Publisher's Description: "November 11, 2002: Grigori Perelman, a famous mathematician, brilliantly establishes his proof of the Poincaré Conjecture. A few years later, he is widely acclaimed for his research. However, he declines the prestigious Fields Medal and persists in not wanting to leave his native city of Saint Petersburg to attend the International Congress of Mathematicians in Madrid in 2006 where the medal is supposed to be awarded. John Ball, the President of the International Mathematical Union, decided to visit Russia in an attempt to convince Perelman to accept the Fields Medal.
   "This book contains the story, part real, part fictional, of the exchanges between Ball and Perelman. We are immersed in the tormented mind of a person who prefers the simple and secluded life to the prestige of his discoveries. We already know the final outcome of the story, Perelman's perpetual refusal to be glorified by the public, and yet there is still much to learn from this character of astonishing complexity.''
   Retrieved from publisher website on August 30, 2021.

Who is Perelman? What did he prove and how? Answers to these questions and the story of his refusal of the Fields Medal in 2006 are sketched in this paper. The main sources for the author were the paper [S. Nasar and D. Gruber, New Yorker 2006, August 28, 44–57] and the book [M. Gessen, Perfect rigor, Houghton Mifflin Harcourt, Boston, MA, 2009; MR2598223].

This article is a survey of G. Perelman's proof of the Poincaré Conjecture. The introductory section discusses early work of Henri Poincaré on topology which led him to ask the question of whether the 3-sphere is the only closed, simply connected 3-manifold, up to homeomorphism. This question later became known as the Poincaré Conjecture. The authors then quickly review the concepts of differential manifolds, Riemannian metrics, curvature, and Ricci flow. The rest of the text deals with R. Hamilton's work on Ricci flow, Perelman's Ricci flow with surgery, and their application to proving the Poincaré Conjecture, following Colding and Minicozzi rather than Perelman's original argument.

