No big event, no announcement, just a casual oh heres what ive been working on the past few years i solved a millennium problem. Is it possible that the fundamental group of v could be trivial, even though v is not homeomorphic to the 3dimensional sphere. Dec 26, 2007 oshea tells the fascinating story of this mathematical mystery and its solution by the eccentric mr. When a conjecture is rigorously proved, it becomes a theorem. In this paper, we will give an elementary proof by the heegaard splittings of the 3dimentional poincare conjecture in point of view of pl topology. In mathematics, the poincare conjecture is a theorem about the characterization of the 3sphere. Pdf on the proof of the poincare conjecture researchgate. Pdf in 2002, perelman proved the poincare conjecture, building on the work of richard.
Poincare wikipedia since it was proposed by polymath h. One was to combine collapsing theory in riemannian geometry with. Poincares conjecture implies con jecture 1 if the following hypothesis is tme. Recall that in the derivation of the lagrange equations we encountered dq, which we were able to write as d q, because we knew that dq was the derivative of q.
Aug 14, 2007 the proofs of the poincara conjecture and the closely related 3dimensional spherical spaceform conjecture are then immediate. There is one problem on this list in common with hilberts list, the riemann. Poincares conjecture is implied by a conjecture on free groups. Apr 03, 2017 perelman is a russian mathematician and he is quite the character. Poincare conjecture a quick chat with stephen smale in this short clip the fields medallist tells us about his bad marks at school and his fascinatingly eclectic work as a mathematician. Specifically, what the results do or explain such as how it would affect the world and how we would apply it. As mentioned earlier, the poincare conjecture deals with topology. Im doing a project on the poincare conjecture for my algebra 2 honors class and i need some background information on the poincare conjecture. It just appears correct and there is no proof to disprove it. Smooth 4dimensional poincare conjecture open problem. Free group, handlebody, homeomorphism, 3manifold, simply connected, 3sphere. The converse is also valjd modulo a hypothesis on the uniqueness of decomposition of the 3sphere as the union of two handlebodies intesecting in a torus. Mathematically, this was proven to be correct by the mathematician grigori perelman russian jew in 2002.
Poincare conjecture, in topology, conjecturenow proven to be a true theoremthat every simply connected, closed, threedimensional manifold is topologically equivalent to s3, which is a generalization of the ordinary sphere to a higher dimension in particular, the set of points in. The existence of ricci flow with surgery has application to 3manifolds far beyond the poincare conjecture. Conjectures must be proved for the mathematical observation to be fully accepted. In 2003, he proved thurstons geometrization conjecture. It has turned out to be an extraordinarily difficult question, much harder than the corresponding question in dimension five or more,2 and is a key stumbling block in the effort to classify 3dimensional manifolds. Perelmans proof, buliding on the work of hamilton, was based on. The existence of such an l implies that m3 is a 3sphere. In fact, the purpose of this book is to prove the following more general theorem. How to tackle the smooth poincare conjecture mathoverflow.
Conjecture and the closely related 3dimensional spherical spaceform conjecture are then immediate. A conjecture on poincarebetti series of modules of differential operators on a generic hyperplane arrangement snellman, jan, experimental mathematics, 2005. He showed that the poincare conjecture was true and then without really telling anyone just posted his solution online in 2002. The generalized poincare conjecture in higher dimensions. But for milnor and kervaire showed that admits nonstandard smooth structures so the smooth poincare conjecture is false in general. In this paper, we give a complete proof of the poincar. By combining perelmans no local collapsing theorem.
The arguments we present here are expanded versions of the ones given by perelman in his. Centuryold math problem may have been solved artikel aus dem boston globe vom 30. The poincare chair owes its creation and its very existence, for the next five years, to the prize awarded by the clay mathematics institute cmi, providence, rhode island for the solution of the poincare conjecture. The existence of ricci flow with surgery has application to 3manifolds far beyond the poincara conjecture. In search of the shape of the universe donal oshea the poincare conjecture is one of the few mathematical results that has managed to catch the interest of the mainstream media.
Sep 22, 2014 the poincare conjecture and grigori perelman. A hypothesis is actually testable based on accepted grounds. This work depends on the accumulative works of many geometric analysts. The generalized poincare conjecture is an undergraduatelevel pointset topology problem for.
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 3manifold triangulated is homeomorphic to the 3sphere, see i81 for example. Derivation of the poincar e equations we will derive the poincar e equations from the action principle. Poincare conjecture is im plied by a single grouptheoretic conjecture. This paper is of the same theory in 41983 excluding the last three lines of the proof of the main theorem. Is there any relation between poincare conjecture and general. In the 1980s, hamilton initiated the program of using ricci. In 2006 the russian mathematician grigori perelman was awarded the mathematical equivalent of the mathematical nobel prize the fields medal. In particular, if m4 is a homotopy sphere, then h2 0 and.
Poincare in 1904, who sought to mathematically define the. He has made contributions to riemannian geometry and geometric topology. By now the situation seems to be that the experts are pretty convinced of the details of perelmans proof for the poincare conjecture. Oshea tells the fascinating story of this mathematical mystery and its solution by the eccentric mr. Suppose m, g is complete, connected and noncompact with sectional curvature k.
No one would deny that perelmans proof makes crucial use of mathematics developed by hamilton about. Towards the poincare conjecture and the classification of 3manifolds. Apr 23, 2014 the famed poincare conjecture the only millennium problem cracked thus far. Among the contenders are shingtung yau and grigory perelman. One of the clay millennium problems is the poincare conjecture.
