What Is the Role of Algebra in Applied Mathematics? David A. Cox
hen I first encountered abstract al- science, engineeering, and operations research gebra in the late 1960s, I was drawn departments. to its power and beauty. One of the I begin with examples from geometric modeling, liberating ideas of the subject is that economics, and splines to illustrate the possible Win dealing with algebraic structures, applications of algebra. I then discuss computer you do not need to worry about what the objects algebra and conclude with remarks on the role of are; rather, it is how they behave that is important. algebra in the applied mathematics curriculum. Algebra is a wonderful language for describing the behavior of mathematical objects. Geometric Modeling, Cramer’s Rule, My fascination with algebra led me to algebraic and Modules geometry, which was then among the most ab- My first example concerns an unexpected applica- stract areas of pure mathematics. At the time, I tion of Cramer’s Rule and the Hilbert-Burch Theorem would never have predicted that twenty-five years on the structure of certain free resolutions. While later I would be writing papers with computer sci- Cramer’s Rule should be familiar, free resolutions entists, where we use algebraic geometry and com- (whatever they are) may sound rather abstract. As mutative algebra to solve problems in geometric we will see, there are questions in geometric mod- modeling. The algebra that I learned as pure and eling where these topics arise naturally. abstract has come to have significant applications. My part of the story begins with the research of What do these and other applications imply Tom Sederberg and Falai Chen on parametric curves about the relation between algebra and applied in the plane. If we are given relatively prime poly- mathematics? The purpose of this essay is to ex- nomials a(t), b(t), and c(t) of degree n, then the plore some aspects of this relation in the hope of parametric equations provoking useful discussions between pure and a(t) b(t) applied mathematics. Here, “applied mathemat- (1) x = and y = ics” includes not only what students learn in math- c(t) c(t) ematics and applied mathematics departments, describe a curve in the plane. To think about this but also the mathematics learned in computer geometrically, Sederberg and Chen [23] considered lines defined by David A. Cox is the Thalheimer Professor of Mathematics at Amherst College. His email address is dac@cs. (2) A(t)x + B(t)y + C(t)=0, amherst.edu. This article is a slightly edited version of an article that where A(t),B(t), and C(t) are polynomials depend- appeared in the Gazette des Mathématiciens (April 2005, ing on the parameter t. As we vary t, the line varies no. 104, p. 13–22). It is also to appear in a future issue of also, hence the name moving line. A moving line the Gaceta de la Real Sociedad Matemática Española. follows a parametrization if for every value of the
NOVEMBER 2005 NOTICES OF THE AMS 1193 parameter, the point lies on the corresponding h1p + h2q =0, where h1 and h2 are polynomi- line. In other words, (1) is a solution of (2) for all als in t. t. When we perform this substitution and clear de- • If a, b, and c have degree n in t, then p has de- nominators, we get the equation gree µ in t and q has degree n − µ, where µ is an integer satisfying 0 ≤ µ ≤ n/2 . (3) A(t)a(t)+B(t)b(t)+C(t)c(t) ≡ 0, • Up to a constant and appropriate signs, the where ≡ indicates that equality holds for all t, i.e., polynomials a(t), b(t), and c(t) are the 2 × 2 A(t)a(t)+B(t)b(t)+C(t)c(t) is the zero polynomial. minors of the matrix What algebraic structures are being used here? R A1(t) B1(t) C1(t) Since we are working over the real numbers , we . have the polynomial ring R = R[t]. Then, moving A2(t) B2(t) C2(t) lines A(t)x + B(t)y + C(t)=0 that follow (1) corre- • The equation of the curve is given by the resul- spond to elements of the set tant {(A(t),B(t),C(t)) ∈ R3 | Resultant(p, q, t) = 0. (4) A(t)a(t)+B(t)b(t)+C(t)c(t) ≡ 0}. The moving lines p and q are called a µ-basis. Thus This is an R-submodule of the free R-module R3 . the µ-basis determines both the parametrization In commutative algebra, we say that (3) is a syzygy (via the third bullet) and the equation of the curve and that (4) is a syzygy module. (via the fourth). See [7] for the details. In the mid-1990s, Sederberg asked me if I had From the point of view of commutative algebra, seen equation (3). I said, “Yes, this equation defines the first bullet says that the syzygy module is free the