Since this blog is about Mathematical Reviews and MathSciNet, let me point out that Gauss also has 238 Related Publications in MathSciNet. What are “related publications” in an Author Profile? Thank you for asking: These are other items (articles, books, proceedings) connected in a significant way to the author, but for which the person was not the author. The most common examples are items for which the person was an editor, a translator, or was the subject of a biography.

People worry about citation counts. The esteemed Gauss has 345 citations to his work. I am told that deans prefer to use citation counts from broad-based sources, such as Web of Science. If that is the case, Gauss is in trouble since I can only find two items for him in Web of Science. Both are from the *Journal für die reine und angewandte Mathematik*, aka, *Crelle’s Journal*. Neither paper has any citations in the database. As a result, Gauss’s citation count in Web of Science is 0. A search for “Carl Friedrich Gauss” brings up 25,200 results in Google Scholar. However, since Gauss never created a profile for himself on Google Scholar, you have to compute his citation count by hand. (He has a lot.)

In the Mathematical Reviews Database, Gauss’s most cited work is

MR0197380

Gauss, Carl Friedrich

Disquisitiones arithmeticae.

Translated into English by Arthur A. Clarke, S. J. Yale University Press, New Haven, Conn.-London 1966 xx+472 pp.

The short review is by W.J. LeVeque, a number theorist who used to be Executive Editor of Mathematical Reviews, then later became the Executive Director of the AMS. The gist of the review is astonishment that this is the first published English translation of this famous work.

For fun, searching the Mathematics Subject Classification for Gauss brings up matches in number theory and real functions: Gauss sums and Integral formulas (Stokes, Gauss, Green, etc.), but nothing from differential geometry, probability, mathematical physics, usw.

Happy 241^{st} birthday, Carl Friedrich Gauss.

If you want to explain the program without saying anything, you can say that the Langlands program centers on Langlands reciprocity, which is a generalization of Artin reciprocity, which is, “of course”, a generalization of the familiar quadratic reciprocity. In searching MathSciNet, there are over 450 items with “Langlands” in the title, in various forms: Langlands’ conjecture, Langlands duality, Langlands program, Langlands correspondence, and so on. I’m not sure which of them I could recommend to anyone as a first introduction. I suppose you could just read the reviews of all of them in MathSciNet, one after another. A surer way to learn about the Langlands program is to have someone in your department working on some part of it — then go to their seminars week after week.

Langlands has a great result in representation theory of semi-simple Lie groups, known as the *Langlands classification of irreducible admissible representations*, that illustrates how the world of mathematics has changed in the last forty years or so. This result, which was written in 1973, circulated in mimeographed form for many years.(*) If you wanted a copy, you had to write a letter to the Institute of Advanced Study and ask for one to be mailed to you. Sometimes, your letter would go unanswered (I don’t know why) and you would have to write again. Sometimes you would have to wait until they ran off some more copies. Alternatively, you could find someone who had a copy and try to make a photocopy on a primitive Xerox machine. Since the originals were already somewhat faint, the photocopy was going to suffer from generational degradation quite quickly. Finally, David Vogan and Paul Sally assembled a handful of important unpublished papers in representation theory and published them as a collection in the AMS’s Mathematical Surveys and Monographs series.

The Langlands paper:

MR1011897

Langlands, R. P.(1-IASP)

On the classification of irreducible representations of real algebraic groups. Representation theory and harmonic analysis on semisimple Lie groups, 101–170,

Math. Surveys Monogr., 31, Amer. Math. Soc., Providence, RI, 1989.

The book:

MR1011895

Representation theory and harmonic analysis on semisimple Lie groups.

Edited by Paul J. Sally, Jr. and David A. Vogan, Jr. Mathematical Surveys and Monographs, 31. American Mathematical Society, Providence, RI, 1989. xii+350 pp. ISBN: 0-8218-1526-1

There is another “Langlands Conjecture” (due to Kostant and Langlands), which proposed that the discrete series representations of a semi-simple Lie group $G$ could be realized in $L^2$-cohomology spaces of holomorphic line bundles over the (complex) manifold $G/H$, where $H$ is a compact Cartan subgroup of $G$. This was proved by Wilfried Schmid in

MR0396856

Schmid, Wilfried

$L^{2}$-cohomology and the discrete series.

Ann. of Math. (2) 103 (1976), no. 2, 375–394.

Langlands also wrote a small number of reviews for Mathematical Reviews. His review of the paper in which Ngô gives his proof of the Fundamental Lemma is a tour de force and can be found at the end of this post.

MR2653248

Ngô, Bao Châu(1-IASP)

Le lemme fondamental pour les algèbres de Lie. (French) [The fundamental lemma for Lie algebras]

Publ. Math. Inst. Hautes Études Sci. No. 111 (2010), 1–169.

(*) Besides the five papers assembled by Vogan and Sally, there is another famous instance of a fundamental result in representation theory circulating only in mimeographed form: Roger Howe’s theory of reductive dual pairs.

MR0986027

Howe, Roger

Remarks on classical invariant theory.

Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570.

[written in 1976]

**A review by Robert Langlands**

MR2653248

Ngô, Bao Châu(1-IASP)

Le lemme fondamental pour les algèbres de Lie. (French) [The fundamental lemma for Lie algebras]

Publ. Math. Inst. Hautes Études Sci. No. 111 (2010), 1–169.

The present paper, for which its author was awarded a Fields Medal, had achieved, even before publication, considerable fame and the Proceedings of the 2010 International Congress of Mathematicians in Hyderabad [World Sci. Publ., Hackensack, NJ, 2011] will contain two accounts of it, one by the author himself and one by James Arthur, a laudation delivered at the presentation of the prize. Both accounts are extremely instructive, and I refer the reader to them, as well as to two excellent accounts of the fundamental lemma available as preprints, one by T. C. Hales [“The work of Ngô Bao Châu”, preprint, arxiv.org/abs/1012.0382] and one by D. Nadler [“The geometric nature of the fundamental lemma”, preprint, arxiv.org/abs/1009.1862].

There is a great deal to be said about the fundamental lemma, about its origins, about the methods used to prove it and the developments that preceded the proof itself, and about its consequences or possible consequences, much more than could be accommodated in a normal review. No one is yet familiar with all this material. As a consequence, a good deal has been written about the lemma that, in my view, is misleading. I am convinced that anyone who wants to contribute to the central problems in the contemporary theory of automorphic representations, or, better, to functoriality and matters related to it, will need a better grasp of all these matters than any one person possesses at present. I shall try here to clarify this assertion, although this will entail a risk: not only of false prophecy but also of revealing my own ignorance. I understand the origins of the lemma and I believe I have as much insight into its possible consequences as anyone, but the proof itself, which exploits difficult tools and concepts from both modern algebraic geometry and topology, contains a very great deal of which I have only an uncertain understanding. The reader should take what I say about geometry or topology with a grain of salt.

The origins of the lemma are in the theory of Shimura varieties and in the theory of harmonic analysis on reductive groups over $\Bbb R$. This second source is analytic and algebraic, the theory of the spectral decomposition of invariant distributions on real reductive groups, a theory that we owe almost in its entirety to Harish-Chandra, although the basic idea, that the pertinent eigenfunctions are characters, was introduced in the context of finite groups by Dedekind and Frobenius. What was imposed on our attention by the theory of Shimura varieties and the trace formula was the understanding that for reductive algebraic groups there are two different notions of conjugation invariance: invariance and stable invariance. These are a result of two different kinds of conjugacy in, say $G(\Bbb R)$, but more generally in $G(F)$, where $F$ is a local field, archimedean or non-archimedean. One is conjugacy in $G(F)$ itself, the other is conjugacy in $G(\overline F)$, where $\overline F$ is the (separable) algebraic closure of $F$. It was only as we began the study of the zeta-functions of Shimura varieties with the help of the trace formula that the importance of the distinction, its consequences, and the attendant difficulties were recognized. They led to the fundamental lemma.

The issue, at first, is less the fundamental lemma, which can take diverse forms, than its consequences, not only for Shimura varieties but more importantly for harmonic analysis, both local and global. With the fundamental lemma, it is possible to create a theory of endoscopy that reduces invariant harmonic analysis, even various forms of twisted-invariant harmonic analysis, on arbitrary reductive groups to stably invariant harmonic analysis on quasi-split groups. It is the latter in which the notion of functoriality is best expressed, and it is functoriality, still to a large extent conjectural, that is the source of the arithmetic power of representation theory and harmonic analysis. Specific forms of functoriality have already been used in the course of establishing Fermat’s theorem and other conjectures of considerable interest to arithmeticians.