In 2002, Grigory Perelman announced a proof of Thurston's Geometrization Conjecture, from which followed the proof of the Poincaré Conjecture. It took the mathematical community about three years to understand Perelman's proof and to check all the details. Just after it became clear that the proof was complete, Perelman, who was forty years old, quit mathematics.
   The book under review recounts this story. It is also a book about Perelman's life, from his childhood through the time of his resignation. The author, Masha Gessen, obtained her information through interviews with a certain number of people who knew Perelman (teachers, classmates and colleagues in Russia and the US). The book also reports on the noise that accompanied the announcement of the proof and that lasted for the three years that followed this announcement, including jealousies, rivalries and controversies that were covered in an unusual way by the mass media. The author also comments on the commercial aspect that obviously offended Perelman, including offers by the most prestigious American universities after it became clear that he had proved the Poincaré Conjecture.
   More than relating the facts, the book contains interesting comments on Perelman's character, and an attempt by the author to explain Perelman's deception by the mathematical community and the reasons that led him to turn down the Fields Medal, the Millennium Prize, and to abandon mathematics altogether.
   The book is very interesting, reading it is pleasant, and a general feeling that comes out of it is that the author is sincere and did her best to collect the right information. Besides recounting the story of Perelman and his solution of the Poincaré Conjecture, the author gives a detailed portrait of the situation of mathematics teaching in the Soviet Union, where Perelman grew up. As a matter of fact, the period that is portrayed starts several decades before Perelman was born, namely, the 1920s, and it starts by recalling how Dimitri Egorov, one of the most prominent Russian mathematicians, was arrested and died in exile, his crime being that he was religious. We likewise learn interesting details concerning Luzin, Kolmogorov, P. Alexandrov, Pontryagin, A. D. Alexandrov, and several other first-class Russian mathematicians of the twentieth century. The author also describes the role played by mathematics as a counter-culture in the former Soviet Union, as opposed to the official propaganda, and she gives a very convincing explanation of why the founders of dissident movements were mathematicians and physicists.
   Although the author is not a professional mathematician, she knows well the milieu she describes, since she is also a product of the Soviet Union educational system, where she attended a special school for students gifted in mathematics, only a few years after Perelman. The book contains details on the subtleties of the ethnic quota system (in particular for Jews) in the school and university admission process, and the manner in which Russian mathematics "prodigies'' are grouped together from the age of ten in special clubs that prepare them, during six or more years, for mathematics competitions. The club members are under the supervision of a teacher who meets with them several times a week, training them in problem-solving, making sure that the best ones do not have any other passion than mathematics. This seems to be the reason why Perelman had to stop playing the violin. Perelman's mentor was also at ease with the fact that the boy was not interested in girls (p. 30). Mathematics club members also had to participate in summer camps devoted to mathematical problem-solving. (Similar treatment was reserved for child prodigies in physics and music, and in fact, the actual Russian system has been inherited from this, though probably to a lesser degree.) Being in such a club was also a prelude for entering a "specialized school'', like the famous Leningrad Specialized Mathematics and Physics School Number 239, in which Perelman was enrolled at the age of fourteen.
   The book is valuable and will hold an important place in the literature on mathematics. It can be read with interest by mathematicians and probably by non-mathematicians. I will now make a few points where I am not fully convinced by the author's point of view.
   The first is an important one, and it concerns one of the leitmotifs of the book, which is the presentation of Perelman as a "problem-solver'', and even as "the man who had never met a problem he could not solve'' (p. 146). We are told, for instance, that if Perelman took second place (and not first) at the All-Soviet Olympiad in 1980 in Saratov, this was because "he has not practised enough'', and that "from then on he practiced ceaselessly'' (p. 68). The author describes Perelman, from his childhood, as "a human math project'' (p. 172), and as someone who "wanted to solve the hardest problem he could find'' (p. 173). She ranks Perelman's way of working in a "category essentially similar to solving a mathematical Olympiad problem'' (p. 146). There are several other hints of the same sort. In the prologue (p. xi), the author declares that when she first began to write this book, she wanted to find answers to three questions, the first question being: "Why was Perelman able to solve the Conjecture?'' The author does not claim she answered this question (and she does not say the contrary). But the insistence on problem-solving gives the impression that this is the clue to the answer, and I think this is an exaggeration. Working on a problem like the Poincaré Conjecture does not have much to do with the fact of having been shaped as a problem-solver by mathematics club teachers or in summer mathematics camps, and solving the Conjecture is not a question of being a problem-solver. The impelling force for mathematical discovery is the desire to understand, and it requires taking time for meditation and discovering all the landscape that surrounds the problem. Competition can be a factor in this process, but a minor one. The devotion is for mathematics, and not for problem-solving. Perelman spent