syzygy module of a(t),b(t), and c(t) .” As I ex- with basis (At (i),Bi(t),Ci(t)), i =1, 2 . For anyone pected, Tom asked me what “syzygy” means. I ex- who knows the theory of modules over a principal plained that in astronomy, a syzygy refers to the ideal domain, this is no surprise. But the degree re- alignment of three celestial bodies along a straight striction in the second bullet is quite different: it or nearly straight line. The term was introduced into tells us that we are really working in the homoge- mathematics by Sylvester in 1853 [27]. These days, neous situation, where we replace a(t),b(t), and c(t) a syzygy can refer to either a polynomial relation with homogeneous polynomials a(s,t),b(s,t), between invariants or a linear relation with poly- and c(s,t) of degree n. This gives the ideal nomial coefficients as in (3). = ⊂ = R But then Tom did something unexpected: he I a(s,t),b(s,t),c(s,t) S [s,t], asked me what “module” means! At first, I was shocked that he didn’t know such a basic term, but and as shown in [6] and [7], the first two bullets then I remembered that civil engineers don’t take translate into the following free resolution of the courses in abstract algebra. (Tom’s degree is in ideal I: civil engineering, though he is now in computer sci- ence.) Vectors of polynomials occur frequently in (6) 0 → S(−n + µ) ⊕ S(−2n + µ) Tom’s research, and modules over polynomial rings A1 A2 are the natural language for discussing such ob- B1 B2 jects. Yet Tom had never heard the term “module” C1 C2 until I mentioned it to him. → This made me realize that our conception of “ap- plied algebra” may need to be enlarged. I will say abc S(−n)3 → I → 0 more about this below, but first let me finish the story of moving lines. Sederberg and Chen had the (the notation S(−n) keeps track of degrees), and insight that when two moving lines follow the pa- the third bullet implies that a, b, and c are the rametrization (1), their common point of intersec- 2 × 2 minors of the 3 × 2 matrix in (6). This is a tion describes the curve. Looking deeper, they also special case of the Hilbert-Burch Theorem (see [6]).1 noticed that there are always two moving lines While this might seem sophisticated, some parts
p := A1(t)x + B1(t)y + C1(t) = 0, make perfect sense. If you take the moving line (5) q := A (t)x + B (t)y + C (t) = 0 2 2 2 1This special case was discovered by Franz Meyer in 1887 that follow the parametrization and have the fol- [19], who conjectured similar results for the syzygy mod- ule of a ,...,a ∈ S. Hilbert, in his great paper of 1890 lowing special properties: 1 n [16], created the basic theory of what we now call com- • All moving lines that follow the parametrization mutative algebra. His first application was Meyer’s con- come from polynomial linear combinations jecture, which he proved via the original version of the of p and q, i.e., moving lines of the form Hilbert-Burch Theorem.
1194 NOTICES OF THE AMS VOLUME 52, NUMBER 10 equations (5) and solve for x and y via Cramer’s propensity to save). When linearized at an equi- Rule, you get librium point, the IS-LM model gives the equations −C1 B1 B1 C1 + = o + det −C B det B C sY ar I G x = 2 2 = 2 2 , A B A B − = − o det 1 1 det 1 1 mY hr Ms M , A2 B2 A2 B2 (7) − o o det A1 C1 − det A1 C1 where s, a, m, h, I ,G,Ms ,M are positive parame- A2 −C2 A2 C2 y = = . ters (for example, Ms is the money supply, and s det A1 B1 det A1 B1 A2 B2 A2 B2 is the marginal propensity to save). The goal is to see how Y and r vary when we vary the parame- Comparing this to the original parametrization (1), ters. This is where Cramer’s Rule shines: it gives it should be no surprise that a, b, and c are the formulas for Y and r that make such questions easy 2 × 2 minors of the matrix formed by the µ-basis. to answer. When Sederberg and Chen saw this consequence The link between this example and (7) is that in of Cramer’s Rule, they knew they were on the right both cases we applied Cramer’s Rule to equations track; Hilbert may have been led to his version of depending on parameters. It follows that applied the Hilbert-Burch Theorem by similar reasoning. linear algebra has both numerical and symbolic as- Remember that when Sederberg and Chen con- pects. Do our current linear algebra courses do jectured this in the late 1990s, Sederberg did not justice to both? Right now, the main place a sym- know what a module was. To me, this suggests bolic parameter occurs is in the expression that modules may have a place in the applied math- det(A − λIn) for the characteristic