The fundamental lemma, once proved, offers two methods to attack functoriality: the first more immediate; the second much more encompassing. Although more limited, the first is of great importance, as it has offered to Arthur reasons for developing the general trace formula, which, thanks to him, has been given a chance to demonstrate the enormous power of nonabelian harmonic analysis, of which the trace formula is an expression, for arithmetic. The lemma allows global, and presumably also local, transfer of stable characters from the endoscopic groups $H$ for a given group $G$ provided with a twisting, perhaps trivial, to the group $G$ itself. The best reference for this type of theorem will be Arthur’s book The endoscopic classification of representations: orthogonal and symplectic groups [in preparation]. It promises to increase greatly the confidence of mathematicians at large in the notion of functoriality, even though the functoriality yielded directly by endoscopy is limited. I add that, in my view, the central issue in endoscopy is the theory with no twisting.

After the introduction of endoscopy, there were a good many years during which I did not pay much attention to the attempts to develop it, on one hand, by Waldspurger, Hales, and others, and, on the other hand, by Goresky, Kottwitz, and MacPherson. These contributions not only made possible the final proof of the lemma in the hands of Laumon and then Ngô, but also introduced ideas that will, I expect, play a major role in the continuing attack on functoriality.

The principal tools of Harish-Chandra in the development of harmonic analysis on real reductive groups and then, later, of Shelstad’s treatment of endoscopy were the bi-invariant differential operators on the group. The spectral decomposition amounts to a spectral decomposition of this family of commuting operators on $L^2(G(\Bbb R))$. This is a local theory. Although a great deal of effort has been spent on non-archimedean fields, the theory has not reached the same stage, in good part because the spectral theory could not be reduced to one for a commutative family. My impression on studying the work of Waldspurger, Laumon and Ngô, without yet in any sense mastering it, is that the cohomology theory of perverse sheaves may offer a substitute, so that the possibilities offered by Waldspurger’s reductions have by no means been exhausted.

Without any real knowledge of perverse sheaves as I began the study of Ngô’s proof, and the earlier work with Laumon, and still only superficially informed, I am struck by the advantages of working with them. At the coarsest of levels, the orbital integrals provide over $\Bbb R$ or $\Bbb C$ the transfer that is dual to the transfer of characters from Cartan subgroups $H$ of $G$, or better, although the theory has not been properly developed in this form even over $\Bbb R$, the transfer of characters implied by functoriality. Something similar will, I suppose, be true for non-archimedean fields, but it will be more delicate because some irreducible characters are not associated to a Cartan subgroup, for example, those associated to representations of the local Galois groups as tetrahedral representations. What, in my view, is taking place in Waldspurger’s analysis, although I have yet to examine it with sufficient care, or even any care, is a reduction of the local analysis to the study of orbital integrals on Lie algebras, not over a local field but over a finite field, or, better expressed, in the context of algebraic geometry over a finite field. The asymptotic behavior described by the germs of Shalika becomes at this level a question of direct images of perverse images and their support, thus a behavior that is strictly geometric and strictly within the range of behavior encountered already in the study of these sheaves. I can imagine that the geometric information available through this translation might replace Harish-Chandra’s study of the orbital integrals and their jumps to characters of $G$. Something similar to the jump conditions that Harish-Chandra met, and even something more subtle, might appear. I imagine that, when examining the possible behavior of the direct images with care, one will find behavior that can only be explained with the help of local Galois groups that admit surjective homomorphisms to relatively complex solvable groups. These matters will have to be studied on their own.

This kind of local information will be necessary if the program proposed for the utilisation of the stable trace formula—a formula available only after the fundamental lemma has been established—is to succeed in establishing functoriality. It is to be utilised in combination with the Poisson formula on the Steinberg-Hitchin base, an affine object introduced by myself with E. Frenkel and Ngô [Ann. Sci. Math. Québec 34 (2010), no. 2, 199–243]. The introduction of the Poisson formula was suggested by Ngô’s use of the Hitchin base.

None of this explains the reasons for the success of Ngô nor for the earlier partial successes of Goresky-Kottwitz-MacPherson and Laumon-Ngô. Moreover, with the exception of Arthur’s laudation, little attention has been paid in various expositions to the needs of specialists of the theory of automorphic representations, thus of those to whom the lemma itself is of the most interest and who may, like me, have little, if any, familiarity with stacks, perverse sheaves, or equivariant cohomology. So it may be worthwhile for me to have attempted to describe some glimpses of understanding that I have had while trying to penetrate their thoughts. I still have a long way to go and I am not certain that these glimpses are not will-o’-the-wisps. Waldspurger and one or two others may have clearer notions of the possibilities than I.

The fundamental lemma itself appears in the context of orbital integrals, thus integrals over the conjugacy classes $\{g^{-1}\gamma_G g\}$ defined by elements $\gamma=\gamma_G\in G(F)$, $F$ a local field, for the present non-archimedean. For $\gamma_G$ semisimple and regular, the conjugacy classes within the stable conjugacy class of $\gamma_G$ are parametrized, in essence, by the elements of the abelian group $H^1({\rm Gal}(\overline F/F),T)$, $T$ the centralizer of $\gamma_G$. If $\kappa$ is a character of this group, we may form $\sum\kappa(\gamma’_G)\scr O_G(\gamma’_G, f_G)$, where the sum over conjugacy classes is to be interpreted as a sum over $H^1({\rm Gal}(\overline F/F),T)$, and $f_G$ is the unity element of the Hecke algebra over $G$. Associated to $\kappa$ is an endoscopic group, thus a quasi-split reductive group $H$, and to $\gamma$ a stable conjugacy class $\{\gamma_H\}$ in $H$, for which we can form a stable sum $\scr O^{\rm st}_H=\sum\scr O_H(\gamma’_H,f_H)$, where $f_H$ is the unity element in the Hecke algebra of $H$. The fundamental lemma, in its simplest and earliest formulation, is the equality of these two sums, up to a well-defined constant factor that will necessarily depend on the choice of Haar measure on $G$ and $H$.

After Waldspurger’s reduction, a new, but similar, equality appears with integrals over a set determined by an element $\gamma$, again often semisimple and regular, of the Lie algebra $\frak g$ of $G$ (or $H$) over $F’$, again a local field but of positive characteristic, the ring of formal power series over a finite field $k$. Not having followed the developments over the years, I find the transition from one context to the other abrupt. My intuition is often brought up short. In addition, the proof of the fundamental lemma, like early proofs in local class field theory and occasionally elsewhere, is an argument from a global statement to a local statement, so that the function field $F$ of a complete nonsingular curve $X$ over $k$ of which $F’$ is a completion at some place $v$ is introduced. $G$ is replaced by a group over this new $F$ and $\gamma$ by an element of the Lie algebra $\frak g$ of $G$, or more precisely by a section of the Lie algebra bundle defined by a $G$-bundle over $X$, a section that is allowed to have poles of large but finite order at a certain number, again large but finite, of points. It is in this context—especially difficult for those not sufficiently conversant with the notions of modern algebraic geometry—that the proof functions.

I was first disoriented by the appearance of Picard varieties in this context. They seemed to be of the usual type, thus closely related to abelian varieties. It was only after some time, when I noticed that the point of departure was the first cohomology group of a torus—thus a multiplicative group—the centralizer of $\gamma$, and that it was entirely possible that the transition from the local field $F$ to the function field $F$ of $X$ and from Galois cohomology to étale cohomology or other cohomologies might entail the appearance of Picard varieties, that I began to feel more at ease. Galois cohomology groups have not been for me geometric objects. As descriptions of families of line bundles, thus of cohomology groups with values in ${\rm GL}(1)$ or, possibly, other abelian algebraic groups, Picard varieties (or stacks) may be representable—whether by varieties or by stacks—and thus subject to study by the usual methods of algebraic geometry. Once reoriented, I found it much easier to follow, at least superficially, the presentations by Ngô and others of the geometrical proofs of the fundamental lemma, in the final form as well as in the earlier forms.

