The work of Grigory Perelman

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The work of Grigory Perelman
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Laudatio on the occasion of the Fields medal award to Grigory Perelman.
Computer programming Fields Medal
Geometry Proof theory Metrischer Raum Resultant
Point (geometry) Compact space Three-dimensional space Focus (optics) Sign (mathematics) Manifold Sectional curvature Cartesian coordinate system Spacetime
Geometry Mass flow rate Orientation (vector space) Manifold Right angle Line (geometry) Nichtlineares Gleichungssystem
Logical constant Entropy Group action Evelyn Pinching Multiplication sign Orientation (vector space) Parameter (computer programming) Sectional curvature Mereology Derivation (linguistics) Riemann curvature tensor Volume Roundness (object) Negative number Descriptive statistics Position operator Isochore Compact space Process (computing) Mass flow rate Infinity 3 (number) Lattice (order) Price index Flow separation Sequence Category of being Curvature Uniformer Raum Order (biology) Mathematical singularity Resultant Geometry Point (geometry) Computer programming Three-dimensional space Finitismus Functional (mathematics) Wage labour Divisor Observational study Connectivity (graph theory) Canonical ensemble Surgery Hyperbolischer Raum Helmholtz decomposition Finite set Cylinder (geometry) Operator (mathematics) Manifold Reduction of order Theorem Nichtlineares Gleichungssystem Set theory Condition number Validity (statistics) Neighbourhood (graph theory) Content (media) Mathematical analysis Algebraic structure Volume (thermodynamics) Total S.A. Multilateration Limit (category theory) Numerical analysis Radius Doubling the cube Object (grammar) Game theory Musical ensemble Family Transverse wave Limit of a function Local ring Maß <Mathematik>
Logical constant Geometry Finitismus Graph (mathematics) Invertierbare Matrix Fields Medal Multiplication sign Mathematical analysis Volume (thermodynamics) Parameter (computer programming) Mereology Sectional curvature Hyperbolischer Raum Helmholtz decomposition Manifold Right angle
the fields Buri burial by John will he or list all pending Grigori program born in 1966 received his doctorate from St. Petersburg State University he quickly became renowned for his work in
pneumonia geometry and Alexander geometry as the latter is a form of money geometry for general metric spaces somewhere parliament's work In Alexander geometry was surveyed in his 1994 ICM talk let me just mentioned 1 of its outstanding results in me
money and geometry this is proof of these soul conjecture Edwards conjecture by she eager grow mall in 1972 and finally approved by Parliament in 1994 and a
short and striking paper the statement is it it and is a complete not compactly money amount of gold with nonnegative sectional curvature and if there is a single point were all these sectional curvature is positive and the manifold is due Margaret to Euclidean space Our main then shifted the focus a bit
research to reaching and its applications the apology of three-dimensional manifolds we call the statement of entree conjecture that simply connected compact three-dimensional manifold is Diffee more afraid to be spared a far reaching
generalization this conjecture is due to the Thurston geometric nation conjecture it says that a compact Orient all 3 dimensional manifold can be economically cut along two-dimensional spirited tore right Inter so-called geometric pieces is a a version of the
conjecture of course there is a more precise version Parliament's work on these conjectures is along these lines to Roach the reaching flow equations was introduced by Richard Hamilton in 1982 based on his
work a program of approved conjectures these conjectures using meeting was initiated but Hamilton and Shane tiny out end 2002 and 2003 comment posted 3 papers on the archive which together resent proves a big Monterrey and geometric nation conjecture his 1st paper from November 2002 is on the edge reform his 2nd paper from March 2003 is on me chief lowered surgery and 3rd paper from July 2003 is about finite extinction detailed exposition of Parliament's work have been given by souring 0 by Kleiner and myself and by Morgan Antioch To say a word about the status parliament laborers and is taking us some time to examine them this partly due to the originality of Carmen's were partly due to technical sophistication of its arguments by now several groups of people have gone through Carmen's papers in detail no serious problems have come to light of course caution is in order when discussing major conjecture and more people need to check Parliament's work before there can be any universally accepted verdict however all indications are it from its arguments are correct but they don't give a bit of background a fuller description will be and Professor Hamilton stock they Oh equation is that time derivative Of the matter is mine is twice the Ricci curvature here is 81 parameter family every Monday and about threats on a manifold and notes meeting cancer deal which can be constructed from Amman curvature tensor and in most of the talk I was to be manifold M three-dimensional compact and Oriental in years 1st paper on this subject Hamilton through the following landmark result that if they simply connected Compaq three-dimensional also has a money in with positively teacher richer than it is to be more effective Teresa is this is a version of operate conjecture except with the additional assumption that there's a metric with positive Ricci curvature they give you some idea the approval let's take our manifold with positively teacher richer and Munder Retief love then after a finite amount of time it's however if we we still have a constant volume and Hamilton show that as time goes on the manifold becomes round and round so that limit we can recognize but it is the priest Over the years many important results about were obtained by Hamilton and others let me just mention 1 more landmark result Hamilton supposedly normalized on a manifold has a smooth solutions that exist positive time and has uniformly bounded sectional curvature than the manifold satisfies the geometric Mason said this clearly showed that meaty flow the promising approach to proving the geometric conjecture there were 2 remaining issues had a deal with possible singularities in the and have to remove these assumptions about the sectional curvature about singularities 1 example is the so-called that and supposedly start with manifold reconfiguration Munder cheaper then Akira singular and a finite amount of time and the reason is that there is a truce which is inching down 4 . To deal with this Hamilton introduced the idea of surgery actually in the four-dimensional context there are which was going singular let's take a meeting of the party which is formed a singularity in this case would be in vault prosecutors let's cut that out and then it happened off by 2 3 balls to obtain a new manifold we can then continue the to flow now when we do the surgery of course we change the policy the manifold however it changes in a controllable way these neck pinch was 1 example of a singularity and went ask what are possible singularity wealth it's not that singularities come from a sectional curvature lower Hamilton introduced at least
