Dec 03, 2006 hamiltonperelmans proof of the poincare conjecture and the geometrization conjecture. However, i have critical questions about perelman s proof of poincare conjecture. Most of the project is based on the book by chow and knopf 6, the notes by peter topping 28 which have recently been made into a book, see 29, the papers of richard hamilton in. Aug 05, 2019 after solving the poincare conjecture, perelman ran away from the recognition and fortune that came with it. For example, the following is a picture of one of his lecture in us. The caozhu team is one of three teams formed for this purpose. In this paper, we shall present the hamilton perelman theory of. The international congress of mathematicians is the most significant meeting in pure and applied mathematics, and one of the oldest scientific congresses. This work depends on the accumulative works of many geometric analysts. Among his notable achievements was a short and elegant proof of the soul conjecture. It should be noted that the piecewise linear or di. Introduction the topology of twodimensional manifolds or surfaces was well understood in the 19th century. In 54 perelman states the result for 3manifolds without prime factors that are acyclic.
Zhang yitang zhang, born 1955 was born in china and is now a professor of mathematics at the university of california. Every compact bound e aryless simply connected 3dimensional manifold is homeomor phic to the 3dimensional unit sphere s3. In particular, we give a detailed exposition of a complete proof of the poincare conjecture due to hamilton and perelman. As of 2020, the poincare conjecture is the only solved millennium problem. The clay mathematics institute clay mathematics institute. Statement of perelman s collapsing theorem and the main di culties in its proof. During his stay in berkeley, perelman learned about the ricci ow project from hamilton.
It can be generalised to all dimensions and was proved for n 2 in the 1920s, for n. In 2003, the russian mathematician grigori perelman. Recently, grisha perelman announced a proof of the conjecture in fact of the stronger geometrization conjecture of william thurston from 1982, which states. I cant find any videos of lectures that grigori perelman gave for his proof of poincare s conjecture. And at the same time just a humble resident of a small apartment in the southern outskirts of st.
Perelman poincare pdf in this paper, we shall present the hamiltonperelman theory of. Grigory perelman and a breakthrough in mathematics. In technical terms the conjecture is that if a space is homotopically equivalent to a threedimensional sphere it is homeomorphic to the threesphere. In fact there is a simple list of all possible smooth compact orientable surfaces. The closure of g, written g, is the intersection of all closed sets that fully contain g. Recently, there has been much interest in the derivation of projective, copeanosmale, naturally fclifford primes. In a series of three papers in 20022003, grisha perelman solved the famous. Perelman also made contributions to riemannian geometry and geometric topology, and the poincare conjecture is still the only one of the seven millennium prize problems to have been solved. This has been one of the most famous open problems in mathematics and has been around for about one hundred years. Then, perelman cut his all contacts with the world and started living lonesome life in his mothers apartment in russia. Perelmans proof is quite intricate and uses an induction on the time interval. Very recently, a book 7 containing complete and mostly selfcontained proofs of the poincare. A purported new mathematics proof is impenetrable now what.
However, a variant of the argument shows that any limiting. Grigory perelman and a breakthrough in mathematics discover. In this paper, we present a simple proof of perelman s collapsing theorem for 3manifolds cf. Notes on perelmans paper on the entropy formula for the.
Morgantian have written out an almost 500 page detailed proof of the poincare conjecture providing the background and details for perelman s proof of the poincare conjecture. It has taken two years for the mathematical community to assimilate perelman s ideas and expand his preprints into complete proofs. Hamilton, perelman and the poincare conjecture lehman. This proof should be considered as the crowning achievement of the hamilton perelman theory of ricci flow.
Perelmans proof of the poincare conjecture terence tao. This next meeting is going to happen in saint petersburg, russia in 2022. In this paper, the absence of a coe cient group from the homology or cohomology group notation means that it is implicitly z unless a ring rhas been speci ed. Grigori perelman solved worlds greatest mystery then disappeared.
Perelman and the poincare conjecture not even wrong. Every simply connected, closed 3manifold is homeomorphic to the 3sphere. In this paper, we provide an essentially selfcontained and detailed account of the fundamental works of hamilton and the recent breakthrough of perelman on the ricci flow and their application to the geometrization of threemanifolds. However, as a warm up, ill begin with the simpler and more classical theory of twodimensional manifolds, i. Until the autumn of 2002, perelman was best known for his work in comparison theorems in riemannian geometry. Perelman s proof uses a modified version of a ricci flow program developed by richard s. We discuss some of the key ideas of perelman s proof of poincares conjecture via the hamilton program of using the ricci flow, from the perspective of the modern theory of nonlinear partial differential equations. Perelman s work, although the poincare conjecture is the central question in my special. A lot of lectures have been asked in mse, i hope i wont get closing vote. Dare chain homotopic denoted by fg if there exists a sequence of maps t ft n.