Nevertheless, in Ngô’s proof and in the reflexions of other authors that preceded it, there are several notions of which my grasp is tenuous: equivariant cohomology on the one hand and the apparently related notion of stacks on the other. Some aspects of the structure of the proof are quite clear. At a given place of $X$ that is defined over $k$, in particular at the place with which we began, the orbital integrals, both for $G$ and for $H$, can be interpreted as counts, although the count is a weighted count because centralizers of the elements $\gamma$ interfere. One of the functions of stacks and equivariant cohomology, for those who understand them, is to take this weighting into account. That said, thanks to the passage to a global context, in the sense of algebraic geometry, thus to the passage to $X$ and bundles over $X$, the counting, or rather the equality of two different counts asserted by the fundamental lemma, is replaced, in the spirit of the Weil conjectures and the Lefschetz formula, by an isomorphism of cohomology groups. The global count is, however, a sum over the points of $X$ of local counts, so that, a global equality once established in general, it is necessary to return to $X$ and to the section of the $\frak g$-bundle that replaced the original $\gamma$, and to make choices that allow us to isolate the local contribution with which we began. Most of the effort is expended on the proof of the global cohomological statement, in the context of perverse sheaves for the étale cohomology and in the context of stacks.

I found it difficult to discover and keep firmly in mind the nature of the local count. There are at least two parameters at hand: the point of $X$ and the point $\gamma$, which is now a section $\varphi$ of the Lie algebra of a $G$-bundle $E$ on $X$ the total order of whose poles is controlled by a divisor $D$. The family $\scr M$ of these Hitchin pairs, $(E,\varphi)$, is an essential element of the theory. The family of the classes in the Lie algebra of the group in question, $G$ or one of its endoscopic groups $H$ as the case may be, is the Hitchin base, a designation now familiar, thanks to Ngô, to a wide mathematical audience. The count is made over this base. Rather, the count is made, for both $G$ and $H$, after a projection to this base. The domain of the projection is, to a first approximation, a scheme whose points are, first, a $G$-bundle on the given base $X$ and, second, the section $\gamma$. So, implicit in the discussion is, I suppose, the existence of moduli spaces or stacks and an understanding of the cohomology of perverse sheaves defined on them. Most of this, and much else, I have to take on faith at present.

The Hitchin base is, as an algebraic variety over $k$, an affine space. The count on the fiber is made indirectly, through the direct images of the cohomology of the fiber. This fiber has, I believe, two important features. One feature it shares with the usual Picard varieties, namely an action of a very large connected group, sometimes an abelian variety; this large group is defined over the Hitchin base. If I understand correctly the explanation in Ngô’s Hyderabad lecture, an important consequence is that the action of the full group, a Picard group (rather stack!) $\scr P$ in the sense of Ngô, on the cohomology of the fibers is defined through a discrete quotient, denoted $\pi_0(\scr P)$ by Ngô, a possibility that is certainly plausible from a topological point of view. This discrete quotient is closely related to the Galois cohomology groups $H^1({\rm Gal}(\overline F/F),T)$ with which we began. These things are well explained in Ngô’s Hyderabad lecture, where it is also explained that the local discrete quotients can be patched together, but in the étale topology, to form a sheaf of abelian groups. It is somewhat comforting, and perhaps not altogether incorrect, if we think of this as a patching in the étale topology of the various $H^1({\rm Gal}(\overline F/F),T)$, defined for widely varying tori $T$. In any case this allows the discrete quotient and its characters to be introduced globally, something that was done in a different manner in the original formulation of the lemma.

The result is a sheaf over the Hitchin base that permits an action of the group $\scr P$. Since $\scr P$ acts on the fibres over the base, its action defines an action on the direct image of the cohomology on the Hitchin base, an action that factors through $\pi_0(\scr P)$. Consequently the direct image can be decomposed as a direct sum with respect to the characters $\kappa$ of $\pi_0(\scr P)$. The principal theorem of Ngô, at least in connection with the fundamental lemma, is to establish that each component of the direct sum is isomorphic to a similar component for an endoscopic group $H$ over $X$, a group defined by the character $\kappa$.

There is a fluidity in the development of the proof that Ngô captures in his various expositions. Ideas appear and reveal themselves as suggestive but ultimately inadequate, and then reappear in a different, often more difficult, guise. It is probably impossible to understand the final proof without some feeling for these initial stages, for equivariant cohomology in all its guises, and, above all, for the geometry of the Hitchin fibration. I certainly have a long way to go, but I find the relatively concrete form in which this fibration is used by Laumon-Ngô in the proof of a special case of the fundamental lemma a helpful guide to the general case.

Since the Hitchin fibration and its properties are basic, a word or two about its construction may not be inappropriate. For a vector bundle, thus for a ${\rm GL}(n)$-bundle, one can associate to the section $\gamma$, or better to the point $a$ in the Hitchin base, a matrix-valued function on $X$, and to each point $x\in X$, the $n$ points in an $n$-dimensional space given by its eigenvalues. As $x$ varies, these points trace out a curve, an $n$-fold covering $Y_a=Y_\gamma$ of $X$. With $\gamma$ we can introduce, at least in favorable circumstances, more: for each point $x$ and each of the eigenvalues, a line, the eigenspace corresponding to the eigenvalue. Thus the section $\gamma$ defines a line bundle on $Y_a$. There are questions that arise at the points where the eigenvalues are multiple, but we do see line bundles on the horizon and therefore, perhaps, abelian varieties and cohomology groups in degree $1$, groups related to those with which endoscopy began. The abelian varieties are a sign that, in the new context, these cohomology classes appear as line bundles that give rise to representable functors, whose points can be described geometrically. The Hitchin fibration, as defined by Ngô, provides similar constructions for a general group. Even in the original form, the eigenvalues associated to $\gamma$ define at each point of $X$ a diagonal matrix, but as the order of the eigenvalues is not prescribed, it is in fact only the conjugacy class of this diagonal matrix that is determined.

At the level of groups we cannot, so far as I know, ordinarily find a map from conjugacy classes to matrices that is inverse to that from matrices to conjugacy classes, but at the level of Lie algebras, low characteristics aside, we can. For example, for the group ${\rm SL}(2)$, the conjugacy class is given, at least at the regular elements by the determinant, $a$, and the representative matrix for this class can be taken to have diagonal elements $0$ and off-diagonal elements $1$ and $a$. There are, I believe, various such lifts. Ngô uses the one associated to the name of Kostant. Our original description of the spectral curve $Y_a$ was deliberately vague about its form at those points where eigenvalues coincide and it is best here to pass over in silence the difficulties they entail in Ngô’s definitions. They entail technical difficulties that I have not yet made any attempt to understand. Indeed, I am not much beyond the introduction to his paper. In any case, what results is a lift not only of the regular conjugacy classes of the Lie algebra to the Lie algebra itself, but an abelian group over these lifts. It is closely related to the centralizer of the lifts and yields a fibration in groups over the Hitchin base. The dimension of the fibers is the rank of $G$. In the definition of the Picard variety (stack) relevant to the Hitchin fibration and to Ngô’s analysis, the bundles associated to this fibration in groups replace the line bundles of the classical theory. I have to remind myself constantly that there are two parameters at play in this fibration: the base $a$, given by the class of $\gamma$, and a point $x$ of $X$, at which $\gamma$ is essentially an element in the Lie algebra of $G$, say over the residue field or over the coordinate ring at $x$.

As already observed, the argument for the proof of the fundamental lemma proceeds in two stages: first, for a fixed $a$ and all of $X$, but fortunately only for well-chosen $a$; second, for a suitable $X$ and a suitable point $x$ of $X$. We have already described the projection at the first stage, from the total space of the Hitchin fibration to the Hitchin base, and the decomposition of the direct image according to the characters $\kappa$ of the Picard stack.

There is an equality of sheaves over $X$ to be proven at the first stage. There are two issues in the proof of the equality: the support of the relevant direct images, and the equality on this support. An endoscopy group is so defined that there is a morphism of the Hitchin base $\scr A_H$ to $\scr A_G$. So we can compare the direct image of a sheaf on $\scr A_H$ with a sheaf on $\scr A_G$. The sheaf on $\scr A_G$ is defined by the part of the direct image of the sheaf associated to the character $\kappa$. For $H$, one does the same thing, but the character for $H$ is taken to be trivial. If $H$ is associated to $\kappa$, it has first to be shown that the direct image of the $\kappa$-component for $G$ is supported on the image of the Hitchin base for $H$. This is, in principle, a consequence of the definitions, but it is not an easy consequence. Indeed, the final proof is tremendously daunting.

