The answer is that the first counts publications that you authored, such as a book or an article. The second counts publications that you had a part in, but didn’t author. Examples are books where you were the translator, conference proceedings where you were an editor, or a biography where you were the subject. Loosely speaking, a related publication is one in which your name is part of the bibliographic information, but you were not the author.

**Example**: Aaron Bertram

Bertram has 37 “regular” publications. Let’s look at the first few by clicking on “Publications” in the box just underneath the photo and the counts:

The top seven results are in and we have:

**MR1454400 ** Bertram, Aaron Quantum Schubert calculus. *Adv. Math.* 128 (1997), no. 2,289–305. (Reviewer: Sara C. Billey) 14M15 (14N10)

**MR1092845** Bertram, Aaron; Ein, Lawrence; Lazarsfeld, Robert Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. *J. Amer. Math. Soc.* 4 (1991), no. 3, 587–602. (Reviewer: Marco Andreatta) 14F17 (14J99 14N05)

**MR1320154** Bertram, Aaron; Daskalopoulos, Georgios; Wentworth, Richard Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians. *J. Amer. Math. Soc.* 9(1996), no. 2, 529–571. (Reviewer: Jun Li) 14N10 (14D99 14M15 58D10)

**MR2998828** Arcara, Daniele; Bertram, Aaron Bridgeland-stable moduli spaces for K-trivial surfaces. With an appendix by Max Lieblich. *J. Eur. Math. Soc. (JEMS)* 15 (2013), no. 1, 1–38. (Reviewer: Pawel Sosna) 14F05 (14D20)

**MR1706853** Bertram, Aaron; Ciocan-Fontanine, Ionuţ; Fulton, William Quantum multiplication of Schur polynomials. *J. Algebra* 219 (1999), no. 2, 728–746. (Reviewer: Laurent Manivel) 14N35 (05E05 14N15)

**MR3010070** Arcara, Daniele; Bertram, Aaron; Coskun, Izzet; Huizenga, Jack The minimal model program for the Hilbert scheme of points on ℙ2 and Bridgeland stability. *Adv. Math.* 235 (2013), 580–626. (Reviewer: Yifei Chen) 14E30 (14C05 14D23)

**MR1158344** Bertram, Aaron Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space. *J. Differential Geom.* 35 (1992), no. 2, 429–469. (Reviewer: Arnaud Beauville) 14H60 (14D20)

Note that Aaron Bertram is an author on each of these.

Now let’s try the other list, Related Publications:

The results are:

**MR2483944** Algebraic geometry—Seattle 2005. Part 2. Papers from the AMS Summer Research Institute held at the University of Washington, Seattle, WA, July 25–August 12, 2005. Edited by D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande and M. Thaddeus. Proceedings of Symposia in Pure Mathematics, 80, Part 2. *American Mathematical Society, Providence, RI,* 2009. pp. i–xiv and 489–1004. ISBN: 978-0-8218-4703-9 14-06

**MR2483929** Algebraic geometry—Seattle 2005. Part 1. Papers from the AMS Summer Research Institute held at the University of Washington, Seattle, WA, July 25–August 12, 2005. Edited by D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande and M. Thaddeus. Proceedings of Symposia in Pure Mathematics, 80, Part 1. *American Mathematical Society, Providence, RI,* 2009. xiv+487 pp. ISBN: 978-0-8218-4702-2 14-06

**MR2222641** Snowbird lectures on string geometry. Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on String Geometry held in Snowbird, UT, June 5–11, 2004. Edited by Katrin Becker, Melanie Becker, Aaron Bertram, Paul S. Green and Benjamin McKay. Contemporary Mathematics, 401. *American Mathematical Society, Providence, RI,* 2006. xii+104 pp. ISBN: 0-8218-3663-3 81-06 (14-06)

**MR1942135** Symposium in Honor of C. H. Clemens. A weekend of algebraic geometry in celebration of Herb Clemens’s 60th birthday held at the University of Utah, Salt Lake City, UT, March 10–12, 2000. Edited by Aaron Bertram, James A. Carlson and Holger Kley. Contemporary Mathematics, 312. *American Mathematical Society, Providence, RI,* 2002. x+289 pp. ISBN: 0-8218-2152-0 14-06 (00B30)

In the bibliographic information, you can see that Bertram is an *editor *of each of these.

If you go to the listing for one of the items, you can see a new box labeled “Related”.

This shows all the people associated to the item in some way other than as an author. In this case, it is listing the editors of the volume. The next example shows another possibility.

