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# Four dimensional euclidean space according to poincaré conjecture

In mathematics, the Poincaré conjecture (UK: / ˈ p w æ̃ k ær eɪ /, US: / ˌ p w æ̃ k ɑː ˈ r eɪ /, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere The shape of the universe as it was, is now and may become in the future is very hard for us to discern empirically. Einstein helped us somewhat by showing us that matter and energy (three-dimensional phenomena) in fact may interact with the a four-dimensional phenomenon: time. In this interaction, spacetime may be warped by the presence of mass/energy. Taken together, as far as we can tell, we live in a four-dimensional universe that is susceptible to deformation such as. The French mathematician Henri Poincaré offered a statement known as Poincaré's conjecture without a proof. It states that any 4-dimensional ball is equivalent to 3-dimensional Euclidean space topologically: a continuous mapping exists so that it maps the former ball into the latter space one-to-one. At first glance, it seems to be too paradoxical for the following mismatches: the.

### Poincaré conjecture - Wikipedi

1. The French mathematician Henri Poincaré offered a statement known as Poincaré's conjecture without a proof. It states that any 4-dimensional ball is equivalent to 3-dimensional Euclidean space topologically: a continuous mapping exists so that it maps the former ball into the latter space one-to-one
2. GENERALIZED POINCARE'S CONJECTURE IN DIMENSIONS GREATER THAN FOUR BY STEPHEN SMALE* (Received October 11, 1960) (Revised March 27, 1961) Poincar6 has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [I81 for example. This problem, still open, is usually called Poincar6's conjec
3. Mike Freedman has proven that a -manifold which is homotopy-equivalent to is homeomorphic to , so the smooth 4-dimensional Poincare conjecture is the only outstanding problem among the generalized Poincare conjectures. Moreover, it can be considered to be reduced to the question of if has an exotic smooth structure
4. mapped into the manifold. In dimension 5 or greater, such disks can be put into general position so that they are disjoint from each other, with no self-intersections, but in dimension 3 or 4 it may not be possible to avoid intersections, leading to serious diﬃculties. Stephen Smale announced a proof of the Poincar´e Conjecture in high dimensions in 1960 . He was quickly followed by John Stallings, who used a completel
5. The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions). The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by Donaldson). If the cobordism has dimension 4, then it is unknown whether.
6. a) Have a plan that can be represented as part of a Euclidean plan or space; b) Have a plan that looks like a plan and, in any case, that is of dimension 2, so a surface; c) Have lines that look like straight lines, or at least simple lines, if possible; d) Maintain the usual incidence properties
7. A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space.Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its. R 4 {\displaystyle \mathbb {R} ^ {4}} is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory. Therefore, the winner must prove that: Yang-Mills theory exists and satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular constructive quantum field theory, and of exotic 4-dimensional Euclidean spaces, when com-bined with Freedman's proof of the topological Poincaré conjecture in dimension 4. Since then the study of mani - folds of dimension less than or equal to 4 (e.g., 3- and 4-dimensional topology, knots in 3-manifolds, mapping class groups of surfaces) has formed a new branch o The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions). The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4