Those of us with less than adequate facility with the concepts can best begin with the theorem for unitary groups proved by Laumon-Ngô, because in [G. Laumon and Ngô Báo Châu, Ann. of Math. (2) 168 (2008), no. 2, 477–573; MR2434884] not only does the Picard stack appear in its primitive form in terms of the spectral curve $Y_a$, but, in addition, the proof of the necessary homotopy lemma, which is used to deal with the problem of support, appears to be at an altogether different level of difficulty than the support theorem of the paper under review. In [G. Laumon and Ngô Báo Châu, op. cit.], both $G$ and $H$ are unitary groups. Since a unitary group is a form of ${\rm GL}(n)$, the concept of spectral curve has a more immediate geometric content and there is a more direct relation between the Hitchin fibrations of $G$ and $H$ that appears to simplify the arguments considerably.

I have already adumbrated the final step of the proof. If the curve $X$ and an element $a$ of the Hitchin base are given, they define locally at any point $x$ of $X$ the elements for the original statement of the fundamental lemma for the Lie algebra, an element of the local Lie algebra and a group $G(\scr O_x)$. Moreover, the equality of a $\kappa$-component of the direct image at $a$ with a direct image for an endoscopic group $H$ can be interpreted, thanks to the Grothendieck-Lefschetz theorem, as an equality of the product over the points of $X$ of two counts, one for $\kappa$-components on $G$ and one for $H$. If we can choose $x$, $X$, and $a$ so that they reproduce any arbitrarily given local data and if $X$ and $a$ are also chosen such that the fundamental lemma is true at all points $x’\neq x$ of $X$, we can cancel all terms in the product but those at $x$ and deduce the desired equality at that point. We cannot, apparently, expect to choose $X$ such that the fundamental lemma is utterly obvious away from $X$, but it can be so chosen that it is accessible to direct computation. To establish the existence of $X$ and $a$ with the necessary properties has required, both in the present paper and in the earlier paper on unitary groups, very sophisticated algebro-geometrical methods. It is also important for its existence that the poles of the section defining $a$ are allowed to grow in number.

For the unitary groups, the very last step, the deduction of the fundamental lemma outside of $x$ from the properties of $X$ and $a$ appears almost an elementary exercise in geometry over finite fields. This is not so in general. Further struggles with perverse sheaves await the reader.

It is clear that, for the majority of specialists in nonabelian harmonic analysis and representation theory, thus, in particular, for specialists in the theory of automorphic representations and the associated arithmetic, certainly for me, it will take more than a few weeks, or even a few months, to assimilate the techniques from contemporary algebraic geometry that are required for the proof of the fundamental lemma. How long it might take geometers to understand fully the questions posed by the arithmetic and the analysis, I hesitate to guess. This might be easier. Representation theory has a briefer and, in some respects, narrower history, but it is less familiar to the majority of mathematicians. Time will tell.

Since, as I intimated at the beginning of this review, the fundamental lemma is an essential and fundamental contribution to a theory that will not be developed by specialists in algebraic geometry alone, there will be a need for further, more accessible expositions of the methods of this paper and those that preceded it, with examples, even very simple examples, and with considerably more explanation of the geometric intuition implicit in the abstract theory. An index to definitions and symbols would also be welcome! The present paper is 168 pages long and these pages are large and very full. An exposition genuinely accessible not alone to someone of my generation, but to mathematicians of all ages eager to contribute to the arithmetic theory of automorphic representations, would be, perhaps, four times as long, thus close to 700 pages. It would, I believe, be worth the effort.

]]>Stephen Hawking was one of the most gifted and most famous scientists of the last fifty years. His science demonstrated a blend of technical ability and intuition. Hawking’s best-known results concern black holes. His earliest work was on singularities in general relativity, what became known as the Hawking-Penrose theorems. His discovery of Hawking radiation was a landmark result that fundamentally changed our understanding of black holes. Hawking had a remarkable life story, some of which was represented in the movie *The Theory of Everything*. Hawking had a playful spirit, which served him well and helped him to connect with the general public. It also endeared him to those who saw his appearances on television shows such as *Star Trek: The Next Generation*, *The Simpsons*, *Futurama*, *The Big Bang Theory, *and even *Last Week Tonight.*

MathSciNet has 148 publications for Stephen Hawking, which are cited by over 2000 different authors. Outside academia, Hawking’s most famous book is *A Brief History of Time*. Among physicists and mathematicians, his book with Ellis, *The Large Scale Structure of Space-time* is a true classic, and still a great place to learn the mathematics of general relativity.

Of his eight earliest papers in MathSciNet, six of them have “singularities” as part of the title. Thirty-five later papers have “black holes” in the title. Clearly this was a theme in his work. Hawking’s *Comm. Math. Phys.* paper on radiation of black holes is:

MR0381625

Hawking, S. W.

Particle creation by black holes.

Comm. Math. Phys. 43 (1975), no. 3, 199–220.

The heart of the paper is a rather serious calculation. But if you look at the paper, you see that it is quite dominated by words, not formulas. (The same is true of the shorter, earlier paper in *Nature*.)

As mentioned above, Hawking could be playful. He clearly enjoyed interacting with people. When I was a post-doc in Roger Penrose’s group at Oxford, some members of the group organized a conference at Durham University. Hawking was there for a couple of days. One night, he invited Penrose and some others out for drinks in a local pub. Hawking didn’t drink, but bought drinks for others. Most of the conversation was about the mathematics and physics that was being presented at the conference. But later, Hawking started posing problems — and was goading the more mathematical members of the group. I remember one problem in particular. Consider $x^{x^{x^\cdots}}$. For what positive values of $x$ does this expression converge? The “obvious” answer to a mathematician who has had a couple of beers is $0<x \le 1$. However, that is not quite right as $x$ can actually be a little larger than $1$. For some stupid reason, we were able to get up to $x=\sqrt{2}$. (It is stupid, because once you understand the problem, you shouldn’t be thinking of square roots.) Hawking prodded us, and plied us with more drinks. Eventually we realized that $e$ had to be involved, which led us to $0 < x \le e^{1/e}$, but we couldn’t prove it — even with more beers. The problem goes way back, having been considered by Euler. The earliest published paper on it that I know of is

MR1578416

Eisenstein, G.;

Entwicklung von $\alpha^{\alpha^{\alpha^\cdots}}$. (German)

J. Reine Angew. Math. 28 (1844), 49–52.

A more recent account is in

MR2091543

Anderson, Joel(1-PAS)

Iterated exponentials.

Amer. Math. Monthly 111 (2004), no. 8, 668–679.

Somehow, Hawking had this at his fingertips.

Several obituaries of Hawking have been published in high profile publications: The New York Times, BBC, and The Washington Post. Two that struck me are the obituary in The Guardian written by Roger Penrose, which is quite forthright, and the announcement from his research group at Cambridge University.

**MR0381625**

Hawking, S. W.

Particle creation by black holes.

*Comm. Math. Phys.* **43** (1975), no. 3, 199–220.

83.53

The author demonstrates that black holes are not completely black, but emit thermal radiation with a characteristic temperature of about $10^{-6}(M_\odot/M)^\circ K$ for a Schwarzschild object of mass $M$. This result had been conjectured previously on thermodynamic grounds.

The production of radiation by black holes is a quantum phenomenon caused by the disturbance of the vacuum state by the gravitational field of a collapsing massive object. The nature of the radiation is deduced by studying the effect of the collapse on the normal modes of a massless scalar field that is initially free to propagate through the centre of the collapsing matter and out again. An explicit calculation of the Bogoljubov transformation between the initial (undisturbed) and final states is given under the assumption that null rays are to be treated as in geometrical optics. Only particle states in the asymptotic region (where they are well defined) are discussed. The result is a Planck radiation spectrum.

Some conjecture is given about back-reaction on the metric from the particle production. This should cause the horizon area to shrink (in contrast to the classical case), possibly terminating in a naked singularity.

{For errata to the bibliographic data of the original MR item see E 52 9960 Errata and Addenda in the paper version. See MR0389129}

Reviewed by P. C. W. Davies

**MR0424186**

Hawking, S. W.; Ellis, G. F. R.

**The large scale structure of space-time**.

Cambridge Monographs on Mathematical Physics, No. 1. *Cambridge University Press, London-New York*, 1973. xi+391 pp.

83.58

Despite its imposing title, this book is a text on general relativity with a very mathematical orientation. It is an excellent introduction to the subject for a mathematician interested in relativity, because it is much more rigorous and uses a language much more familiar to the mathematician than that found in the usual texts. The thrust of the book is toward proving the “singularity theorems”, stating that large classes of solutions of Einstein’s equations with reasonable equations of state reach a singularity (of some sort) in a finite time. To deal with these theorems the authors introduce mathematical machinery for those not familiar with the tools they use, and discuss general relativity in this context, arriving at a description of several exact solutions of Einstein’s equations. This discussion takes up half of the book, and the remaining half is dedicated to the discussion of more specialized mathematical tools and to proving the singularity theorems for collapsing stars, and finally for the universe. The book ends with a short discussion of the meaning of singularities.