If we look at Gauss’s profile, we see that he has 36 Publications and 239 Related Publications. Among Gauss’s related publications, a few sample items are:

**MR3576593** Rowe, David E. Looking back on Gauss and Gaussian legends: answers to the quiz from 37(4). *Math. Intelligencer* 38 (2016), no. 4, 39–45. 01A55 (01A70)

**MR0966232** Almkvist, Gert; Berndt, Bruce Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies diary. *Amer. Math. Monthly* 95 (1988), no. 7, 585–608. (Reviewer: R. A. Askey) 01A50 (01A55 01A60 33A25)

**MR2308277** Goldstein, Catherine; Schappacher, Norbert A book in search of a discipline (1801–1860). *The shaping of arithmetic after C. F. Gauss’s Disquisitiones arithmeticae, *3–65, *Springer, Berlin,* 2007. 01A55 (11-03 11A15 11L03)

**MR0392346** Schimank, Hans Carl Friedrich Gauß. (German) *Gauß-Gesellschaft Göttingen, Mitteilungen No. 8, *pp. 6–31. (14 plates) *Gauß-Gesellschaft, Göttingen,* 1971. 01A55

**MR0532650** Wussing, Hans Carl Friedrich Gauss. (German) Second edition. Biographien Hervorragender Naturwissenschaftler, Techniker und Mediziner, 15. *BSB B.G. Teubner Verlagsgesellschaft, Leipzig,* 1976. 100 pp. (1 plate). 01A70

**MR1138221** Schneider, Ivo Gauss’ contribution to probability theory. *Proceedings of the International Symposium on Mathematics and Theoretical Physics (Guarujá, 1989), *72–85, Sympos. Gaussiana Ser. A Math. Theoret. Phys., 1, *Inst. Gaussianum, Toronto, ON,* 1990. 01A55 (60-03)

Notice that Gauss neither wrote nor edited any of these. Rather, each is about him and his work.

I hope this post helps explain the difference between the two types of publications in an author profile.

We have made arrangements for complimentary access to MathSciNet at the ICM venue and at surrounding hotels. No special password is required. You will be able to pair a device with the ICM access, allowing you to have up to thirty days of access to MathSciNet after leaving the Congress. The instructions for pairing your device are the same as in the previous blog post on using MathSciNet on the road. You can come to the AMS booth in the exhibit area to find out more about MathSciNet, including how to personalize your author profile, what new features have been released (or are planned), and how to become a reviewer.

Once again, zbMATH and Mathematical Reviews are hosting a joint reception in honor of the many mathematicians who review. The place and time are:

Rio de Janeiro I & II

Hotel Grand Mercure Rio de Janeiro Riocentro

Av. Salvador Allende, 6555

Barra da Tijuca, Rio de Janeiro

August 4, 2018, from 15:00h to 17:00h.

There will be snacks and refreshments. We hope to see you there!

]]>MR1115118

Kashiwara, M.(J-KYOT-R)

On crystal bases of the Q-analogue of universal enveloping algebras.

Duke Math. J. 63 (1991), no. 2, 465–516,

which provides a canonical base for representations of the quantized universal enveloping algebra $U_q(\scr G)$ associated with a Kac-Moody Lie algebra. In most (maybe all) cases, these bases are the same as Lusztig’s canonical basis (see MR1035415). The algebras have their origins in exactly solvable models in statistical mechanics, and are important in representation theory for Lie groups and Lie algebras. Also in representation theory, Kashiwara and Jean-Luc Brylinski provided a solution to the Kazhdan–Lusztig conjecture in their paper:

MR0632980

Brylinski, J.-L.; Kashiwara, M.

Kazhdan-Lusztig conjecture and holonomic systems.

Invent. Math. 64 (1981), no. 3, 387–410.

Beilinson and Bernstein simultaneously proved the conjecture using similar methods, but with a slightly different take. See MR0610137.

Kashiwara has been good about writing books and long survey articles explaining $\scr D$-modules, microlocal analysis, and related subjects. These are difficult subjects, involving ideas and techniques from several areas of mathematics. So books and surveys are very much appreciated. His little book

MR1943036

Kashiwara, Masaki

D-modules and microlocal calculus. (English summary)

Translated from the 2000 Japanese original by Mutsumi Saito. Translations of Mathematical Monographs, 217. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2003. xvi+254 pp. ISBN: 0-8218-2766-9

is a great introduction to the subject of ${\scr D}$-modules. His book with Schapira

MR1074006

Kashiwara, Masaki(J-KYOT-R); Schapira, Pierre(F-PARIS13)

Sheaves on manifolds.

With a chapter in French by Christian Houzel. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 292. Springer-Verlag, Berlin, 1990. x+512 pp. ISBN: 3-540-51861-4

is a standard reference for microlocal analysis and $\scr D$-modules. His long survey article from the Katata Conference

MR0420735

Sato, Mikio; Kawai, Takahiro; Kashiwara, Masaki

Microfunctions and pseudo-differential equations. Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), pp. 265–529. Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973.

provided the canonical reference on that subject for a long time.