polynomial of an ematics curriculum. n × n matrix A. As the above examples indicate, determinants with symbolic parameters have a Economics, Cramer’s Rule, and Symbolic more substantial role to play, even at the elemen- Linear Algebra tary level. Recently I had a conversation with a colleague The link becomes even deeper when we think about the role of Cramer’s Rule in linear algebra. about what it means algebraically to do linear alge- Although Cramer’s Rule played an important role bra with parameters. For simplicity, assume that the in the historical development of linear algebra, it independent parameters t1,...,tn appear rationally seems less relevant these days, especially with the in the equations (e.g., no square roots or exponentials emphasis on solving equations by Gaussian elim- of parameters). Since linear algebra needs a field, ination. Moreover, Cramer’s Rule is useless when it is natural to work over the rational function field a system of equations has an ill-conditioned coef- K = R(t1,...,tn). However, there are many situations ficient matrix. For these reasons, my colleague was where the parameters appear polynomially wondering if the topic should be omitted—why and denominators cause problems. This means burden students with an unnecessary formula? doing linear algebra over the polynomial ring R = R I’ve always loved Cramer’s Rule for its intrinsic [t1,...,tn]. Here, “vectors” become vectors of poly- beauty, but this approach doesn’t always work for nomials, i.e., elements of a free module over R. For ∈ = R those students who prefer to view mathematics example, given a, b, c R [t], the polynomial ∈ 3 through its applications. After some thought, I re- solutions (A, B, C) R of the linear equation alized that while Cramer’s Rule may be useless for Aa + Bb + Cc =0form the syzygy module (4). numerical linear algebra, it has a valid place in the Splines and Modules larger world of applied linear algebra. To see the difference, note that the use of For another example of how linear algebra relates Cramer’s Rule in (7) is applied—it’s part of geo- to modules, we turn to the study of multivariate metric modeling—and is nonnumerical. For an- splines. Consider the spline illustrated in Figure 1. other nonnumerical application of Cramer’s Rule, consider the following elementary example from economics. The IS-LM model analyzes the interac- tion between total national income and the money supply, following ideas of John Maynard Keynes. As explained in [24, pp. 115–7], we want to un- derstand Y = total national product and r = interest rate in terms of policy parameters (e.g., the money sup- Figure 1. A polynomial spline. ply) and behavioral parameters (e.g., the marginal
NOVEMBER 2005 NOTICES OF THE AMS 1195 Here, we have a square centered at the origin, to- spline module (9) has Hilbert function (10), which gether with a spline built using the polynomials is a polynomial in k for k ≥ 2. f1,f2,f3,f4 ∈ R = R[x, y] indicated in the figure. In The examples discussed so far were chosen to practice, the degrees of the polynomials are fixed illustrate how in advance, but we will postpone doing so in order the theory of modules over polynomial rings to reveal the underlying algebraic structure. To ob- = linear algebra with parameters tain a C1 spline, the polynomials must satisfy = symbolic linear algebra 2 f1 − f2,f3 − f4 ∈x , (8) arises naturally in a variety of applied contexts. This 2 f2 − f3,f1 − f4 ∈y , subject also has a rich theoretical side that uses sophisticated tools from algebraic geometry and 2 where x is the ideal of R = R[x, y] generated by commutative algebra, yet its central problem is 2 2 x , and similarly for y . Then solving linear equations over polynomial rings. A 4 clear statement of this philosophy appears in the (9) (f1,...,f4) ∈ R | f1,f2,f3,f4 satisfy (8) introduction to Eisenbud’s recent book The Geom- is easily seen to be a submodule of R4. Furthermore, etry of Syzygies [12, p. xii]. one can show (see Exercise 5 of [6, Sec. 8.3]) that this submodule is free of rank four with basis Computer Algebra and Applied Algebra