eight or nine years in isolation, while he worked on the proof, and he created a new theory. A "problem-solver'', in the sense described in the first chapters of the book, is not allowed to spend much time on a single problem. Problem-solving, in this sense, is related to exercising and finding the solution, in a short amount of time, to a problem which is NOT an open problem, and the approach includes trying as many problems as possible. The spirit is far from working on ONE major problem, like the Poincaré Conjecture. Let me recall here a detail that is mentioned in the book (p. 66), namely, that by chance, one of the problems that was proposed at the 1982 All-Soviet Mathematical Olympiad had been solved and discussed at length at Rukshin's math club (the one Perelman attended). Alexander Levin, part of the Leningrad team in that year, had been absent on the day the problem was discussed at the club, and therefore he did not solve the problem at the competition. We learn by the way that Levin had the handicap of having parents who "apparently insisted he pay as much attention to his school work as to the math club''.
   The question of "Why was Perelman able to solve the Conjecture?'' has no easy response. On the other hand, I think the author gives in this book a clear answer to the two other questions that she addresses in her Prologue, namely, why did Perelman abandon mathematics, and why did he refuse the Clay Prize. The author records, for instance, the words that Perelman exchanged in December 2005 with the director of the Steklov Institute, when he decided to resign: "I have nothing against the people here, but I have no friends, and anyway, I have been disappointed in mathematics and I want to try something else. I quit.'' In trying to analyze the reasons that led Perelman to refuse the positions that were offered to him, the author writes (p. 164) that Perelman "abhorred the idea of being some department's prize possession'', and in any case, money seems to be something that Perelman profoundly dislikes, except for what is needed for (a very sober) living. We learn for instance, about an open scandal that Perelman created at the Steklov Institute in 2005 when he noticed that his pay slip was higher than usual, and insisted on returning the money (p. 184). All the offers, grants and awards he was offered after proving the Poincaré Conjecture were insulting to him. Besides this, the suggestion (which the author had heard from other people) that "recognition along with three other mathematicians, none of [whom] had accomplished anything as momentous as the proof of the Poincaré Conjecture'' (p. 193) could have played a role in Perelman's refusal of the Fields Medal sounds strange, given the humbleness that usually characterizes a great mind.
   Another point in the book which deserves to be regarded with care is the fact that the capability to reject received ideas and to create new ones, and the ability to wholly concentrate on an abstract problem is connected with a diagnosis of Asperger's syndrome, which is a form of autism. Indeed, the relation between mathematics and autism has been pointed out on a few occasions. For instance, there have been at least two papers on that subject in the Mathematical Intelligencer (2003 and 2010). The author spends several pages on this, and apparently she considers it an important element in understanding Grigory Perelman; from that point of view, this form of autism is considered as a form of pure and original intelligence. Using the word "autism'' to explain the unusual behaviour of some mathematicians is a bit condescending, but this is fine. The problem from my point of view is that people usually consider autism as a mental disorder, and they will not give responsibilities to people tainted by mental disorders. Furthermore, the common approach to autism is to try to cure it, which is at odds with the desire to nurture mathematical creativity.
   There is a chapter (16 pages) on mathematics, in which the author tries to describe geometrization and the Poincaré Conjecture. The chapter is an honest attempt to present these problems to the general reader, although I was uncomfortable with some statements. For instance, the definition given of Riemannian geometry on p. 135 is mistaken. (It may be correct to call "Riemannian geometry'' the geometry of the sphere, and this was even common at the end of the nineteenth century, but the statement that this is the geometry that is used in Einstein's "general theory of relativity'' is false.) The statement made on p. 134 that Gauss was the teacher of János Bolyai is also false. It is probable that the two mathematicians never met and never exchanged any letters.
   Let me also note that a passage (p. 158) about Perelman's quarrel with his mentor, Yuri Burago, and concerning another researcher at the same laboratory (Burago was the head of that laboratory) is simply incomprehensible. We learn that it concerns "footnoting practices'', whose meaning is completely unclear to the reader (mathematician or not).
   The words "perfect rigor'' in the title are the best words that the author found to describe Perelman. Perfect rigor and a profound honesty, as Gromov puts it: "Perelman has moral principles to which he holds. And this surprises people. They often say that he acts strangely because he acts honestly, in a nonconformist manner, which is unpopular in this community—even though it should be the norm. His main peculiarity is that he acts decently. He follows ideals that are tacitly accepted in science.'' (p. 111)
   The book was published before the Millennium Prize Ceremony took place, in June 2010, in Paris. Perelman, as expected, did not show up at the ceremony. I would like to quote Thurston who, at that occasion, said something like the following (I cite from memory): "Perelman taught us a lot of beautiful mathematics. But there is something more we should learn from him. He thought profoundly about who he is.''
   Despite my reservations, I recommend this book. It is a good and unique record of an exceptional moment and of an exceptional person in the history of mathematics.