### The Poincaré Conjecture

• d, used to his three.
• The realm which remains the most mysterious, even today, is 4-dimensional space. No exotic spheres have yet been found here. At the same time no-one has managed to prove that none can exist. The assertion that there are no exotic spheres in four dimensions is known as the smooth Poincaré conjecture. In case anyone has got this far and is still not sure, let me make this clear: the smooth Poincaré conjecture is not the same thing as the Poincaré conjecture! Among other.
• he Poincaré Conjecture was posed ninety-nine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3-dimensional manifolds. The final paragraph provides a brief description of the.
• The Poincaré conjecture is a topological problem established in 1904 by the French mathematician Henri Poincaré. It characterises three-dimensional spheres in a very simple way. This problem was directly solved between 2002 and 2003 by Grigori Perelman, and as a consequence of his demonstration of the Thurston geometrisation conjecture
• The Poincaré Conjecture is first and only of the Clay Millennium problems to be solved. It was proved by Grigori Perelman who subsequently turned down the $1 million prize money, left mathematics, and moved in with his mother in Russia. Most explanations of the problem are overly-simplistic or overly-technical, but on this blog I try to • Moreover the space-time must be globally hyperbolic with Cauchy surfaces which, subject to the truth of the Poincaré conjecture, are diffeomorphic to R3. View Show abstrac • Poincaré conjecture, in topology, conjecture—now proven to be a true theorem—that every simply connected, closed, three-dimensional manifold is topologically equivalent to S 3, which is a generalization of the ordinary sphere to a higher dimension (in particular, the set of points in four-dimensional space that are equidistant from the origin) ### The space-time interpretation of Poincare's conjecture • During the 1920s Poincaré's conjecture became a well known problem. In particular H. Kneser mentioned it in a talk he delivered to the Versammlung Deutscher Naturforscher und Ärzte (joint meeting with the DMV) in 1928; [].After citing Dehn's manifold, Kneser stated that this example is also related to the 120-cell (one of the six regular polytopes of 4-space) in ordinary 4-space • ent close to the Ox axis and as one approaches infinity on the Oy axis. These characteristics of the two models of hyperbolic space will become useful later on, when we will try to infer some properties for a point based on where. • of the Poincar´e conjecture and the geometrization conjecture of Thurston. While the complete work is an accumulated eﬀorts of many geometric analysts, the major contributors are unquestionably Hamilton and Perelman. An important problem in diﬀerential geometry is to ﬁnd a canonical metric on a given manifold. In turn, the existence of a canonical metric often has profound topological. • The Poincaré conjecture concerns the three-dimensional equivalent of this situation. It asserts that if any loop in a closed three-dimensional space without boundary can be shrunk to a point without tearing either the loop or the space, then the space is equivalent to a three-dimensional sphere. Posed in 1904 by Henri Poincaré, the leading. In mathematics, the Poincaré conjecture (UK: / ˈ p w æ̃ k ær eɪ /, US: / ˌ p w æ̃ k ɑː ˈ r eɪ /, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space Generalized Poincaré conjecture In mathematics , the Poincaré conjecture ( UK : / ˈ p w æ̃ k ær eɪ / ,  US : / ˌ p w æ̃ k ɑː ˈ r eɪ / ,   French: [pwɛ̃kaʁe] ) is a theorem about the characterization of the 3-sphere , which is the hypersphere that bounds the unit ball in four-dimensional space The Poincare Conjecture is essentially the first conjecture ever made in topology; it asserts that a 3-dimensional manifold is the same as the 3-dimensional sphere precisely when a certain algebraic condition is satisfied. The conjecture was formulated by Poincare around the turn of the 20th century. A solution, positive or negative, is worth US$1,000,000 , since it is one of th

1. The conjecture states that there is only one shape possible for a finite universe in which every loop can be contracted to a single point. Poincaré's conjecture is one of the seven millennium problems that bring a one-million-dollar award for a solution. Grigory Perelman, a Russian mathematician, has offered a proof that is likely to win the.
2. He observed that the Euclidean group, selected after many conventions, can be seen as acting on a space of three, four or five dimensions. The choice of a three-dimensional space is justified by considerations of commodity. Unfortunately, Poincaré's argument is viciously circular because the choice of the Euclidean group was grounded on Lie's classification of transformation-groups.
3. It should be chosen for Poincaré's conjecture [ 34] proved by G. Perelman [35-37]. If that condition misses, the topological structure is equivalent to any of both almost disjunctive domains. 4 of Minkowski's space of special relativity5 rather than to a 4D Euclidean ball. The two domains of Minkowsk
4. dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in. 4. In mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. For example, a point on the unit.

The Poincaré conjecture can be understood by analogy with the case in two dimensions. A two-dimensional space, or surface, is like a bubble made from an infinitely thin film of soap. If the. In this quotation from Henri Poincaré's essay Non-Euclidean Geometry published in Nature in 1892 (No. 1165, Vol 45, p. 406), he refers to a theorem of Sophus Lie. Does anyone know a source for this theorem, or one that discusses it? reference-request geometry euclidean-geometry riemannian-geometry projective-geometry. Share. Improve this question. Follow edited Mar 13 '19 at 9:43. Conifold. The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations Jeffrey Weeks 610 NOTICES OF THE AMS VOLUME 51, NUMBER 6 Introduction Because of the finite speed of light, we see the Moon as it was roughly a second ago, the Sun as it was eight minutes ago, other nearby stars as they were a few decades ago, the center of our Milky Way Galaxy as it was 30,000 years ago, nearby. Therefore, in dimension three, the Poincaré Conjecture and the Generalized Poincaré Conjecture for Alexandrov spaces are no longer necessarily equivalent, as opposed to the manifold case. In this section, we prove the Generalized Poincaré Conjecture for compact Alexandrov three-spaces (cf. Proposition 1.4). We also provide examples of. Calculate Euclidean distance between 4-dimensional vectors. Ask Question Asked 7 The function/method/code above will calculate the distance in n-dimensional space. a and b are arrays of floating point number and have the same length/size or simply the n. Since you want a 4-dimension, you simply pass a 4-length array representing the data of your 4-D vector. Share. Improve this answer.