2 2 2 2 2 2 The last forty years have witnessed the emergence (1, 1, 1, 1), (0,x ,x , 0), (y ,y , 0, 0), (x ,y , 0, 0). of substantial computational power and the dis- In other words, every spline in (9) can be written covery (in some cases, rediscovery) of fundamen- uniquely as a polynomial linear combination of tal algorithms for symbolic computation. These the above four splines. interlinked developments have resulted in a tremen- To see how this relates to linear algebra, consider dous burst of research, both pure and applied. C1 splines of degree ≤ k. It is straightforward to Books in this area range from undergraduate reduce (8) to a system of linear equations involv- textbooks to technical monographs, some aimed at researchers in algebraic geometry and commuta- ing the coefficients of f1,f2,f3,f4 . Solving these equations by standard techniques gives a good tive algebra, others intended for more diverse method for finding splines. But since we understand audiences. the module (9), we can answer this question in- Computer algebra encompasses a wide variety stantly: the splines of degree ≤ k are given uniquely of subjects, including coset enumeration, Galois by theory, modular arithmetic, symbolic integration, 2 2 g1 · (1, 1, 1, 1) + g2 · (0,x ,x , 0) symbolic summation, difference equations, power series, and special functions, among others. As + g · (y2,y2, 0, 0) + g · (x2,y2, 0, 0), 3 4 one might expect, polynomials play an important where g1,g2,g3, and g4 have degrees at most role in computer algebra, where one finds algo- k, k − 2,k− 2, and k − 4, respectively. Since the rithms for greatest common divisors, factoriza- space of polynomials in x, y of degree ≤ k has tion, Gröbner bases, resultants, characteristic sets, dimension k+2 , it follows that the space of C1 quantifier elimination, and cylindrical algebraic 2 decomposition. Introductions to computer alge- polynomial splines for Figure 1 of degree ≤ k has bra include the survey [2] and the texts [3], [4], [9], dimension [13], [14], [20], [28], and [29]. Computer algebra k +2 k k − 2 can be used to solve problems in robotics, splines, (10) +2 + . 2 2 2 integer programming, differential equations, sta- tistics, coding theory, computational chemistry, More generally, in 1973 Strang [26] conjectured a computer-aided geometric design, geometric r formula for the dimension of the space of C poly- theorem proving, and systems of polynomial equa- nomial splines for a given triangulation of the tions. These and other applications are described plane. This was proved in 1988 by Billera [1] using in [2], [6], [8], [10], [15], and [28]. The bibliographies the above methods. An introduction to splines of these books reveal a vast literature of applica- from the module point of view can be found in [6]. tions of computer algebra. In geometric modeling, the idea of fixing the de- How does computer algebra relate to the appli- gree of a moving line or surface has also proved cations of polynomial algebra described earlier in to be useful; see [5] for a survey. In general, there the article? For those parts of the applied mathe- is a lovely interplay between working with poly- matics community where the computational tools nomials of fixed degree (studied via linear algebra) are of paramount importance, the computer alge- and polynomials of arbitrary degree (studied via bra books mentioned above may be the solution. modules). This leads to the notions of Hilbert func- (Not yet the perfect solution, since the books don’t tions and Hilbert polynomials. For example, the always make full use of the language of abstract
1196 NOTICES OF THE AMS VOLUME 52, NUMBER 10 algebra, and those that do sometimes assume too example, given the increasing importance of much or require a steep learning curve.) But for noncommutative structures (e.g., vertex algebras), many applied communities, computational tools are there is the worry that algebra texts may devote too not the central issue; rather, the focus is on un- much space to the commutative case. So there derstanding the overall structure of the problems might not be time to discuss topics like Gröbner being studied. This is where the language of alge- bases, even if they are in the book. My guess is that bra can be helpful. This is why applied mathe- in order for algebra courses to serve the needs of matics may want to think about how to give its stu- all students, there should be a greater emphasis on dents better access to algebra. examples that illustrate the usefulness of algebra as a language for describing mathematical objects The Algebra Curriculum in pure and applied situations. But as above, the Where does a student of applied mathematics learn biggest weakness is that applied mathematics algebra? This question can have many different students will never see such a course unless they answers: are in a department that includes both pure and • Some students learn algebra as an undergradu- applied mathematics. In applied