Reviewed by Michael P. Ryan Jr.

]]>Andrew Ranicki has died. Ranicki was a topologist, with particular expertise in algebraic surgery. Indeed, Ranicki had the unusual title of Professor of Algebraic Surgery at the University of Edinburgh. (Andrew was a special case for almost everything.) His two papers on surgery in Proc. London Math. Soc. [MR0560997 and MR0566491 (*****)] were among his most cited papers. He was also familiar to many for his work in $K$-theory, including his work as an editor for leading journals in the subject.

Ranicki received his doctorate from Cambridge University under the supervision of Adams and Casson with a dissertation titled “Algebraic L-Theory”. After Cambridge, he had positions at IHES, Princeton University, and the Institute for Advanced Studies. He came to Edinburgh in 1982. He had eleven PhD students. In MathSciNet, he has 23 coauthors.

The sharing of mathematics was important to Ranicki. Many years ago, he told me about a project for videos of lectures on topology. Ranicki and others were gathering existing videos, plus recording new videos to cover essential topics. They needed a convenient way to catalogue the URLs for the videos. Ranicki and his collaborators turned to Pinterest (which must have been just out of beta), creating the board Surgery Theory in Topology. I had never heard of Pinterest, so this was my only exposure to it and figured it was some variant on math.stackexchange. I was surprised when my sister-in-law told me that she was active on Pinterest for information on recipes. She was more appropriately surprised to learn that I had an account and that I thought it was for mathematics and science. But, it worked for Ranicki — at least for a while. His account went dormant after a few years.

Ranicki also shared what he knew through his writing, having written a half dozen books. A list of them is at the end of this post. He also edited or contributed to fifteen other books, collections, and conference proceedings.

Ranicki was a remarkable mathematician, but perhaps an even more remarkable person. Many people would comment on his infectious laugh. And he laughed often. On my various visits to the University of Edinburgh to work with Toby Bailey, Andrew would go out of his way to make me feel welcome. His home with his wife Ida Thompson was the site of many a gathering of mathematicians in Edinburgh.

Andrew’s father was Marcel Reich-Ranicki, a very well-known literary critic and essayist, who, among many other things, was a writer and editor for the *Frankfurter Allgemeine Zeitung*. Both of Andrew’s parents were survivors of the Holocaust. Their personal history was hugely important to Andrew. Through them, he knew directly of incredible horrors. Yet, Andrew was an incredibly positive person who loved life and loved people.

MR0620795

Ranicki, Andrew

Exact sequences in the algebraic theory of surgery.

Mathematical Notes, 26. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. xvii+864 pp. ISBN: 0-691-08276-6

MR1208729

Ranicki, Andrew(4-EDIN)

Lower K- and L-theory.

London Mathematical Society Lecture Note Series, 178. Cambridge University Press, Cambridge, 1992. vi+174 pp. ISBN: 0-521-43801-2

MR1211640

Ranicki, A. A.(4-EDIN)

Algebraic L-theory and topological manifolds.

Cambridge Tracts in Mathematics, 102. Cambridge University Press, Cambridge, 1992. viii+358 pp. ISBN: 0-521-42024-5

MR1410261

Hughes, Bruce(1-VDB); Ranicki, Andrew(4-EDIN)

Ends of complexes.

Cambridge Tracts in Mathematics, 123. Cambridge University Press, Cambridge, 1996. xxvi+353 pp. ISBN: 0-521-57625-3

MR1713074

Ranicki, Andrew(4-EDIN-MS)

High-dimensional knot theory.

Algebraic surgery in codimension 2. With an appendix by Elmar Winkelnkemper. Springer Monographs in Mathematics. Springer-Verlag, New York, 1998. xxxvi+646 pp. ISBN: 3-540-63389-8

MR2061749

Ranicki, Andrew(4-EDIN-MS)

Algebraic and geometric surgery.

Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2002. xii+373 pp. ISBN: 0-19-850924-3

He was also an author on

MR1434100

Ranicki, A. A.(4-EDIN-MS); Casson, A. J.(1-CA); Sullivan, D. P.(1-CUNY); Armstrong, M. A.(4-DRHM); Rourke, C. P.(4-WARW-MI); Cooke, G. E.

The Hauptvermutung book.

A collection of papers of the topology of manifolds. K-Monographs in Mathematics, 1. Kluwer Academic Publishers, Dordrecht, 1996. vi+190 pp. ISBN: 0-7923-4174-0

(*****) For the joint review of the two Proc. London Math. Soc. papers, I recommend the PDF version, as the commutative diagram in the review does not render in MathJax.

Meza did his PhD (find his thesis here) in the Computational and Applied Mathematics Department at Rice University with Bill Symes. (CAAM is in Duncan Hall, which is an exquisite building.) Symes did his PhD at Harvard with Phillip Griffith in algebraic geometry, though the vast majority of his publications are in PDEs and inverse problems.

When using MathSciNet, clicking on the “**Author**” tab switches you from doing a search for published items to doing a search for authors. Author searches are very important, but getting the right name can be tricky. (See this earlier post on the subject of authors and names.) When you start typing a name in the Author search, after a couple of letters have been entered, MathSciNet begins offering suggestions. The suggestions are coming from instant queries of the Mathematical Reviews author database. For example, this is what what I now see when typing “Serre” into an Author search:

When you have a list of suggestions, you can either hit “Enter” to ignore the suggestion and search using what you have typed so far or click on one of them to execute an Author search using those terms. It is important to note that the result of choosing a selection is launching a search. Many times, you will have a unique match, but other times there will be multiple matches. For instance, selecting “Serre F” from the example above matches two authors: “Serre, F.” and “Serre, François“.

Help with Journal searches is particularly useful since many journal titles are similar. We have roughly 1,800 active journals in MathSciNet, plus many historical journals (which might just be older versions of current journals). Lots of journals have the word “Journal” in the title. “Mathematical” is also popular. Skimming this list can give you an idea of similarities of titles.

The auto-suggest feature for Journal searches works similarly to that for Author searches. Click on the “**Journal**” tab to switch to doing a search for journals. When typing in a search word or phrase, after you type a couple of letters, MathSciNet starts offering suggestions. The suggestions are coming from instant queries of the Mathematical Reviews journal database. For example, here is what I see after typing “functional” into a Journal search:

Once again, when you have a list of suggestions, you can either hit “Enter” to ignore the suggestion and use what you have typed so far or click on one of the suggestions to execute a Journal search using those terms. Again, the result of choosing a selection is launching a search. Many times, you will have a unique match, but other times, there will be multiple matches. For instance, clicking on “Journal of Functional Analysis” in the example produces six results. Some are older incarnations of the current “Journal of Functional Analysis“, but you also get Journal of Nonlinear Functional Analysis and Differential Equations.

We hope these two new features help you to use MathSciNet more effectively. We have been beta testing them in-house at Mathematical Reviews for a while now and have found both to be tremendously helpful.

And, if you are going to the Joint Mathematical Meetings in San Diego, I hope you will come see Mathematical Reviews and MathSciNet at the AMS booth in the exhibit hall. My previous post has information about our activities there.

]]>**Complimentary MathSciNet at the meeting**. By special arrangement, MathSciNet will be available when using the wifi at the conference site. This year, access to MathSciNet should also be available in the hotels!

**Demos of MathSciNet**. Wednesday, Thursday, and Friday of the meeting, at 2:15pm at the Mathematical Reviews area of the AMS booth, there will be demonstrations of how to use MathSciNet. The demos will be by experts, who know lots of great ways to use MathSciNet. Fill out the questionnaire and have a chance to win a $100 gift card!

**Update your Author Profile Page. **Representatives will be available at the booth to help you update your author profile page on MathSciNet, including the opportunity to add a photograph or your name in its native script. If you are an AMS member and have signed up for the professional photograph service at the JMM, we can use that picture. We will also be set up to take a picture, in case you don’t have a favorite photo available.

**Working at Mathematical Reviews. **If you are interested in possibly working at Mathematical Reviews as an Associate Editor, the JMM would provide a good opportunity to find out more about what we do. We are looking for someone with expertise in algebra and an interest in a range of algebraic topics. My earlier post explains the open position. Stop by the booth to talk with us!