All elementary mathematics online mathematical school. In this paper, we give a complete proof of the poincar. In mathematics, the poincare conjecture is a theorem about the characterization of the 3sphere several teams of mathematicians verified that perelmans proof was correct. Perelman poincare pdf perelman poincare pdf perelman poincare pdf download. The poincare conjecture, proposed by french mathematician henri poincare in 1904, was the most famous open problem in topology. Perelmans proof of the poincare conjecture is wrong, my own. To complete geometrization, in his second paper, perelman develops a. Regardless of his desire to live a reclusive lifestyle, mathematicians awarded him the fields medal in 2006, which is the most sought after prize in mathematics. Muhammad alkhwarizmi timeline of mathematics mathigon. In mathematics, the poincare conjecture is a theorem about the characterization of the 3sphere, which is the hypersphere that bounds the unit ball in fourdimensional space. Perelman introduced a series of highly original ideas and powerful techniques to complete hamiltons programme. So far most of the sections 111 of 12 is covered in our notes. May 01, 2014 grigory perelman a superstar of modern mathematics, the author of the proof of the poincare s theorem. Right now, mathematicians are boycotting icm in russia due to the governments treatment of its political prisoners one of the most outrageous cases is that of azat.
Grigori perelman solved worlds greatest mystery then. The work of grigory perelman university of california, berkeley. Perelman was born in 1966 and received his doctorate from st. Perelmans proof of the poincare conjecture is wrong, my. Introduction three dimensions the positivity of curvature was used in an essential way in conjunction with a sophisticated version of the maximum principle. Based on it, we shall give the first written account of a.
The best summary ive read of perelmans proof of the. The poincare conjecture states that a homotopy 3sphere is a genuine 3sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence. In particular, we will not discuss the important and quite different results on this conjecture in four and higher dimensions by smale, freedman, etc.
They initially implied the proof was their own achievement based on the hamilton perelman theory, but later retracted the original version of their paper, and posted a revised version, in which they referred to their work as the more modest exposition of hamilton perelman s proof. To complete geometrization, in his second paper, perelman develops a clever thickthin decomposition of the manifold. Terence tao perelman s proof of the poincare conjecture. Zhu xiping born 1962 in shixing, guangdong is a chinese mathematician. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The publics view of the mathematician as a reclusive genius toiling away for years at. Dare chain homotopic denoted by fg if there exists a sequence of. This conjecture has been near the center of a maelstrom of activity in topology, geometry, analysis, and many allied and subdisciplines for a hundred years. Hamiltonperelmans proof of the poincare conjecture and the geometrization conjecture. Hamiltons fundamental idea is to formulate a dynamical process in which a given threemanifold is geometrically distorted, with the distortion process governed by a differential equation analogous to the heat. These are notes on perelman s paper the entropy formula for the ricci.
I was studying his papers so i thought maybe i can learn much more through his lectures. One of the great stories of mathematics in recent years has been the proof of the poincare conjecture by grisha perelman. Perelman, the ricci flow and the poincare conjecture. Now a reclusive yet respected japanese mathematician has put forth a solution to another notorious problem. Dec 22, 2015 grigori perelman s proof in 2003 of the poincare conjecture comes to mind as well. He followed this up by answering questions in detail on a lecture tour of the us, and confirmation of the validity of his proof is now emerging from various research groups around the world. The goal of this notes is to give more details to some theorems and arguments that appear in perelman s paper. May 03, 2014 a few years ago a brilliant mathematician grigori perelman solved a very famous 100 years old problem in topology called poincare conjecture. Hamiltonperelmans proof of the poincare conjecture and the. Perelmans wfunctional on manifolds with conical singularities. Zhu are based principally on the first two papers of perelman and follow his proof of the geometrization conjecture, with.
The proof requires only slight adaptation of the arguments presented here. The millennium prize problems are seven problems in. Every compact, connected 3manifold can be cut along embedded spheres and tori into pieces each of which has a geometric structure from the eightfold path. The actual solution was not found until grigori perelman published his papers. Genius of grigory perelman perelman studied at the mathematical school and already then showed an unprecedented mathematical mind. The poincare conjecture clay mathematics institute. Poincare conjecture, perelman way, and topology of social. The work of grigory perelman john lott grigory perelman has been awarded the fields medal for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the ricci.
1756 1532 215 1231 987 1276 579 1185 954 161 1177 1116 663 626 1556 345 1751 1585 895 1742 505 1503 1198 1749 967 80 1016 1720 1365 1271 80 345 1416 563 760 966 345 108 1358