A 64-dimensional two-distance counterexample to Borsuk's conjecture. In 1933 Karol Borsuk asked whether each bounded set in the n-dimensional Euclidean space can be divided into n+1 parts of smaller diameter. The diameter of a set is defined as the supremum (least upper bound) of the distances of contained points Steiner Tree Heuristics in Euclidean d-Space Andreas E. Olsen 1, Stephan S. Lorenzen , Rasmus Fonseca1, and Pawel Winter1 Department of Computer Science, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen O, Denmark mr.foxhide@hotmail.com, stephan.lorenzen@gmail.com, frfonseca,pawelg@di.ku.dk Abstract. We present a class of heuristics for the Euclidean Steiner tree problem in a d.

### Smooth 4-dimensional Poincare conjecture Open Problem Garde

1. e the position of an element (i.e., point).This is the informal meaning of the term dimension.. In mathematics, a sequence of n numbers can be understood as a location in n-dimensional space
2. The Poincaré conjecture is a topological problem established in 1904 by the French mathematician Henri Poincaré. It characterises three-dimensional spheres in a very simple way. It uses only the first invariant of algebraic topology - the fundamental group - which was also defined and studied by . Poincaré. The conjecture implies that if a space does not have essential holes, then it is.
3. Euclidean geometry, his discovery of chaos (in celestial mechanics), and his creation of algebraic topology (in which the Poincaré conjecture was the central unsolved problem for almost a century). These topics also belong to the three main areas of Poincaré's research that have been translated into English, and I discuss them further.
4. The book presents ideas by H. Poincare and H. Minkowski according to those the essence and the main content of the relativity theory are the following: the space and time form a unique four-dimensional continuum supplied by the pseudo-Euclidean geometry. All physical processes take place just in this four-dimensional space. Comments to works and quotations related to this subject by L. de.
5. Theorem 4. The piecewise-linear Poincaré hypoth-esis is true for n-dimensional manifolds except pos-sibly when n = 4. That is, any closed PL manifold of dimension n 6=4 with the homotopy type of an n-sphere is PL-homeomorphic to the n-sphere. For n > 4 this was proved by Smale ; while for n = 3 it follows from Perelman's work, together.