Between November 2002 and July 2003, Grigori Perelman posted on the ArXiv three major papers ["The entropy formula for the Ricci flow and its geometric applications'', preprint, arxiv.org/abs/math/0211159; "Ricci flow with surgery on three-manifolds'', preprint, arxiv.org/abs/math/0303109; "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds'', preprint, arxiv.org/abs/math/0307245]. They contained a sketchy (but concise) proof of the Poincaré conjecture and Thurston's geometrization conjecture. The approach uses the Ricci flow, introduced by Richard Hamilton in the 1980's. The texts, still available on the web, are revolutionary in many ways. Kleiner and Lott started to work on these articles in order to fill in the details and check the various statements. Preliminary notes were soon posted and made available to everyone interested in understanding this major breakthrough; they were also regularly updated. The present article is the final version crowning their work. It contains a very complete description of the first two papers posted by Perelman. The proof of Thurston's geometrization is given. It is certainly an unavoidable reference.
   The authors have made the choice of following as closely as possible Perelman's approach. Each chapter, section or subsection refers to a part of Perelman's articles. They start with a beautiful and precise overview of the Ricci flow approach to the geometrization. Although the proof of the Poincaré conjecture is not directly addressed in these notes, the authors include a rough outline of it. Let me recall that there are two ways to derive the Poincaré conjecture from the Ricci flow. Either one proves the geometrization conjecture and Poincaré's question is answered as a by-product, or one shows that if one starts with a simply-connected manifold the Ricci flow stops in finite time (we say that it becomes extinct). This last approach is a short-cut since one does not have to go through the painful long time study of the Ricci flow. The finite time extinction is the third paper posted by Perelman [op. cit., "Finite extinction time … '']. The argument is detailed in the book by J. W. Morgan and G. Tian [Ricci flow and the Poincaré conjecture, Amer. Math. Soc., Providence, RI, 2007; MR2334563] which focuses on the proof of the Poincaré conjecture. A somewhat different solution for showing the finite time extinction was given by T. H. Colding and W. P. Minicozzi, II [J. Amer. Math. Soc. 18 (2005), no. 3, 561–569; MR2138137]. The overview of the geometrization given in that text is sufficient for a reader who would only want to get the general picture. It describes, for example, the structure of singularities obtained by the blow-up technique relying on the compactness theorem and the non-collapsing property. Then follows a more precise summary of the first paper. After these very useful surveys the authors proceed to the description of each section of [G. Perelman, op. cit., "The entropy formula … '']. The computations are given with full details, sometimes with alternative proofs, and some very nice examples enlightening the basic notions introduced by Perelman.
   The description of the second paper, done equally carefully, is again preceded by a useful overview. The reader should be aware that this second part is technically difficult since it contains the construction of the Ricci flow with surgery and the study of its behaviour during long time evolution. After some practice one gets used to the proofs (mostly by contradiction) and the treatment given by the authors appears to be very clear. This should be read while looking also at the original papers by Perelman; the picture then clarifies progressively. In the last part the geometrization is proven. When the Ricci flow with surgery evolves, the manifold splits into a so-called thick part, that is, a non-collapsing part, which ought to become hyperbolic (with finite volume) after rescaling, and a thin part. The thin part is a graph manifold (with boundary). One important issue is that the hyperbolic pieces, if noncompact, are bounded by incompressible tori, that is tori whose fundamental group injects in the ambient manifold. Several proofs are given. One relies on an idea of Hamilton using minimal surfaces. Two others use the monotonicity of some Riemannian invariants: the first eigenvalue of a Schrödinger operator or the normalised minimum of the scalar curvature. The proof that the thin part is a graph manifold relies on a work by T. Shioya and T. Yamaguchi [Math. Ann. 333 (2005), no. 1, 131–155; MR2169831]. Although this last paper concerns closed manifolds the method seems to go through in the case of manifolds with boundary. More precise treatments were proposed in [B. Kleiner and J. Lott, "Locally collapsed 3-manifolds'', to appear; J. Morgan and G. Tian, "Completion of the proof of the geometrization conjecture'', preprint, arxiv.org/abs/0809.4040]. A completely different approach can be read in the article [L. Bessières et al., Invent. Math. 179 (2010), no. 2, 435–460; MR2570121] and the book [L. Bessières et al., Geometrisation of 3-manifolds, European Mathematical Society (EMS), to appear]. Another article detailing the proof of the geometrization conjecture has been published by H.-D. Cao and X. P. Zhu [Asian J. Math. 10 (2006), no. 2, 165–492; MR2233789; erratum, Asian J. Math. 10 (2006), no. 4, 663; MR2282358].
   It is particularly interesting to use all these references, each of them adding a stone to the building.