**Mathematical Reviews Reception**: Friday, 6:00 pm– 7:00 pm. All friends of the Mathematical Reviews (MathSciNet) are invited to join reviewers and MR editors and staff (past and present) for a special reception in honor of all of the efforts that go into the creation and publication of the Mathematical Reviews database. Refreshments will be served. The location is the Presidio Room in the Marriott.

The move to the cloud means a new address for MathSciNet: https://mathscinet.ams.org/mathscinet. The old address redirects you to this URL, so there is no immediate need to change links or bookmarks — but updating links is not a bad idea. This change did affect some proxy settings for off-site users, however. We have contacted librarians and have provided instructions on how to reset the proxy settings. If you are having a problem, write to msn-support@ams.org.

The move to the cloud also means that we will be retiring the five mirror servers at Rice University (Houston), IMPA (Rio de Janeiro), IRMA (Strasbourg), MPI (Bonn), and the University of Bielefeld (Bielefeld). For twenty years, these institutions provided better global access to the Mathematical Reviews database by hosting servers around the world. Over time, though, the technology of the web has evolved to a point where distributed access is now best provided by hosting sites on cloud servers. We are truly grateful to the five hosts. Over the next six months, as the mirror sites retire, their URLs will begin redirecting to the cloud servers.

This is the latest of the enhancements rolling out for MathSciNet. Sometimes they are quite visible, like the faceted searches released in January. Sometimes they are infrastructure, like this move to cloud servers. The internal changes are often best appreciated by not being noticed at all, except that your searches are a little faster and MathSciNet is always available.

]]>The 2017 Nobel Prize in Physics was awarded to Rainer Weiss, Barry C. Barish, and Kip S. Thorne for their work on the detection of gravitational waves. (See Note 1.) The physics and engineering that go into this accomplishment are truly impressive. However, before anyone could imagine setting up the experiment, some mathematical questions needed to be answered. There are two articles in the August 2017 issue of the *AMS Notices* that give an overview the mathematics of gravitational waves. In this post, I crib from those two articles and provide a literature tour of some of the significant papers by relying on MathSciNet. A longer article by Bieri just published in the AMS Bulletin goes into more detail on a selection of the topics.

General Relativity (GR) depends heavily on mathematics, in particular, differential geometry and PDEs, since Einstein’s equations are nonlinear PDEs involving the curvature and the metric itself. The equations hold an inherent beauty that manifests itself not just in the mathematical description of space-time, but also in a rich realm of geometry: Einstein manifolds. This is digression from the theory of gravitational waves, but it is, in the language of Michelin guides, “Worth a trip.” As a tour guide for this trip, I heartily recommend the book *Einstein manifolds* written by a group of geometers under the pseudonym Arthur L. Besse.

There are many places for a mathematician to learn general relativity (GR). For a first read, Einstein’s own account in *The Meaning of Relativity* is extremely good. However it is deceptive: it seems like easy reading, but he is describing very deep ideas that are overturning our view of the world. You need to pay attention to understand what is going on. I didn’t pay attention the first time through and had to read it a second time. Here are three books that are suited to mathematicians for learning about general relativity.

- Hawking and Ellis,
*The large scale structure of space-time*. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. xi+391 pp. MR0424186 - Barrett O’Neill,
*Semi-Riemannian Geometry*, With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp. MR0719023 - Misner, Thorne, and Wheeler,
*Gravitation*. W. H. Freeman and Co., San Francisco, Calif., 1973. ii+xxvi+1279+iipp. MR0418833.

The book by Hawking and Ellis assembles the tools you need to understand general relativity, then applies them to discuss black holes, in particular the Hawking and Penrose singularity theorems. O’Neill’s book is written by a mathematician for mathematicians. The bulk of it is about differential geometry with a semi-Riemannian metric. You could give a complete course on differential geometry from O’Neill’s book and not cover relativity. But why would you leave it out? He sets everything up for you, then puts it to use. This is probably not the introduction that a physicist would want, but since it begins on familiar territory, it suits many mathematicians. Misner, Thorne, and Wheeler is often referred to as “the Bible” for studying general relativity. It is cited so often, that people often refer to it as MTW. It is a monster-sized book, but it is really two books in one: a one-semester introduction and a deeper look at the subject. MTW is the only one of these three books that really treats gravitational waves.

These three books provide a general introduction to GR. The two *Notices* papers are describing results that depend on the modern approach of geometric analysis in general relativity. For that, a good place to start would be the book:

MR2391586

Christodoulou, Demetrios(CH-ETHZ)

Mathematical problems of general relativity. I.

Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+147 pp. ISBN: 978-3-03719-005-0.

I have not read it – I am relying on the review in MathSciNet! A short history of the mathematical study of Einstein’s equations can be found in the quite readable and very enjoyable article by Choquet-Bruhat:

MR3467361

Choquet-Bruhat, Yvonne(F-IHES)

Beginnings of the Cauchy problem for Einstein’s field equations.

Surveys in differential geometry 2015. One hundred years of general relativity, 1–16,

Surv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.

The articles in the AMS Notices are:

MR3676432

Hill, C. Denson(1-SUNYS); Nurowski, Paweł(PL-PAN-CTP)

How the green light was given for gravitational wave search.

*Notices Amer. Math. Soc.* 64 (2017), no. 7, 686–692.

and

MR3676433

Bieri, Lydia(1-MI); Garfinkle, David(1-OAKL-P); Yunes, Nicolás(1-MTS-P)

Gravitational waves and their mathematics.

*Notices Amer. Math. Soc.* 64 (2017), no. 7, 693–707.

As the *Notices * articles point out, there are two big questions that had to be answered before we could have the spectacular results from LIGO. The paper by Hill and Nurowski addresses the first question: **1. Do gravitational waves exist?** The paper by Bieri, Garfinkle, and Yunes answers the second question: **2. How might one detect gravitational waves?**

As mentioned above, Einstein’s equations are nonlinear PDEs:

(1) $R_{i,j} – \frac{1}{2}R g_{i,j} = \kappa T_{i,j}$

where $g$ is the metric, $R_{i,j}$ is the Ricci curvature (which is a bunch of second derivatives of the metric), $R$ is the scalar curvature, and $T_{i,j}$ is the energy-momentum tensor that describes the matter or energy present in the space-time. There is a linearized form, which was known already to Einstein. First, think of the metric as a perturbation of the metric $\eta_{i,j}$ on Minkowski space:

$g_{i,j} = \eta_{i,j} + \epsilon h_{i,j}$.

To linearize, develop the left-hand side of (1) in powers of $\epsilon$, neglecting terms of order $2$ and above. Away from sources, the energy-momentum tensor is zero. So in these regions, the Einstein equations linearize to a system of decoupled relativistic wave equations in the unknowns $h_{i,j}$. Thus, at least in linearized GR, there are plane waves traveling at the speed of light. Because these are linear equations, superposition applies and you can add these plane waves to make any wave you want . These are the *gravitational waves* first described by Einstein. Far from sources, the linearized theory should coincide closely with the nonlinear theory. So one hopes to be able to detect gravitational waves by looking for phenomena that behave like the waves in the linearized model. But first one has to answer Question 1.

The question is whether or not waves exist *in the full theory*, not just the linearized version. Hill and Nurowski reduce this question to seven sub-questions. (Well that’s simpler, isn’t it?)

- What is the definition of a
*plane*gravitational wave in the full theory? - Does the so-defined plane wave exist as a solution to the full Einstein system?
- Do such waves carry energy?
- What is a definition of a gravitational wave with
*nonplanar front*in the full theory? - What is the energy of such waves?
- Do there exist solutions to the full Einstein system satisfying this definition?
- Does the full theory admit solutions corresponding to the gravitational waves emitted by bounded sources?

Sub-questions 1 and 4 look surprising at first – the answers ought to be “Of course!” But the mathematics of general relativity can be subtle, and simple things like bad choices of coordinates can mask important phenomena, as Einstein and Rosen discovered:

MR3363463

Einstein, A.; Rosen, N.

On gravitational waves.

J. Franklin Inst. 223 (1937), no. 1, 43–54.

(Well, actually, they made the bad choice and Howard Robertson discovered the badness of it when he refereed the paper.) Ignoring Robertson’s observation led Einstein and Rosen to conclude that plane waves don’t exist in *physical* space-times, since their coordinate choice indicated that the space-time had singularities.