Poincaré's conjecture proved by G. Perelman by the isomorphism of Minkowski space and the separable complex Hilbert space March 2019 DOI: 10.13140/RG.2.2.33374.4896 The Poincare Conjecture is essentially the first conjecture ever made in topology; it asserts that a 3-dimensional manifold is the same as the 3-dimensional sphere precisely when a certain algebraic condition is satisfied. The conjecture was formulated by Poincare around the turn of the 20th century. A solution, positive or negative, is worth US$1,000,000 , since it is one of th As a direct corollary of Theorems 6.4 and 6.3, when $$\mathcal {X}$$ is a Banach space, we conclude that, similarly to the general case, the set of probability measures $$\mu$$ satisfying the dimension free convex concentration property $$\mathbf {CCI}_2^\infty (\alpha )$$ with exponential profile coincides with the set of probability measures verifying the convex Poincaré inequality According to Newtonian physics and tradition-al quantum physics, the three-dimensional space where everything happens is ﬁxed and immutable. Einstein's theory of general relativity, in contrast, S HENRI POINCARÉ conjectured in 1904 that any three-dimensional object that shares certain properties of the three-dimensional sphere can be morphed into a 3-sphere. It took 99 years for. ### Low-dimensional topology - Wikipedi of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincaré group. The spacetime interval between two events in Minkowski space is either: 1. space-like, 2. light-like ('null') or 3. time-like. Contents 1 History 2 Structure 2.1 The Minkowski inner product 2.2 Standard basis 3 Alternative definition 4 Lorentz transformations and symmetry 5 Causal structure 5.1. The Poincaré conjecture is best described by stepping down a dimension and looking at the surface of a sphere, which lives in three-dimensional space but itself is two-dimensional. This surface has two important properties: it has no boundary — when you walk around on it you will never fall over an edge — and when you tie a piece of string around a sphere you can always slide it off. 3.1 Euclidean Dimension. A space is a collection of entities called points. Both terms are undefined but their relation is important: space is superordinate while point is subordinate. Our everyday notion of a point is that it is a position or location in a space that contains all the possible locations. Since everything doesn't happen in exactly the same place, we live in what can rightly be. We consider time-dependent solutions of the Einstein-Maxwell equations using anti-de Sitter (AdS) boundary conditions, and provide the first counterexample to the weak cosmic censorship conjecture in four spacetime dimensions. Our counterexample is entirely formulated in the Poincaré patch of AdS. We claim that our results have important consequences for quantum gravity, most notably to the. ### Four-dimensional space - Wikipedi (as well as the stronger Baum Connes conjecture ) for groups which act both isometrically and metrically properly on Euclidean space, in the sense of . This applies, for example, to countable amenable groups . The contents of the paper are as follows. In Section 2 we shall give an account of the standard Bott periodicity theorem, from the point of view of C*-algebra theory, and with. Dimension-Cruncher: Exotic Spheres Earn Mathematician John Milnor an Abel Prize. His discovery that some seven-dimensional spheres look different under the lens of calculus spurred decades of. The proofs of the Poincaré Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate. The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincaré Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3. Knot invariants of finite type, also called Vassiliev invariants, were initially conceived to study knots in euclidean 3-space , . Many important knot invariants are of finite type, most notably the coefficients of the Alexander-Conway and the Jones polynomial, after a suitable change of variables  ,  198 Scott Walter There are two aspects to Poincaré's conjecture I want to underline. First, scientists are free in Poincaré's scheme to choose between the two couples: Euclidean geometry and non-Maxwellian optics, or non-Eu- clidean geometry and Maxwellian optics. Either way, the geometry of space and the laws of optics result from a convention. In essentials, as Torretti notes (1984. nonpositive according as n = 0 or 2 mod 4. This conjecture for « = 4 was proved first by J. W. Milnor and then by S. S. Chern by a different method. The main object of this paper is to prove this conjecture for a general n under an extra condition on higher order sectional curvature, which holds automatically for n = 4. Similar results are obtained for Kahler manifolds by using holomorphic. Its existence prompted the still open 3-dimensional Poincaré conjecture and had an enormous influence on the subsequent development of topology. Poincaré obtained his example from two solid double tori identified along their boundary surfaces of genus 2. Another construction that yields the same space is due to Threlfall and Seifert. Their description of the Poincaré sphere as the spherical. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this property. For this purpose, we introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n-dimensional. between the two spaces, and is the heart of Freedman's proof of the 4-dimensional Poincare Conjecture. I will talk a little about the design and some related topics (gropes, kinks, shrinkability) and how to really visualize what's going on 2.3. The Bernstein conjecture and domain walls. From a physical point of view, a minimal surface is a mathematical idealization of something with finite thickness. A model which incorporates this is a nonlinear Laplace equation of the form. If V () has two critical points at = ±1, say, at which Poincaré conjecture. In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: Every simply connected, closed 3- manifold is homeomorphic to the 3-sphere. This mathematics-related article is a stub Idea General. A spacetime is a manifold that models space and time in physics.. This is formalized by saying that a spacetime is a smooth Lorentzian space (X, μ) (X,\mu) equipped with a time orientation (see there).. Hence a point in a spacetime is called an event.. In the context of classical general relativity a spacetime is usually in addition assumed to be connected and four-dimensional ### Yang-Mills existence and mass gap - Wikipedi The Poincaré Conjecture has Been Proved 307. Posted by chrisd on Sunday April 07, 2002 @04:43AM from the np-is-the-problem-for-me dept. Martin Dunwoody, a famous mathematician who works in the field of topology has a preprint that provides a proof of the Poincaré conjecture. This was one of the seven Clay Mathematics Institute millenium prize. Solomon Lefschetz, in History of Topology, 1999. 