Then every hodge class on x is a linear combination with rational coefficients of chern classes of vector bundles on x. Mike freedman has proven that a manifold which is homotopyequivalent to is homeomorphic to, so the smooth 4dimensional poincare conjecture is the only outstanding problem among the generalized poincare conjectures. Perelmans results go well beyond a solution to the problem at hand, as did those of dr. Recent progress on the poincare conjecture and the classification of 3manifolds, by john morgan, bull. Especially since hamilton wasnt even willing to meet with perelman to discuss the latters ideas that led to the solution of the poincare conjecture, it would be a particularly cruel miscarriage of fairness to attribute the proof to hamilton. This proof should be considered as the crowning achievement of the hamiltonperelman.
A conjecture is an important step in problem solving. The poincare conjecture and grigori perelman ib maths. Russian reports he has solved a celebrated math problem. Poincares conjecture is implied by a conjecture on free. It should be noted that the piecewise linear or di. Hodge conjecture for kahler varieties, vector bundle version. Is there any relation between poincare conjecture and. As mathbalarka has noted, the 4d smooth poincare conjecture is unsolved. This manuscript contains a detailed proof of the poincare conjecture.
Ricci flow and the poincare conjecture clay mathematics. The conjecture that this is indeed the case has come to be known as the poincare conjecture. It is not known whether every smooth homotopy 4sphere is di. Towards the poincare conjecture and the classification of 3. When i addressed peters statement that there was no mystery for perelman to solve, my response hinted towards the smooth version of the poincare conjecture, which perelman solved in 3d. Hodge conjecture for kahler varieties, coherent sheaf version. The poincare conjecture is a topological problem established in 1904 by the french mathematician. Consider a compact 3dimensional manifold v without boundary. The poincare conjecture, by john milnor in the millennium prize problems, clay mathematics institute and the american mathematical society, 2006.
The extended entries include pontrjagins article on smooth manifolds and their application in homotopy theory. Novikovs paper on homotropy properties of tom complexes. Russian reports he has solved a celebrated math problem the. Thus hamiltons program, when carried out successfully, will give a proof of the poincar. Generalized poincares 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 3manifold triangulated is homeomorphic to the 3sphere, see i81 for example. Of the dollars and the glory for mathematicians, the million dollars that the clay mathematics institute cmi has offered for the solution of the poincare conjecture is mere icing on the cake.
Mar, 2010 the poincare conjecture is first and only of the clay millennium problems to be solved. Perelmans results say he has proved a much broader conjecture about the geometry of three. The full geometrization conjecture requires some more argument and i gather that perelman is supposed to at some point produce another preprint with more about this. Perelman and the poincare conjecture not even wrong. It was the most stubborn mathematical problem to be solved ever since poincare brought it to the surface almost a century ago. In this short clip the fields medallist tells us about his bad marks at school and his fascinatingly eclectic work as a mathematician. Generalized poincares conjecture in dimensions greater. Perelman, if i remember correctly, studies ricci curvature flows which is a large area of mathematics in its own right. Nov 24, 2009 this video presents a visual overview of the poincare conjecture, background, solution, mathematics, ensuing controversy and a few figures surround the solution. In these papers he sketched a proof of the poincare conjecture and a more general.
Strong generic vanishing and a higherdimensional castelnuovode franchis. Thoms work on global properties of differential manifolds. Wall street journal in 1904, henri poincare, a giant among mathematicians who transformed the fledging area of topology into a powerful field essential to all mathematics and physics, posed the poincare conjecture, a tantalizing puzzle that speaks to the possible shape of the. This work depends on the accumulative works of many geometric. First of all, i highly doubt that all of those pages are devoted to solving the poincare conjecture. In the derivation of the poincar e equations we encounter. Poincare recurrence theorem article about poincare.
There is one problem on this list in common with hilberts list, the riemann hypothesis. Nevertheless, the theorem is insufficient for predicting the time for a system to return, that is, the recurrence. It forms the heart of the proof via ricci flow of thurstons geometrization conjecture. The clash over the poincare conjecture the new yorker. Generalized poincares conjecture in dimensions greater than. It is the activity in which the human mind seems to take least from the. The poincare conjecture is one of the most important conjectures in modern mathematics, and has currently been demonstrated adequately to the point that it is considered a full theorem.
In the course of his research, he discovered some things that led to this proof of the poincare conjecture. The poincar e conjecture at the end of the 5th supplement to analysis situs, poincar e asks. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Perelman is a russian mathematician and he is quite the character. Poincares conjecture article about poincares conjecture. It is one of the seven millennium prize problems, stated by the clay mathematics institute in 2000. A complete proof of the poincare and geometrization conjectures application of the hamiltonperelman theory of the ricci flow pdf. In mathematics, the hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a nonsingular complex algebraic variety to its subvarieties. Aug 28, 2006 david gruber and sylvia nasar on the math worlds war over who solved the poincare conjecture. Unlike 150year old riemann hypothesis or the twin prime conjecture whose age is measured in millennia, the abc conjecture was discovered in the rather recent and mundane year of 1985. In everyday problem solving, it is very rare that a problems solution is immediately apparent.
279 896 1194 239 542 1080 1112 1413 1225 1118 1173 59 1173 488 155 916 1473 1412 1054 1178 489 1102 168 145 1092 137 188 851 660 658 483 306 1096 1082 1537 260 1485 1363 348 230 603 730