mathematics, ate mathematics major. engineering, or operations research departments, • Some learn algebra in a graduate algebra course. students are unlikely to encounter a general course • Some learn algebra in specialized courses in an in abstract algebra. Furthermore, when such stu- area of applied mathematics where algebra is es- dents try to take abstract algebra in a pure math- pecially relevant. ematics department, they are likely to encounter • Some learn algebra later in their research career. a version of the course aimed at future algebraists. • Some never need algebra. Specialized courses in areas of applied mathe- The last bullet reflects the reality that applied matics may be one of the best ways to introduce mathematics is an immense enterprise. In spite of students to the algebra relevant for their research. my belief in the value of abstract algebra, it is not This works well in areas of applied mathematics essential that all applied mathematicians learn the where the use of algebra has a clear utility and a subject. good track record. But who was the first person to For the first four bullets, let me offer some re- teach such a course? Where did that person learn flections on the role of algebra in each situation. algebra? Also, if such courses become the main Virtually all undergraduate mathematics ma- source of algebra in applied mathematics, then we jors study abstract algebra. While this seems to run the risk of limiting the scope of how algebra solve the problem for these students, let me point can be used. out that in the U.S., such courses rarely discuss Some of the most interesting uses of algebra are modules, and they often don’t say much about the unexpected ones. This is part of what makes polynomial rings in several variables (the focus mathematics so much fun. As happened with Seder- is more on the univariate case). Furthermore, the berg, applied mathematicians sometimes find that abstraction can make the course seem less relevant algebra is relevant to the problem they are work- to students whose interests lie in applied mathe- ing on. Sometimes the tools of algebra solve the matics. In awareness of this problem, some algebra problem, while other times the language of alge- texts focus on applications (see [17] for a recent ex- bra clarifies the issues and structures involved and ample), and some “pure” texts include numerous helps the researcher focus on what is essential applied examples and a few (such as [18] and [22]) (which in turn can generate juicy problems for the have sections on Gröbner bases. But the biggest algebraists to explore). But how does the researcher weakness of any undergraduate algebra course is come to realize that algebra is relevant? I see two that many applied mathematicians have back- ways this can happen: grounds in subjects like physics, engineering, or • Talk to a mathematician who knows algebra. operations research. They never see such a course. Here, the challenge is on the pure side of mathe- A graduate course in abstract algebra is more matics: are algebraists trained in such a way that they can talk to nonexperts and make likely to cover modules and polynomial rings in sev- sense? Given the importance of what is called eral variables, and graduate texts like [11] and [21] “technology transfer” in the U.S., how do we have sections on Gröbner bases. But does such a make students in pure mathematics better able course seem relevant to students in applied math- to communicate with people far outside their ematics? Or is it just something they need for the speciality? qualifying exams?2 Another complication is that Remember some algebra learned earlier. Aside there is more algebra that people need to know. For • from the algebra courses or specialized courses 2One answer is that in the U.S., many applied mathemati- discussed above, applied mathematicians could cians end up at four-year colleges, where they teach a also be exposed to algebra by having certain wide variety of courses, often including abstract algebra. applied courses take the opportunity to use the