3.9. Examples. In Analysis Situs Poincaré constructed eight examples of 3-spaces by matching appropriate faces of a cube (first four examples) or of a regular octahedron. His purpose was to obtain explicit 3-manifolds whose Betti numbers and groups π 1 could be computed. The second example is to be rejected as not corresponding to an M 3 According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. Philosophy of space Leibniz and Newton. Gottfried Leibniz. In the seventeenth century, the philosophy of space and time emerged as a. According to a standard definition of Penrose, a space-time admitting well-defined future and past null infinitiesI + andI − is asymptotically simple if it has no closed timelike curves, and all its endless null geodesics originate fromI − and terminate atI +. The global structure of such space-times has previously been successfully investigated only in the presence of additional constraints ### 4-manifold - Wikipedi There are four axioms: W0 (assumptions of relativistic quantum mechanics) Quantum mechanics is described according to von Neumann; in particular, the pure states are given by the rays, i.e. the one-dimensional subspaces, of some separable complex Hilbert space. The Wightman axioms require that the Poincaré group acts unitarily on th The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to describe string theory, eleven.  ### How Grigori Perelman solved one of Maths greatest mystery We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the borderline case. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in. The physical interpretation of Poincare's conjecture is what is meant In technical terms the conjecture is that if a space is homotopically equivalent to a three-dimensional sphere it is homeomorphic to the three-sphere. In less technical terms it says that if you have a bounded three-dimensional space in which all loops can be shrunk down to points, it has to be the three-sphere. In dimensions other than three the analog conjecture has been proved, but the case. Poincaré's conjecture was subsequently generalized to any number of dimensions, but in fact the three-dimensional version has turned out to be the most difficult of all cases to prove. In 1960. Poincar¶e before topology 3 For a three-dimensional manifold M one can also consider the maximum number P2 of disjoint closed surfaces in M that fail to separate M as the \two- dimensional connectivity number of M.The idea of separation fails to explain the \one-dimensional connectivity of M, however, since no ﬂnite set of curves can separate M.Instead, one takes the maximum number of. 19TH CENTURY MATHEMATICS Approximation of a periodic function by the Fourier Series The 19th Century saw an unprecedented increase in the breadth and complexity of mathematical concepts. Both France and Germany were caught up in the age of revolution which swept Europe in the late 18th Century, but the two countries treated mathematics quite differently. [ ### Exotic spheres, or why 4-dimensional space is a crazy Poincaré proved the two-dimensional case and he guessed that the principle would hold in three dimensions. Determining if the Poincaré conjecture is correct has been widely judged the most important outstanding problem in topology - so important that, in 2000, the Clay Mathematics Institute in Boston named it as one of seven Millennium Prize Problems and offered a$1 million prize for its. Footnote 6 Thanks to Princet, Picasso is supposed to have taken into consideration the new concept of space-time visualization, modeled on non-Euclidean geometries; it may have been thanks as well to the representations on the two-dimensional plane of hypercubes and other four-dimensional complex polyhedra that appeared in the Traité élémentaire de géométrie à quatre dimensions (1902. Thirdly, the redundant high-dimensional Euclidean space of original financial system is built using the technique of phase space reconstruction, with the delay time τ d = 1. According to the correlation integral, the correlation dimension is C = 4.5, and the embedding dimension is m = 9 (Figure 4). Figure 4. The plot correlation dimension  ### Poincaré conjecture - Revista Mètod

We formulate a model of noncommutative four-dimensional gravity on a covariant fuzzy space based on SO(1,4), that is the fuzzy version of the $\\text{dS}_4$. The latter requires the employment of a wider symmetry group, the SO(1,5), for reasons of covariance. Addressing along the lines of formulating four-dimensional gravity as a gauge theory of the Poincaré group, spontaneously broken to the. According to GTR, the geometry of four-dimensional space-time is non- Euclidean. Physical space is described as possessing variable curvature,8 the curvature depending on the distribution of masses and fields. The theory transforms the action of gravity into a feature of space-time geometry. Freely falling bodies are simply describing geodesics (shortest paths) in a four-dimensional space-time.   The Poincaré conjecture is currently the only millennium prize problem that currently has a verifiable solution, making it the easiest one out of the seven problems. All jokes aside, I do want to take time to recognize Grigori Perelman , one of th.. Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean. Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids 207 222; The concept of residue after Poincaré: Cutting across all of mathematics 225 240; The proof of the Poincaré conjecture, according to Perelman 243 258; Henri Poincaré and the partial differential equations of mathematical physics 257 27

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