Kashiwara is known for connecting deep ideas from algebraic geometry, homological algebra, and microlocal analysis. His work can be seen as an abstract approach to differential equations. If you look carefully at his papers, though, you see that his work is often grounded in very concrete examples. His famous paper [MR0485861] with Kowata, Minemura, Okamoto, Ōshima, and Tanaka solving the Helgason Conjecture writes out some very explicit calculations on the upper half-plane. Part III of his famous series of papers (Part I = MR0370665; Part II = MR0511186) on holonomic systems with regular singularities starts by considering a particular ODE. (It is not a simple ODE, but it is in one way very concrete.) I recall years ago asking him a question about holonomic systems and ${\scr D}$-modules. It is too long ago for me to remember the exact question, but I do remember how he answered. I had already talked to my former thesis advisor about the question, who suggested I talk to Kashiwara — and he just happened to be visiting at the time. Kashiwara thought a little, then said, “Let’s write down an ODE.” Momentarily, I was quite deflated. I thought my question was hard — and we’re writing down an ODE? But then Kashiwara modified the equation and demonstrated how it had picked up a property. Then he added some other complication. Then he put the equation on a Riemann surface, not the complex plane. Then it became harder again by some other tweak. Finally he pointed out that you could make a system of such equations, but in a way that they were really PDEs. And then he answered my question (whatever it was).

]]>Thanks are due, in particular, to Heath O’Connell from FermiLab who worked with our IT department to set up the matching.

Every listing in INSPIRE contains bibliographic information for the paper. For a published paper, there will be a link to that version through the DOI. If there is a version on the arXiv, there will be a link to that. When possible, INSPIRE will add the abstract, information about figures, some keywords, and other information. Below the main box are some links, such as to the cross-listing on the ADS website seen below and the export options, such as BibTeX, MARC, and others. When it exists, the link to MathSciNet is in the bottom left, as indicated in the screenshot below. Clicking on that link brings you to the listing of the paper in MathSciNet.

The following examples were found by exploring a spreadsheet with one third of all the matches in it. The only information was the DOI, the INSPIRE ID, and the MR Number for the item in MathSciNet. In other words, they are fairly random.

- Field of a particle in uniform motion and uniform acceleration

Nathan Rosen (Technion)

1962 – 7 pages

Annals Phys. 17 (1962) no.2, 269-275

http://inspirehep.net/record/1481114

https://mathscinet.ams.org/mathscinet-getitem?mr=134740 - A Technique for the Numerical Solution of Certain Integral Equations of the First Kind

David L. Phillips (Argonne (main))

1962 – 14 pages

J.Assoc.Comput.Machinery 9 (1962) no.1, 84-97

(1962-01)

http://inspirehep.net/record/1280488

https://mathscinet.ams.org/mathscinet-getitem?mr=134481 - Mass differences and Lie Algebras of Finite Order

L. O’Raifeartaigh

1965

Phys.Rev.Lett. 14 (1965) no.14, 575

(1965-04-05)

http://inspirehep.net/record/1474752

https://mathscinet.ams.org/mathscinet-getitem?mr=176775 - Curvatures of Left Invariant Metrics on Lie Groups

J. Milnor (Princeton, Inst. Advanced Study)

1976 – 37 pages

Adv.Math. 21 (1976) 293-329

http://inspirehep.net/record/114923

https://mathscinet.ams.org/mathscinet-getitem?mr=425012 - Non-existence of time-periodic vacuum space-times

Spyros Alexakis, Volker Schlue (Toronto U., Math. Dept.)

Apr 17, 2015 – 62 pages

J.Diff.Geom. 108 (2018) no.1, 1-62

(2018)

http://inspirehep.net/record/1361906

https://mathscinet.ams.org/mathscinet-getitem?mr=3743702 - Global anomalies on Lorentzian space-times

Alexander Schenkel (Nottingham U.), Jochen Zahn (Leipzig U.)

Sep 21, 2016 – 22 pages

Annales Henri Poincare 18 (2017) no.8, 2693-2714

(2017-05-19)

http://inspirehep.net/record/1487568

https://mathscinet.ams.org/mathscinet-getitem?mr=3671548 - Central charges, black hole entropy and geometrical structure of N extended supergravities in D = 4

Laura Andrianopoli (Genoa U. & INFN, Turin)

Sep 1997 – 6 pages

Fortsch.Phys. 47 (1999) 101-107

http://inspirehep.net/record/453364

https://mathscinet.ams.org/mathscinet-getitem?mr=1717312 - Seiberg-Witten theory, matrix model and AGT relation

Tohru Eguchi, Kazunobu Maruyoshi (Kyoto U., Yukawa Inst., Kyoto)

Jun 2010 – 20 pages

JHEP 1007 (2010) 081

http://inspirehep.net/record/857146

https://mathscinet.ams.org/mathscinet-getitem?mr=2719970 - Adiabatic dynamics of instantons on S4

Guido Franchetti, Bernd J. Schroers (Heriot-Watt U. & Maxwell Inst. Math. Sci., Edinburgh)

Aug 26, 2015 – 44 pages

Commun.Math.Phys. 353 (2017) no.1, 185-228

(2016-10-15)

http://inspirehep.net/record/1389884

https://mathscinet.ams.org/mathscinet-getitem?mr=3638313

Thank you again to the people at INSPIRE for helping with this project.

]]>For more details about Otto, see the post from two years ago: Happy Birthday Otto Neugebauer.

]]>

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.