From the text: "Grigory Perelʹman has been awarded the Fields Medal for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.
   "Perelʹman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Aleksandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelʹman's results in Aleksandrov geometry are summarized in his 1994 ICM talk [G. Ya. Perelʹman, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 517–525, Birkhäuser, Basel, 1995; MR1403952]. We state one of his results in Riemannian geometry. In a short and striking article, Perelʹman proved the so-called Soul Conjecture.
   "Soul Conjecture (conjectured by J. Cheeger and D. Gromoll in 1972 [Ann. of Math. (2) 96 (1972), 413–443; MR0309010], proved by Perelʹman in 1994 [J. Differential Geom. 40 (1994), no. 1, 209–212; MR1285534]): Let M be a complete connected noncompact Riemannian manifold with nonnegative sectional curvatures. If there is a point where all of the sectional curvatures are positive, then M is diffeomorphic to Euclidean space.
   "In the 1990s, Perelʹman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints posted on the arXiv in 2002–2003 ["The entropy formula for the Ricci flow and its geometric applications'', preprint, arxiv.org/abs/math/0211159; "Ricci flow with surgery on three-manifolds'', preprint, arxiv.org/abs/math/0303109; "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds'', preprint, arxiv.org/abs/math/0307245], Perelʹman presented proofs of the Poincaré conjecture and the geometrization conjecture.
   "The Poincaré conjecture dates back to 1904 [H. Poincaré, Rend. Circ. Mat. Palermo 18 (1904), 45–110; JFM 35.0504.13]. The version stated by Poincaré is equivalent to the following.
   "Poincaré conjecture: A simply connected closed (= compact boundaryless) smooth 3-dimensional manifold is diffeomorphic to the 3-sphere.
   "Thurston's geometrization conjecture is a far-reaching generalization of the Poincaré conjecture. It says that any closed orientable 3-dimensional manifold can be canonically cut along 2-spheres and 2-tori into `geometric pieces' [W. P. Thurston, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381; MR0648524]. There are various equivalent ways to state the conjecture. We give the version that is used in Perelʹman's work.
   "Geometrization conjecture: If M is a connected closed orientable 3-dimensional manifold, then there is a connected sum decomposition M=M1#M2#⋯#MN such that each Mi contains a 3-dimensional compact submanifold-with-boundary Gi⊂Mi with the following properties:

1.

Gi is a graph manifold.

2.

The boundary of Gi, if nonempty, consists of 2-tori that are incompressible in Mi.

3.

Mi−Gi admits a complete finite-volume Riemannian metric of constant negative curvature.

   "In the statement of the geometrization conjecture, Gi is allowed to be ∅ or Mi. (For example, if M=S3 then we can take M1=G1=S3.) The geometrization conjecture implies the Poincaré conjecture. Thurston proved [op. cit.] that the geometrization conjecture holds for Haken 3-manifolds. Background information on the Poincaré and geometrization conjectures is in [J. W. Milnor, Notices Amer. Math. Soc. 50 (2003), no. 10, 1226–1233; MR2009455].
   "Perelʹman's papers have been scrutinized in various seminars around the world. At the time of this writing, the work is still being examined. Detailed expositions of Perelʹman's work have appeared in [H. D. Cao and X. P. Zhu, Asian J. Math. 10 (2006), no. 2, 165–492; MR2233789; erratum, Asian J. Math. 10 (2006), no. 4, 663; MR2282358; B. Kleiner and J. Lott, "Notes on Perelman's papers'', preprint, arxiv.org/abs/math/0605667; J. W. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Amer. Math. Soc., Providence, RI, 2007; MR2334563].''

Let {Xi}3i=1 be a global frame on S3 with [X1,X2]=2X3, [X2,X3]=2X1 and [X3,X1]=2X2. Let {θi}3i=1 be the dual co-frame on S3. The author constructs the following metric on [0,∞)×S3: g=dt2+∑3i=1A2i(t)θi. Take where ϕ,ψ∈C∞[0,∞) have the following properties: (i) ϕ(t)=0 for t∈[0,a], ϕ(t)>0 for t∈(0,∞), 0≤ϕ′(t)≤t−2 and |ϕ′′(t)|≤t−3; (ii) ψ(t)=0 for t∈[0,a/2], ψ′(t),ψ′′(t)>0 for t∈(a/2,a), and ψ′(t)=(tln3/2t)−1 for t>a. The constant a is a sufficiently large number. The author shows that |K|=O(t−2) and Ric≥C/(t2ln3/2t). Clearly, the vertex can be smoothed off so that the curvatures have the same properties. He asserts that the asymptotic cone of this space is not unique. Here an asymptotic cone of a complete Riemannian space M is the pointed Gromov-Hausdorff limit of ((1/rn)M,p) for some sequence rn→∞.

Summary: "It is shown that a connected sum of an arbitrary number of complex projective planes carries a metric of positive Ricci curvature with diameter one and, in contrast with the earlier examples of Sha-Yang and Anderson, with volume bounded away from zero. The key step is to construct complete metrics of positive Ricci curvature on the punctured complex projective plane, which have uniform euclidean volume growth and almost contain a line, thus showing topological instability of the splitting theorem of Cheeger-Gromoll, even in the presence of the lower volume bound. In the absence of such a bound, the topological instability was earlier shown by Anderson; metric stability holds, even without the volume bound, by the recent work of Colding-Cheeger.''

Convergence with collapsing dimension of Aleksandrov spaces is an important extension of the widely considered problem of collapse in Riemannian manifolds. The author states that it is "likely'' that, if the limiting Aleksandrov space has no proper extremal subsets, then the converging spaces are fiber bundles over the limit space. The author proves a weaker version of this result, namely that the homotopy groups of the spaces are connected by a Serre exact sequence. As an application to Riemannian geometry the author proves that if N is a complete noncompact Riemannian manifold of nonnegative sectional curvature that does not admit an isometric splitting and is not diffeomorphic to Rn, then the radius of its ideal boundary is at most π/2. This statement was conjectured by T. Shioya in 1993 [Math. Z. 212 (1993), no. 2, 223–238; MR1202809] and, according to the author, proved independently by S. Mendonça.

In this report under review, the author discusses the following problem: Let X be the limit space of a sequence of collapsing complete n-manifolds Mi with sectional curvature KMi≥−λ. What can we say about the geometric/topological structure of X? What is the relationship between Mi and X for large i?
   It is known that X has curvature ≥−λ in the sense of Aleksandrov. Thus all statements for Aleksandrov spaces with curvature bounded below are true for X. Roughly speaking, X has a natural stratification with totally geodesic strata, the strata being topological manifolds, but not smooth in general. The author outlines the proof of this result here.
   In order to find the relationship between X and Mi for large i, the author explains why one should study the extremal subsets in X. For example, the set of all topological singularities of X is an extremal subset. The outline of the proof of this fact is given in this report. Finally, the author asserts that there is a fibration over an open subset of X that consists of "weakly singular points'' (including regular points), the fibre Fi being a manifold.

The k-dimensional Uryson width wk(X) of a metric space X is defined as the lower bound of those δ>0 for which there is a k-dimensional space P and a continuous map f:X→P such that diamf−1(p)≤δ for all p∈P. The purpose of this paper is to prove the following conjecture of Gromov: namely, that for an n-dimensional closed Riemannian manifold M with nonnegative sectional curvature, there exists a constant c(n) depending only on dimension such that c−1Vol(M)≤∏n−1k=0wk(M)≤cVol(M). The author introduces the packing widths of a metric space and then proves the above relationship, replacing Uryson width with packing width. The equivalence between Uryson width and packing width is then shown.

The author gives a short geometric proof of the following theorem: For given n≥2 and K∈R there exists ε=ε(n,K)>0 such that any closed Riemannian manifold Mn with sectional curvature secM≥−K, Ricci curvature RicM≥n−1 and diameter diamM≥π−ε is a twisted sphere. Using direct geometric methods the author shows that for given p,q∈M with d(p,q)=diamM≥π−ε and any point τ∈M the inequality \anglearc\,(prq)>\pi/2 holds. Then the proof can be completed using a well-known argument of Grove and Shiohama.