killing method to analyze the possible singularity To illustrate this year's our manifold which was going singular that a sequence based on points with a sectional curvature they're blowing up them but we still so that after a when the sectional curvature of our base the unit and value than this case you see that we can form limit namely a cylinder are right soda summarize the idea blowup analysis is to take a convert and subsequent these rescale solutions to try to get a limiting solution this will model the formation of the singularity a basic question is whether such a limit exists if it does it'll be very special 1st live all negative time with carbon ancient solutions and secondly it has not negative curvature from a result of Hamilton United now Hamilton's compacted serum gives sufficient conditions to extract such a converted subsequent and these these rescale solutions 1 needs to thank 1st when a uniform curvature bounce on false as 2nd base when needs a uniform lower bound on the interactivity radius at the base when can get the needed curvature bound by carefully choosing these blowup points there were too many obstacles 1st have to get the needed and activity radius around and 2nd what are the possible blew limits these problems were solved by Perelman let me be discussed 3 games comments were being no local collapse and from chief with surgery and a longtime behavior Our men's 1st great food and flight was no local collapsing from from its 1st paper the precise date is written at the bottom but let me just like to summarize the content of it I suppose we have Ricci that exist for a finite time and then if we have a ball which has sectional curvature bounce that implies that we have the needed ejected 80 radius found at the center of that ball the implication is that 1 can take these blow limits To say a word about parliament method of approved he introduces do monotonic quantities for which calls W entropy and reduced body these quantities arrives From a deep new understanding the underlying structure of the Ricci flowing operation to illustrate the idea that proved let's look W functional as a function of time is 93 and however parliament shows that it if there were not that is if 1 have local collapsing and that would force the W functional to go to minus infinity which contradicts the modest visited Burma then give the following classification of possible lower limits their art some compact possibilities it could be a finite of the round making or it could be a few more effective or objectives for the not compact possibilities it could be Iran shrinking cylinder or it's easy to close or it could be did you more for 2 or 3 days and after scaling each time slice looks like Ed and senators Our meant then canonical neighborhood theorem which says that any region of high scalar curvature is modeled after scaling by 1 of these lower limits let me now past meeting slow surgery there are 2 main issues here 1st defined the along which 1 wants to cut and a data show that these surgery times do not accumulate if the surgery times accumulated 1 might never get time for went recognize the quality of the manifold here's a picture what manifold might look like at the 1st singularity there's a region called Omega With a scalar curvature is not too big coming off this there is a finite number of so-called epsilon horns and then there could be other connected components such as this picture so-called double epsilon haunts there is a our amends surgery procedure be take each of these Absolon horns you take a point within it and you rescale To get something which looks cylindrical then you slices along a transverse to spare and blue and 3 Reebok now I want to do this at the 1st singularity time it here is the main problem at later singularity times we saw declined to spirit along with we're gonna come In order to do this we still need to know the validity of the canonical neighborhood there antinode local collapsing on the problem is the earliest surgeries invalidate the truth at least their arms let me just described 1 ingredient of Parliament's solution to this problem it is to perform the surgery deep within these epsilon next garment shows that if 1 does want and that during surgery In the fact I'm very long as is illustrated here so to summarize parliamentary through the following technically difficult surgery 1 can choose me surgery parameters so that there is a well-defined we surgery that exist for all time and in particular has only a finite number surgeries any finite time and now there could be a neighbor number of total surgeries want comets inside 1 extract topological consequences nevertheless each so finally let me say something about long-time behavior of the chief of the 1st His Wendy starting manifold is simply connected In this case we find extinction time around says that after for nighttime there's nothing left b remaining manifold is just the empty set as a consequence the manifold is a connected some of the various pieces disappeared in other words it's eclectic some of standard pieces namely quotient of around 3 or as 1 process to back but were similarly original manifold simply enacted a policy that market Reese which is operated Thatcher Taylor now In the general case the study manifold may not be simply connected in order to see long-time behavior the Ricci flower let's divide time to met by these factors and let we say that acts as a factor component of the time t manifold here is the desired picture for acts we would like to say that it has incompressible to rely so that if we can't among those to try he remaining pieces army the hyperbolic or graft at here hyperbolic means it admits a finite volume band finite volumetric of constant negative sectional curvature and the other hand wrap manifolds a very special type of 3 manifolds In order to achieve this Parliament introduces I think the decomposition I won't give their precise definition but the acts as a properties that it is locally volume not collapsed and has local two-sided sectional curvature about on the other hand then part of X is locally by collapsed and has a local lower sectional curvature each the the statement
about the the party is that it becomes hyperbolic that is from large time they part artifacts approach is that they are part of a finite volume manifold of constant sectional curvature minus 1 4 and in addition because right if there are any are incompressible annex the privileges is based partly on arguments from Hamilton's nonsingular paper for the party the statement is for large time is a graph manifold putting this together the abstract Is it the original manifold and is connected some pieces at each other a hyperbolic graph decomposition this implies geometric they should conjecture the inconclusive Gregory torment has revolutionized the fields geometry and policy His work at flood is a spectacular achievement in geometric analysis garment makers show profound originality and enormous technical skills we were certainly exploring comments ideas for many years to come thank you I saw