The positive answer to sub-questions 1, 2, and 3 comes in a paper by Bondi in *Nature* and the paper:

MR0106747

Bondi, H.; Pirani, F. A. E.; Robinson, I.

Gravitational waves in general relativity. III. Exact plane waves.

Proc. Roy. Soc. London Ser. A 251 1959 519–533.

See also the paper:

MR0096537

Pirani, F. A. E.

Invariant formulation of gravitational radiation theory.

Phys. Rev. (2) 105 1957 1089–1099.

Hill and Nurowski point out that there was a parallel discovery much earlier by a mathematician that included the Bondi-Pirani-Robinson waves as a special case, but the gravitational theorists overlooked it:

MR1512246

Brinkmann, H. W.;

Einstein spaces which are mapped conformally on each other.

Math. Ann. 94 (1925), no. 1, 119–145.

Oops.

Now you have a definition of a plane wave, but you still have a problem because full GR is nonlinear, so superposition doesn’t work — you cannot build arbitrary waves out of plane waves. However, Pirani gives a geometric definition of nonplanar waves by considering the principal null directions, which are the eigendirections of the Weyl tensor (the traceless part of the Riemann tensor). Specifically, waves occur for space-times that are (asymptotically) *algebraically special:*

MR0096537

Pirani, F. A. E.

Invariant formulation of gravitational radiation theory.

Phys. Rev. (2) 105 1957 1089–1099.

Positive answers to sub-questions 4 and 5 come from Andrzej Trautman in his two papers:

MR0097265

Trautman, A.

Boundary conditions at infinity for physical theories.

Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 1958 403–406.

and

MR0097266

Trautman, A.

Radiation and boundary conditions in the theory of gravitation.

Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 1958 407–412.

The basic idea is to say that a radiative spacetime should satisfy certain boundary conditions at infinity, in analogy with radiative fields in Maxwell’s theory of electromagnetism. (At some point, I probably should have introduced null infinity, but this post is already long enough.)

That leaves us with questions 6 and 7. These were answered by Robinson and Trautman by actually producing some solutions with the desired property.

MR0135928

Robinson, I.; Trautman, A.

Some spherical gravitational waves in general relativity.

Proc. Roy. Soc. Ser. A 265 1961/1962 463–473.

**Review**: The authors obtain a class of metrics of which some represent, in their own words, “a very simple kind of spherical radiation”. They begin by establishing

$ds^2=-\rho^2p^{-2}\{(d\xi-ad\sigma)^2+(d\eta-bd\sigma)^2\}+2d\rho\,d\sigma+c\,d\sigma^2$

($a$, $b$, $c$, $p$ functions of the coordinates $\xi$, $\eta$, $\rho$, $\sigma$, but with $\partial p/\partial\rho=0$) as a canonical form for a metric which admits a shear-free, diverging and hypersurface-normal null vector field $\sigma_i$ that satisfies $R_{ik}\sigma^i\sigma^k=0$. They then solve the remaining field-equations for empty space and show that the solutions define two families of $V_2$ and admit a number of local and integral invariants. Algebraic properties of the curvature tensor and its rate of change along propagation-rays are discussed, and several explicit solutions of the Einstein and Maxwell-Einstein equations are obtained. These include the Schwarzschild metric as well as some static degenerate solutions of Levi-Civita.

Reviewed by H. S. Ruse

So, after some hiccups and a lot of hard work, we know what gravitational waves are and even have some information about their existence.

Gravity appears to us as a fairly strong force. It sticks us to the Earth. The sun has a massive gravitational field that bends light (one thing that makes eclipses interesting to physicists). In the event detected by LIGO, about 3 times the mass of the sun was converted into gravitational waves in a fraction of a second. So if a gravitational wave is going to come our way, especially one caused by the collision of two black holes, you might expect it to arrive like the big waves on the North Shore of Oahu. But, alas, we are observing the waves far from the source, in an area that we are assuming is essentially flat. So, in order to observe gravitational waves, LIGO needed to detect a displacement that was on the scale of 1/1000 the charge diameter of a proton over the course of a 4 km baseline, or a change of about one part in $10^{21}$. In order to do this, the experimenters need to have a very precise picture of what to look for.

A good way to find solutions to the Einstein equations is to solve a Cauchy problem. The Einstein equations split into a set of evolution equations and a set of constraint equations. You want to solve this system by specifying initial data. Typically, you either prescribe data on a space-like hypersurface (the classical case) or on a null hypersurface (the characteristic case). In the classical case, the hypersurface is known as a Cauchy surface, which, at least locally, you can picture as a slice corresponding to a fixed time that will then evolve under the equation. For gravitational waves, though, the characteristic case is more suitable, as explained in the *Bulletin *paper by Bieri. In any case, a priori, there is no guarantee that if your initial data satisfy the constraints that the solution of the evolution equation also satisfies the constraints. But in the case of the Einstein equations, Choquet-Bruhat showed exactly that. This is great! Her local solution is in the paper

MR0053338

Fourès-Bruhat, Y.

Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. (French)

Acta Math. 88, (1952). 141–225.

The global solution is in her paper with Geroch:

MR0250640

Choquet-Bruhat, Yvonne; Geroch, Robert

Global aspects of the Cauchy problem in general relativity.

Comm. Math. Phys. 14 1969 329–335.

Once we know that solutions exist, we need to know more about them because we need to know what we are looking for. You can read about the long-term existence of solutions in the book by Christodoulou and Klainerman

MR1316662

Christodoulou, Demetrios(1-PRIN); Klainerman, Sergiu(1-PRIN)

The global nonlinear stability of the Minkowski space.

Princeton Mathematical Series, 41. Princeton University Press, Princeton, NJ, 1993. x+514 pp. ISBN: 0-691-08777-6

where they also investigate the asymptotic structure of these spacetimes. As stated in the review of the book, “It is shown that the laws of gravitational radiation discovered by Bondi and others more than thirty years ago using formal power series expansions are rigorously true in this class of spacetimes.”

So, now we are getting closer to the question of how you might detect gravitational waves. There is a good mathematical description of the waves. What is needed next is a description of what might actually be observable. The behavior of neighboring geodesics in any geometric setting is governed by the Jacobi equation. In typical differential geometry courses, the Jacobi equation is used to identify conjugate points. Here, though, the equation allows for a computation of the (small) displacement of test masses when a gravitational wave passes. Since gravitational waves move at the speed of light, it is helpful to know that there is a *memory effect* that lingers and can be computed. See, for instance,

MR1144215

Thorne, Kip S.(1-CAIT-TA)

Gravitational-wave bursts with memory: the Christodoulou effect.

Phys. Rev. D (3) 45 (1992), no. 2, 520–524

as well as more recent work as found in

MR3467364

Bieri, Lydia(1-MI); Garfinkle, David(1-OAKL-P); Yau, Shing-Tung(1-HRV)

Gravitational waves and their memory in general relativity. (English summary) Surveys in differential geometry 2015. One hundred years of general relativity, 75–97,

Surv. Differ. Geom., 20, Int. Press, Boston, MA, 2015.

What we need, though, is to be able to observe it and to measure it in the real world, thus confirming that it was caused by a gravitational wave. This uses the technique of *matched filtering*, meaning you have a collection of templates (derived from a solution) and you match the collected data to these templates.

Once again, the nonlinearity of the Einstein equations means that we don’t have many closed-form solutions to use for templates. Rather, we need simulations. The Choquet-Bruhat result tells us that for actual solutions, initial data that satisfy the constraints will evolve and still satisfy the constraints. But in a simulation — an approximation, even a tiny deviation can cause a violation of the constraint and invalidate the simulation. Fortunately, mathematical physicists are good at numerical solutions and simulations and came up with a suite of three techniques that overcome this: hyperbolicity, constraint damping, and excision. Here are some papers that are representative of each:

**Hyperbolicity**

MR2254287

Pretorius, Frans(3-AB-P)

Simulation of binary black hole spacetimes with a harmonic evolution scheme.

Classical Quantum Gravity 23 (2006), no. 16, S529–S552.

**Constraint damping**

MR3079062

Gundlach, Carsten(4-SHMP-SM); Martín-García, José M.(F-PARIS6-IAP); Garfinkle, David(1-OAKL-P)

Summation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions.

Classical Quantum Gravity 30 (2013), no. 14, 145003, 31 pp.

**Excision**

MR2079939

Thornburg, Jonathan(D-MPIGP)

Black-hole excision with multiple grid patches.