Let M denote a complete noncompact manifold of nonnegative sectional curvature with soul S. It is well known that if u is a vector tangent to S, and v is orthogonal to S, then the plane spanned by u and v has zero curvature. In a short geometric argument based on the second Rauch theorem, Perelʹman shows that this is also true of the parallel translate of this plane along the geodesic in direction v. As a fundamental consequence, he obtains the existence of a Riemannian submersion from the ambient space onto S. This answers in particular a question asked some twenty years earlier by Cheeger and Gromoll: If the curvature is strictly positive at some point, then M is diffeomorphic to Euclidean space.

This paper gives an affirmative answer to the conjecture that a complete Riemannian manifold Mn, n≥2, having Ricci curvature ≥n−1 and volume close (depending only on n) to that of the standard sphere, must be homeomorphic to the sphere. Using the work of Hamilton and Smale it was sufficient for the author to prove that πi(M)=0 for i<n. This result, in turn, follows from the main lemma of the paper, which states: For any c2>c1>1 and integer k, if the volume of every metric ball contained in a ball Bp(c2R)⊂M is almost maximal (in terms only of c1,c2, and k) then any map of a k-sphere into Bp(R) can be continuously extended to a map of the (k + 1)-disk into Bp(c1R) (with a similar statement for maps into the complement of Bp(R)). The proof is by induction on k, extending the map to the (k−1)-skeleton of finer and finer triangulations of the disk. The construction of the extensions uses the Abresch-Gromoll inequality and a corollary of the Bishop-Gromov volume comparison. The author also states modifications of the main lemma for nonnegative Ricci curvature and Ricci curvature ≥−(n−1). In the first case it follows that if Ric(M)≥0 and the volumes of all balls centered at a fixed point are almost maximal, then M is contractible. A consequence of the second modified statement is a homotopy-type finiteness theorem for manifolds of fixed dimension, lower Ricci curvature bound, and having all balls of almost maximal volume.

The present paper is concerned with fine stratification of FSBCC [G. Ya. Perelʹman, Algebra i Analiz 5 (1993), no. 1, 232–241; see the preceding review], taking into account both topological and metric singularities. First, "extremal subsets'' are introduced and studied. Their definition is linked to elementary Morse theory: F⊂M is extremal iff for any distance function f=dist(q), q∈M, f(p)=|pq|, if p≠q is a local minimum on F then it is a critical point on M of maximum type, i.e. lim sup(f(pi)−f(p))/|ppi|≤0. An extremal subset is called primitive if it does not contain an extremal subset with nonempty (relative) interior. Principal parts of a primitive extremal set (which consist of points which are not contained in any other primitive extremal set) form the desired stratification. It is finer than topological stratification; moreover, each stratum of the fine stratification is a topological manifold. An interesting example: the stratification of a quotient (by an isometric action of a compact group) FSCBB by primitive extremal sets is finer than the stratification by orbit type. Finally, extremal sets have nice extrinsic properties: shortest curves in the intrinsic metric of F are quasigeodesics in M.

This paper addresses the issue of the local topological structure of Aleksandrov spaces. It is proved that FSCBB is MCS, that is, a finite-dimensional space with curvature bounded below has multiple conic singularities (i.e. every point has an open neighborhood homeomorphic to Rm×Cone(Σ), where Σ is a lower-dimensional MCS). In particular, an FSCBB admits a canonical stratification with topological manifolds as strata.
   The method of proof is to imitate Morse theory: a class of "admissible functions'', derived from the distance function, is introduced and a local analysis of these functions is done, deeply enough so that the levels of regular values of admissible functions are under control; they are shown to be lower-dimensional MCSs. Moreover, away from critical points, admissible functions are shown to be projections of locally trivial bundles.

