The Differences of Ricci Flow Between Perelman and Hamilton About the Poincaré Conjecture and the Geometrization Conjecture Of
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Millennium Prize for the Poincaré
FOR IMMEDIATE RELEASE • March 18, 2010 Press contact: James Carlson: [email protected]; 617-852-7490 See also the Clay Mathematics Institute website: • The Poincaré conjecture and Dr. Perelmanʼs work: http://www.claymath.org/poincare • The Millennium Prizes: http://www.claymath.org/millennium/ • Full text: http://www.claymath.org/poincare/millenniumprize.pdf First Clay Mathematics Institute Millennium Prize Announced Today Prize for Resolution of the Poincaré Conjecture a Awarded to Dr. Grigoriy Perelman The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads: The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude. The award of the Millennium Prize to Dr. Perelman was made in accord with their governing rules: recommendation first by a Special Advisory Committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles), then by the CMI Scientific Advisory Board (James Carlson, Simon Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with final decision by the Board of Directors (Landon T. -
The Work of Grigory Perelman
The work of Grigory Perelman John Lott Grigory Perelman has been awarded the Fields Medal for his contributions to geom- etry and his revolutionary insights into the analytical and geometric structure of the Ricci flow. Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture. Soul Conjecture (conjectured by Cheeger–Gromoll [2] in 1972, proved by Perelman [19] in 1994). 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, Perelman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints [21], [22], [23] posted on the arXiv in 2002–2003, Perelman presented proofs of the Poincaré conjecture and the geometrization conjecture. The Poincaré conjecture dates back to 1904 [24]. 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 Poin- caré conjecture. It says that any closed orientable 3-dimensional manifold can be canonically cut along 2-spheres and 2-tori into “geometric pieces” [27]. -
A Survey of the Elastic Flow of Curves and Networks
A survey of the elastic flow of curves and networks Carlo Mantegazza ∗ Alessandra Pluda y Marco Pozzetta y September 28, 2020 Abstract We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In par- ticular, we give a complete proof of global existence and smooth convergence to critical points of the solution of the elastic flow of closed curves in R2. In the last section of the paper we also discuss a list of open problems. Mathematics Subject Classification (2020): 53E40 (primary); 35G31, 35A01, 35B40. 1 Introduction The study of geometric flows is a very flourishing mathematical field and geometric evolution equations have been applied to a variety of topological, analytical and physical problems, giving in some cases very fruitful results. In particular, a great attention has been devoted to the analysis of harmonic map flow, mean curvature flow and Ricci flow. With serious efforts from the members of the mathematical community the understanding of these topics grad- ually improved and it culminated with Perelman’s proof of the Poincare´ conjecture making use of the Ricci flow, completing Hamilton’s program. The enthusiasm for such a marvelous result encouraged more and more researchers to investigate properties and applications of general geometric flows and the field branched out in various different directions, including higher order flows, among which we mention the Willmore flow. In the last two decades a certain number of authors focused on the one dimensional analog of the Willmore flow (see [26]): the elastic flow of curves and networks. -
Non-Singular Solutions of the Ricci Flow on Three-Manifolds
COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 7, Number 4, 695-729, 1999 Non-singular solutions of the Ricci flow on three-manifolds RICHARD S. HAMILTON Contents. 1. Classification 695 2. The lower bound on scalar curvature 697 3. Limits 698 4. Long Time Pinching 699 5. Positive Curvature Limits 701 6. Zero Curvature Limits 702 7. Negative Curvature Limits 705 8. Rigidity of Hyperbolic metrics 707 9. Harmonic Parametrizations 709 10. Hyperbolic Pieces 713 11. Variation of Area 716 12. Bounding Length by Area 720 1. Classification. In this paper we shall analyze the behavior of all non-singular solutions to the Ricci flow on a compact three-manifold. We consider only essential singularities, those which cannot be removed by rescaling alone. Recall that the normalized Ricci flow ([HI]) is given by a metric g(x,y) evolving 695 696 Richard Hamilton by its Ricci curvature Rc(x,y) with a "cosmological constant" r = r{t) representing the mean scalar curvature: !