a curve, SIAM Journal of Applied Mathematics 45 665-673. TOMOGRAPHY AND FRITZ JOHN 5 [3] P. Glendinning (2006) View from the Pennines: Using mathematics I, tomog- raphy and Johann Radon, Mathematics Today, 42 132-134. [4] S. Hildebrandt (1998) Remarks on the Life and Work of Fritz John, Communi- cations on Pure and Applied Mathematics LI 971-989. [5] F. John (1938) The ultrahyperbolic equation with four independent variables Duke Mathematics Journal 4 300-322. [6] A.A. Kirillov (1961) On a problem of I.M. Gel'fand, Soviet Mathematics Doklady 2 268-269.