This is an important paper in many respects. It contains a careful and fairly detailed discussion of basic facts of the theory, including various equivalent forms of definitions. It recognizes that the home of various important theorems of Riemannian geometry is the theory of Aleksandrov spaces, that both statements and proofs become more satisfactory (but not necessarily easier) in this context, and other theorems emerge naturally to complete the picture. It develops useful tools for studying Aleksandrov spaces with curvature bounded below in full generality. Finally, it contains an ample discussion of further results and open problems.
   There are many useful concepts introduced and striking results proved in this paper: globalization of the conclusion of Toponogov's theorem (this enters the very definition of Aleksandrov space but only in the small); the introduction of the concept of burst points and the proof that the set of burst points is open dense; careful treatment of the tangent cone, in particular a proof of the fact that various constructions of the tangent cone agree; the compactness theorem (the set of spaces with dimension ≤n, curvature ≥k and diameter ≤D is compact); the almost isometry theorem (roughly: Hausdorff close spaces are Lipschitz close); volume comparison theorem, estimate of the Hausdorff dimension of singular points and convergence of Hausdorff measures under taking the limit of Aleksandrov spaces; and the introduction of directional derivatives and the study of levels of "almost regular maps'', roughly the distance functions from compact sets.
   The present-day study of Aleksandrov spaces is motivated by the emergence of such spaces as the limits of collapsing families of Riemannian manifolds. This significantly widens the scope of the theory, which traditionally fed on the tension between synthetic and analytic approaches to geometry and on (important) methodological questions.

This paper considers the realization of complete, 2-dimensional polyhedral metrics of negative curvature by polyhedral saddle surfaces in Euclidean space; here a metric is polyhedral when there is a neighborhood of each of its points which is isometric to the neighborhood of the vertex of a cone; the metric has negative curvature when the sum of the conical angles at any vertex is no less than 2π. A polyhedral surface is a saddle surface if at no point is there a locally strongly supporting plane. The author proves: (1) Any complete polyhedral metric of negative curvature specified over the plane can be imbedded in R3 as a polyhedral saddle surface; (2) corresponding to each complete polyhedral metric of negative curvature, specified on a cylinder having closed geodesics, there is a polyhedral saddle surface in R3 which realizes this metric.

Let M⊂Rn be a compact convex body. Its inner k-radius rk (1≤k≤n) is defined as the radius of the greatest k-dimensional sphere contained in M. The outer k-radius Rk of M (1≤k≤n) is the "radius'' of the smallest spherical cylinder containing M, where the dimension of the sphere defining the cylinder is n+1−k. In this paper it is proved that Rk/rk≤k+1 for any compact convex body. The maxima of R1/r1 and Rn/rn, respectively, are achieved by regular simplices. Exact upper estimates for Rk/rk seem not to be known for k=2,⋯,n−1. Using Rk and rk the author also answers two questions (asked in the geometric seminar of the Novosibirsk Mathematics Institute [Working paper, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1984; per bibl.]) on inscribing spheres in M and inscribing M in cylinders, where M is a convex body having C2 boundary.
   English translation: Siberian Math. J. 28 (1987), no. 4, 665–666.

The paper contains an axiom system of the real Euclidean plane in Hilbert's style except that all axioms are formulated in terms of compact sets (segments, etc.). The first part develops the mathematical theory and the second part discusses historical and philosophical points that make such a finitistic axiomatic desirable. The appendix by Perelʹman discusses the equivalence of a Pasch-style axiom of Aleksandrov and some of its consequences.
   English translation: Siberian Math. J. 28 (1987), no. 4, 523–539.

The following theorem is proved. Let M be a bounded family of compact convex sets in the Euclidean space En such that μn(⋂M)=0. Here μn denotes the n-dimensional volume. Then for every ε>0 there are n+1 sets F1,⋯,Fn+1 in the family M such that μn(⋂n+1i=1Fi)<ε. Moreover, if dim⋂M=l,0≤l<n, then there are n−l+1 sets in the family M such that μn(⋂n−l+1i=1Fi)≤ε.

Every abstract (k−1)-dimensional simplicial complex (abstract k-skeleton, in the author's terminology) can be realized as the nerve of a system of convex (2k−2)-dimensional polyhedra in R2k−1.