*<XfY) = 2 ^rg{X,Y) - Rc(X,Y) where -h/h- This differs from the unnormalized flow (without r) only by rescaling in space and time so that the total volume V = /1 remains constant. Definition 1.1. A non-singular solution of the Ricci flow is one where the solution of the normalized flow exists for all time 0 < t < oo, and the curvature remains bounded |jRra| < M < oo for all time with some constant M independent of t. For example, any solution to the Ricci flow on a compact three-manifold with positive Ricci curvature is non-singular, as are the equivariant solutions on torus bundles over the circle found by Isenberg and the author [H-I] which has a homogeneous solution, or the Koiso soliton on a certain four-manifold [K]; by contrast the solutions on a four-manifold with positive isotropic curvature in [H5] definitely become singular, and these singularities must be removed by surgery. -
Ricci Flow and Nonlinear Reaction--Diffusion Systems In
Ricci Flow and Nonlinear Reaction–Diffusion Systems in Biology, Chemistry, and Physics Vladimir G. Ivancevic∗ Tijana T. Ivancevic† Abstract This paper proposes the Ricci–flow equation from Riemannian geometry as a general ge- ometric framework for various nonlinear reaction–diffusion systems (and related dissipative solitons) in mathematical biology. More precisely, we propose a conjecture that any kind of reaction–diffusion processes in biology, chemistry and physics can be modelled by the com- bined geometric–diffusion system. In order to demonstrate the validity of this hypothesis, we review a number of popular nonlinear reaction–diffusion systems and try to show that they can all be subsumed by the presented geometric framework of the Ricci flow. Keywords: geometrical Ricci flow, nonlinear bio–reaction–diffusion, dissipative solitons and breathers 1 Introduction Parabolic reaction–diffusion systems are abundant in mathematical biology. They are mathemat- ical models that describe how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are converted into each other, and diffusion which causes the substances to spread out in space. More formally, they are expressed as semi–linear parabolic partial differential equations (PDEs, see e.g., [55]). The evolution of the state vector u(x,t) describing the concentration of the different reagents is determined by anisotropic diffusion as well as local reactions: ∂tu = D∆u + R(u), (∂t = ∂/∂t), (1) arXiv:0806.2194v6 [nlin.PS] 19 May 2011 where each component of the state vector u(x,t) represents the concentration of one substance, ∆ is the standard Laplacian operator, D is a symmetric positive–definite matrix of diffusion coefficients (which are proportional to the velocity of the diffusing particles) and R(u) accounts for all local reactions. -
Mathematics Opportunities
Mathematics Opportunities will be published by the American Mathematical Society, NSF Integrative Graduate by the Society for Industrial and Applied Mathematics, or jointly by the American Statistical Association and the Education and Research Institute of Mathematical Statistics. Training Support is provided for about thirty participants at each conference, and the conference organizer invites The Integrative Graduate Education and Research Training both established researchers and interested newcomers, (IGERT) program was initiated by the National Science including postdoctoral fellows and graduate students, to Foundation (NSF) to meet the challenges of educating Ph.D. attend. scientists and engineers with the interdisciplinary back- The proposal due date is April 8, 2005. For further grounds and the technical, professional, and personal information on submitting a proposal, consult the CBMS skills needed for the career demands of the future. The website, http://www.cbms.org, or contact: Conference program is intended to catalyze a cultural change in grad- Board of the Mathematical Sciences, 1529 Eighteenth Street, uate education for students, faculty, and universities by NW, Washington, DC 20036; telephone: 202-293-1170; establishing innovative models for graduate education in fax: 202-293-3412; email: [email protected] a fertile environment for collaborative research that tran- or [email protected]. scends traditional disciplinary boundaries. It is also intended to facilitate greater diversity in student participation and —From a CBMS announcement to contribute to the development of a diverse, globally aware science and engineering workforce. Supported pro- jects must be based on a multidisciplinary research theme National Academies Research and administered by a diverse group of investigators from U.S. -
AROUND PERELMAN's PROOF of the POINCARÉ CONJECTURE S. Finashin After 3 Years of Thorough Inspection, Perelman's Proof of Th