NOVEMBER 2005 NOTICES OF THE AMS 1197 language of algebra. This could happen in an [11] D. DUMMIT and R. FOOTE, Abstract Algebra, third ed., applied linear algebra course that addresses the John Wiley & Sons, New York, 2004. numerical and symbolic aspects of the subject [12] D. EISENBUD, The Geometry of Syzygies, Springer- or in a numerical analysis course that discusses Verlag, New York, Berlin, and Heidelberg, 2005. [13] J. VON DER GATHEN and J. GERHARD, Modern Computer some topics from Stetter’s recent book Numer- Algebra, Cambridge Univ. Press, Cambridge, 2003. ical Polynomial Algebra [25].3 [14] K. O. GEDDES, S. R. CZAPOR, and G. LABAHN, Algorithms In some ways, it all comes down to communication for Computer Algebra, Kluwer, Dordrecht, 1992. between pure and applied mathematics. There is [15] R. GOLDMAN and R. KRASAUSKAS (eds.), Topics in Alge- already a lot, but there needs to be more. braic Geometry and Geometric Modeling, Contemp. The questions and suggestions posed in this Math., Vol. 334, Amer. Math. Soc., Providence, RI, 2003. article are preliminary and should be taken with a [16] D. HILBERT, Ueber die Theorie der algebraischen For- grain of salt. Their main purpose is to stimulate men, Math. Annalen 36 (1890), 473–534. conversations about the proper role of algebra in [17] D. JOYNER, R. KREMINSKI, and J. TURISCO, Applied Abstract Algebra, Johns Hopkins Univ. Press, Baltimore, MD, applied mathematics. I am firmly convinced that 2004. algebra has a lot to offer, once we collectively fig- [18] N. LAURITZEN, Concrete Abstract Algebra, Cambridge ure out the best way to make use of this amazing Univ. Press, Cambridge, 2003. language. [19] F. MEYER, Zur Theorie der reducibeln ganzen Func- tionen von n Variabeln, Math. Annalen 30 (1887), Acknowledgements 30–74. I am grateful to D. Barbezat, E. Lamagna, J. Reyes, [20] B. MISHRA, Algorithmic Algebra, Springer-Verlag, New and T. Leise for helpful suggestions and conversa- York, Berlin, and Heidelberg, 1993. tions. Thanks also go to M.-F. Roy for her encour- [21] J. J. ROTMAN, Advanced Modern Algebra, Prentice- agement and to G. Avrunin, F. Chen, R. Goldman, Hall, Upper Saddle River, NJ, 2002. [22] , A First Course in Abstract Algebra, second ed., P. Lax, T. Sederberg, and B. Turkington for useful ——— Prentice-Hall, Upper Saddle River, NJ, 2000. comments on early drafts of this essay. Finally, I [23] T. W. SEDERBERG and F. CHEN, Implicitization using mov- owe a great debt to my collaborators in geometric ing curves and surfaces, in Proceedings of SIGGRAPH, modeling for helping me to see the wonderful 1995, 301–8. consequences of the algebra I love. [24] C. P. SIMON and L. BLUME, Mathematics for Economists, W. W. Norton, New York and London, 1994. References [25] H. STETTER, Numerical Polynomial Algebra, SIAM, [1] L. BILLERA, Homology of smooth splines: Generic tri- Philadelphia, 2004. angulations and a conjecture of Strang, Trans. Amer. [26] G. STRANG, The dimension of piecewise polynomial Math. Soc. 310 (1988), 325–40. spaces, and one-sided approximation, in Conference [2] A. M. COHEN, H. CUYPERS, and H. STERK (eds.), Some Tapas on the Numerical Solution of Differential Equations of Computer Algebra, Springer-Verlag, Berlin, Heidel- (Univ. Dundee, Dundee, 1987), Lectures Notes in Math., berg, and New York, 1999. Vol. 363, Springer-Verlag, New York, Berlin, and Hei- [3] J. S. COHEN, Computer Algebra and Symbolic Compu- delberg, 1974, pp. 144–52. tation: Elementary Algorithms, A K Peters, Wellesley, [27] J. J. SYLVESTER, On a theory of syzygetic relations of MA, 2002. two rational integral functions, comprising an appli- cation of the theory of Sturm’s functions, and that of [4] ——— , Computer Algebra and Symbolic Computation: Mathematical Methods, A K Peters, Wellesley, MA, the greatest algebraic common measure, Philos. Trans. 2003. Roy. Soc. London 143 (1853), 407-548. ANG [5] D. COX, Curves, Surfaces, and Syzygies, in [15, 131–50]. [28] D. W , Elimination Theory, Springer-Verlag, Wien [6] D. COX, J. LITTLE, and D. O’SHEA, Using Algebraic Geom- and New York, 2001. etry, second ed., Springer-Verlag, New York, Berlin, and [29] F. WINKLER, Polynomial Algorithms in Computer Al- Heidelberg, 2005. gebra, Springer-Verlag, Wien and New York, 1996. [7] D. COX, T. SEDERBERG, and F. CHEN, The moving line ideal basis of planar rational curves, Comput. Aided Geom. Des. 15 (1998), 803–27. [8] D. COX and B. STURMFELS (eds.), Applications of Compu- tational Algebraic Geometry, Amer. Math. Soc., Prov- idence, RI, 1998. [9] J. H. DAVENPORT, Y. SIRET, and E. TOURNIER, Computer Al- gebra, second ed., Academic Press, London, 1993. [10] A. DICKENSTEIN and I. Z. EMIRIS (eds.), Solving Polyno- mial Equations: Foundations, Algorithms, and Appli- cations, Springer-Verlag, Berlin, Heidelberg, and New York, 2005.
3This is easier said than done, since as in pure mathematics, there is more and more that students in applied mathe- matics need to know. However, it is worth thinking about.
1198 NOTICES OF THE AMS VOLUME 52, NUMBER 10