Classical Quantum Gravity 21 (2004), no. 15, 3665–3691.

Finally, all the mathematical ingredients are in place: a detailed picture of gravitational waves in the full, nonlinear theory and a means of computing samples of gravitational waves to the level of accuracy needed for matched filtering. Performing the experiment requires turning the mathematics into machinery, which is in itself an exquisite feat of science and engineering.

Feynman used to boast that QED (quantum electrodynamics) was the physical theory that was able to make the most accurate predictions and that had been tested to the greatest level of precision, phrasing it as like measuring the distance from New York to LA to within the width of a human hair (one part in $10^{12}$). With the LIGO experiments, the general relativists have verified a theory using an instrument that is sensitive on the order of one part in $10^{21}$. (See Note 4).

**Acknowledgment**: I am grateful to Lydia Bieri for some extra comments on the work involved, in particular for helping me to understand Choquet-Bruhat’s work better.

**Notes**:

(1) The paper announcing the LIGO results is

MR3707758

Abbott, B. P.(1-CAIT-LIG); et al.;

Observation of gravitational waves from a binary black hole merger.

Authors include B. C. Barish, K. S. Thorne and R. Weiss.

Phys. Rev. Lett. 116 (2016), no. 6, 061102, 16 pp.

(2) Before Choquet-Bruhat, various people made some headway on the existence problem for Cauchy data for the Einstein equations. One such person was Cornelius Lanczos, who had a remarkable life. His mixture of bad luck and good luck led to him making contributions to several different areas of mathematics, including general relativity, but also numerical analysis and linear algebra. The AMS published a nice biography of Lanczos by Barbara Gellai where you can read the details.

(3) The experiments to detect gravitational waves are impressive and *expensive*. CERN’s search for the Higgs boson was even more expensive. These are two major accomplishments in experimental physics that don’t get off the ground without some good mathematics to indicate that there was something interesting out there and giving a clue as to how to look for it. But there are some mathematically inspired physics experiments that are much less expensive. For instance, you can build a magnetic monopole detector for about the cost of a cup of coffee at Starbucks. It is described in

MR0856880

Clifford Henry Taubes

Physical and Mathematical Applications of Gauge Theories,

*Notices Amer. Math. Soc.*, 33 (1986), no. 5, 707–715.

All you need is a battery, a lightbulb, and some wire. Oh, and the patience to sit unblinking for a very long time. The paper provides a very accessible introduction to gauge theory suitable for a general mathematical audience. Taubes also has a little fun with the writing. The article has passages like:

*These are the plans, good luck in the chase,
*

Earlier, discussing decay of quarks, Taubes writes:

*Please, don’t blink, as you might miss the prize*

*of seeing the lash of this year’s lone quark’s predicted demise.*

*After five years of patience, statistics are small;*

*no announcement’s been made of matter’s downfall.*

Other passages are in blank verse. That era of the AMS *Notices* is not (yet) available online. If you want to read the article, I am afraid you will have to go to the library. It is at least worth a detour as you walk from Starbucks to your office.

(4) I’ve skipped over something important in comparing the accuracy of QED to the sensitivity of the LIGO experiments. I refer you to a discussion on physics.stackexhange.com to address that.

]]>Emmanuel Candès has won a prestigious MacArthur Fellowship. The official announcement is here. The LA Times has a nice write-up. Both the Los Angeles Times and the MacArthur announcement highlight Candès’s work on compressed sensing. Terry Tao has a spot-on reaction to this work, quoted in the LA Times, typical of most mathematicians when you first hear about the method: *you can’t be getting be getting something for nothing. This can’t work.* But it does! Tao finally came around to believe it, as has the rest of the world.

Candès is a collaborative researcher. The work on compressed sensing came out of conversations he was having initially with researchers in imaging science (MRIs). He tested out his ideas in conversations with Tao (while picking up their children at daycare, according to the legend). In MathSciNet, you can see that Candès has 48 coauthors, representing a wide variety of people: statisticians, analysts, younger collaborators, older collaborators.

One of his key publications on compressed sensing is his paper with Justin Romberg and Terry Tao in *Communications on Pure and Applied Mathematics,* Stable signal recovery from incomplete and inaccurate measurements. For your reading enjoyment, our review of it is reproduced below. The complete article is here.

Congratulations, Emmanuel Candès!

**MR2230846**

Candès, Emmanuel J.(1-CAIT-ACM); Romberg, Justin K.(1-CAIT-ACM); Tao, Terence(1-UCLA)

Stable signal recovery from incomplete and inaccurate measurements.

*Comm. Pure Appl. Math.* 59 (2006), no. 8, 1207–1223.

94A12

The authors consider the problem of recovering an unknown sparse signal $x_0(t)inBbb{R}^m$ from $nll m$ linear measurements which are corrupted by noise, building on related results in [E. J. Candès and J. K. Romberg, Found. Comput. Math. 6 (2006), no. 2, 227–254; MR2228740; E. J. Candès, J. K. Romberg and T. C. Tao, IEEE Trans. Inform. Theory 52 (2006), no. 2, 489–509; MR2236170; E. J. Candès and T. C. Tao, IEEE Trans. Inform. Theory 51 (2005), no. 12, 4203–4215; MR2243152; “Near optimal signal recovery from random projections: universal encoding strategies?”, preprint, arxiv.org/abs/math.CA/0410542, IEEE Trans. Inform. Theory, submitted; D. L. Donoho, Comm. Pure Appl. Math. 59 (2006), no. 6, 797–829; MR2217606]. Here $x_0(t)$is said to be sparse if its support $T_{0}=lbrace tcolon x_0(t)neq0rbrace$ has small cardinality. The measurements are assumed to be of the form $y=Ax_0+ e$, where $A$ is the $ntimes m$ measurement matrix and the error term $e$ satisfies $Vert eVert_{l_{2}}leepsilon$. Given this setup, consider the convex program $$ minVert xVert_{l_{1}} {rm subject to};Vert Ax-y Vert_{l_{2}}leepsilon.;;;(textrm {P}_{2}) $$

The authors define $A_{T}$, $Tsubsetlbrace 1,ldots,mrbrace$, to be the $ntimesvert Tvert$ submatrix obtained by extracting the columns of $A$ corresponding to the indices in $T$. Their results are stated in terms of the $S$-restricted isometry constant $delta_S$ of $A$ [E. J. Candès and T. C. Tao, op. cit., 2005], which is the smallest quantity such that $$ (1-delta_{S})Vert cVert_{l_{2}}^{2}leVert A_{T}c Vert_{l_{2}}^{2}le(1+delta_{S})Vert cVert_{l_{2}}^2 $$ for all subsets $T$ with $vert Tvertle S$ and coefficient sequences $(c_{j})_{jin T}$.

Their main results are below. The first describes stable recovery of sparse signals, while the second yields stable recovery for approximately sparse signals by focusing on the $S$ largest components of the signal $x_0$.

Theorem 1. Let $S$ be such that $delta_{3S}+3delta_{4S}<2$. Then for any signal $x_{0}$ supported on $T_0$ with $vert T_0vertle S$ and any perturbation $e$ with $Vert eVert_{l_{2}}leepsilon$, the solution $x^{sharp}$ to $({rm P}_2)$ obeys $$ vert x^{sharp}-x_0vert_{l_{2}}le C_{S}cdotepsilon, $$ where the constant $C_S$ depends only on $delta_{4S}$. For reasonable values of $delta_{4S}$, $C_S$ is well behaved; for example, $C_Sapprox8.82$ for $delta_{4S}=frac{1}{5}$ and $C_{S}approx 10.47$ for $delta_{4S}=frac{1}{4}$.

Theorem 2. Suppose that $x_0$ is an arbitrary vector in $Bbb{R}^m$, and let $x_{0,S}$ be the truncated vector corresponding to the $S$ largest values of $x_0$ (in absolute value). Under the hypotheses of Theorem 1 in the paper, the solution $x^{sharp}$ to $({rm P}_{2})$ obeys $$ Vert x^{sharp}-x_0Vert_{l_{2}}le C_{1,S}cdotepsilon+ C_{2,S}cdotfrac{Vert x_0-x_{0,S}Vert_{l_{1}}}{sqrt{S}}. $$

For reasonable values of $delta_{4S}$, the constants in the equation labelled 1.4 in the paper are well behaved; for example, $C_{1,S}approx 12.04$ and $C_{2,S}approx8.77$ for $delta_{4S}=frac{1}{5}$.

Reviewed by Brody Dylan Johnson

]]>