AROUND PERELMAN'S PROOF OF THE POINCARE¶ CONJECTURE S. Finashin Abstract. Certain life principles of Perelman may look unusual, as it often happens with outstanding people. But his rejection of the Fields medal seems natural in the context of various activities followed his breakthrough in mathematics. Cet animal est tr`esm¶echant, quand on l'attaque, il se d¶efend. Folklore After 3 years of thorough inspection, Perelman's proof of the Poincar¶eConjecture was ¯nally recognized by the consensus of experts. Three groups of researchers not only veri¯ed its details, but also published their own expositions, clarifying the proofs in several hundreds of pages, on the level \available to graduate students". The solution of the Poincar¶eConjecture was the central event discussed in the International Congress of Mathematicians in Madrid, in August 2006 (ICM-2006), which gave to it a flavor of a historical congress. Additional attention to the personality of Perelman was attracted by his rejec- tion of the Fields medal. Some other of his decisions are also non-conventional and tradition-breaking, for example Perelman's refusal to submit his works to journals. Many people ¯nd these decisions strange, but I ¯nd them logical and would rather criticize certain bad traditions in the modern organization of science. One such tradition is a kind of tolerance to unfair credit for mathematical results. Another big problem is the journals, which are often transforming into certain clubs closed for aliens, or into pro¯table businesses, enjoying the free work of many mathemati- cians. Sometimes the role of the \organizers of science" in mathematics also raises questions. -
The Mean Curvature Flow of Submanifolds of High Codimension
The mean curvature flow of submanifolds of high codimension Charles Baker November 2010 arXiv:1104.4409v1 [math.DG] 22 Apr 2011 A thesis submitted for the degree of Doctor of Philosophy of the Australian National University For Gran and Dar Declaration The work in this thesis is my own except where otherwise stated. Charles Baker Abstract A geometric evolution equation is a partial differential equation that evolves some kind of geometric object in time. The protoype of all parabolic evolution equations is the familiar heat equation. For this reason parabolic geometric evolution equations are also called geometric heat flows or just geometric flows. The heat equation models the physical phenomenon whereby heat diffuses from regions of high temperature to regions of cooler temperature. A defining characteris- tic of this physical process, as one readily observes from our surrounds, is that it occurs smoothly: A hot cup of coffee left to stand will over a period of minutes smoothly equilibrate to the ambient temperature. In the case of a geometric flow, it is some kind of geometric object that diffuses smoothly down a driving gradient. The most natural extrinsically defined geometric heat flow is the mean curvature flow. This flow evolves regions of curves and surfaces with high curvature to regions of smaller curvature. For example, an ellipse with highly curved, pointed ends evolves to a circle, thus minimising the distribution of curvature. It is precisely this smoothing, energy- minimising characteristic that makes geometric flows powerful mathematical tools. From a pure mathematical perspective, this is a useful property because unknown and complicated objects can be smoothly deformed into well-known and easily understood objects. -
CHAPTER 6: RICCI FLOW Contents 1. the Hamilton–Perelman Program
CHAPTER 6: RICCI FLOW DANNY CALEGARI Abstract. These are notes on Ricci Flow on 3-Manifolds after Hamilton and Perelman, which are being transformed into Chapter 6 of a book on 3-Manifolds. These notes are based on a graduate course taught at the University of Chicago in Fall 2019. Contents 1. The Hamilton–Perelman program1 2. Mean curvature flow: a comparison8 3. Curvature evolution and pinching 15 4. Singularities and Limits 33 5. Perelman’s monotone functionals 41 6. The Geometrization Conjecture 54 7. Acknowledgments 61 References 61 1. The Hamilton–Perelman program In this section we give a very informal overview of the Hamilton–Perelman program proving the Poincaré Conjecture and the Geometrization Conjecture for 3-manifolds. 1.1. What is Ricci flow? There’s lots of different ways to answer this question, depending on the point you want to emphasize. There’s no getting around the precision and economy of a formula, but for now let’s see how far we can get with mostly words. First of all, what is curvature? To be differentiable is to have a good linear approximation at each point: the derivative. To be smooth is for successive derivatives to be themselves differentiable; for example, the deviation of a smooth function from its derivative has a good quadratic approximation at each point: the Hessian. A Riemannian manifold is a space which is Euclidean (i.e. flat) to first order. Riemannian manifolds are smooth, so there is a well-defined second order deviation from flatness, and that’s Curvature. In Euclidean space of dimension n a ball of radius r has volume rn times a (dimension dependent) constant. -
Annual Report 1996–1997
Department of Mathematics Cornell University Annual Report 1996Ð97 Year in Review: Mathematics Instruction and Research Dedication Wolfgang H. J. Fuchs This annual report is dedicated to the memory of Wolfgang H. J. Fuchs, professor emeritus of mathematics at Cornell University, who died February 24, 1997, at his home. Born on May 19, 1915, in Munich, Germany, Wolfgang moved to England in 1933 and enrolled at Cambridge University, where he received his B.A. in 1936 and his Ph.D. in 1941. He held various teaching positions in Great Britain between 1938 and 1950. During the 1948Ð49 academic year, he held a visiting position in the Mathematics Department at Cornell University. He returned to the Ithaca community in 1950 as a permanent member of the department, where he continued to work vigorously even after his ofÞcial retirement in 1985. He was chairman of the Mathematics Department from 1969 to 1973. His academic honors include a Guggenheim Fellowship, a Fulbright-Hays Research Fellowship, and a Humboldt Senior Scientist award. Wolfgang was a renowned mathematician who specialized in the classical theory of functions of one complex variable, especially Nevanlinna theory and approximation theory. His fundamental discoveries in Nevanlinna theory, many of them the product of an extended collaboration with Albert Edrei, reshaped the theory and have profoundly inßuenced two generations of mathematicians. His colleagues will remember him both for his mathematical achievements and for his remarkable good humor and generosity of spirit. He is survived by his wife, Dorothee Fuchs, his children Ñ Annie (Harley) Campbell, John Fuchs and Claudia (Lewis McClellen) Fuchs Ñ and by his grandchildren: Storn and Cody Cook, and Lorenzo and Natalia Fuchs McClellen. -
LETTERS to the EDITOR Responses to ”A Word From… Abigail Thompson”
LETTERS TO THE EDITOR Responses to ”A Word from… Abigail Thompson” Thank you to all those who have written letters to the editor about “A Word from… Abigail Thompson” in the Decem- ber 2019 Notices. I appreciate your sharing your thoughts on this important topic with the community. This section contains letters received through December 31, 2019, posted in the order in which they were received. We are no longer updating this page with letters in response to “A Word from… Abigail Thompson.” —Erica Flapan, Editor in Chief Re: Letter by Abigail Thompson Sincerely, Dear Editor, Blake Winter I am writing regarding the article in Vol. 66, No. 11, of Assistant Professor of Mathematics, Medaille College the Notices of the AMS, written by Abigail Thompson. As a mathematics professor, I am very concerned about en- (Received November 20, 2019) suring that the intellectual community of mathematicians Letter to the Editor is focused on rigor and rational thought. I believe that discrimination is antithetical to this ideal: to paraphrase I am writing in support of Abigail Thompson’s opinion the Greek geometer, there is no royal road to mathematics, piece (AMS Notices, 66(2019), 1778–1779). We should all because before matters of pure reason, we are all on an be grateful to her for such a thoughtful argument against equal footing. In my own pursuit of this goal, I work to mandatory “Diversity Statements” for job applicants. As mentor mathematics students from diverse and disadvan- she so eloquently stated, “The idea of using a political test taged backgrounds, including volunteering to help tutor as a screen for job applicants should send a shiver down students at other institutions. -
New Geometric Flows
National Science Review REVIEW 5: 534–541, 2018 doi: 10.1093/nsr/nww053 Advance access publication 12 September 2016 MATHEMATICS New geometric flows Gang Tian1,2 ABSTRACT Downloaded from https://academic.oup.com/nsr/article/5/4/534/2615238 by guest on 27 September 2021 This is an expository paper on geometric flows. I will start with a brief tour on Ricci flow, thenIwill discuss two geometric flows introduced in my joint work with J. Streets and some recent workson them. Keywords: Ricci flow, metrics, AMMP, Kahler,¨ Hermitian, symplectic RICCI FLOW AND ITS APPLICATION First, we recall that a metric g is Einstein if Ric(g) = λ g, where λ is a constant, often nor- We will assume that M is a closed manifold of dimen- malized to be −(n − 1), 0 or n − 1. Note sion n and g is fixed. The Ricci flow was introduced 0 that Ric(g) = (R ) denotes the Ricci curvature by Hamilton in [1]: ij of g and measures the deviation of volume form from the Euclidean one. In dimension 2, an Ein- ∂g stein metric is simply a metric of constant Gauss ij =−2R , g (0) = a given metric. ∂t ij curvature. (1) Now we assume that M is a closed 3-manifold. It Hamilton and later DeTurck proved that for any ini- is known that any Einstein metric on a 3-manifold tial metric there is a unique solution g(t)onM × [0, is of constant sectional curvature. Therefore, it fol- T)forsomeT > 0. Hamilton also established an an- lows from the classical uniformization theorem that alytic theory for Ricci flow.