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285G, Lecture 19: The structure of Ricci flow at the singular time, surgery, and the Poincaré conjecture

In the previous lecture, we studied high curvature regions of Ricci flows on some time interval , and concluded that (as long as a mild topological condition was obeyed) they all had canonical neighbo… more →

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285G, Lecture 19: The structure of Ricci flow at the singular time, surgery, and the Poincaré conjecture — 9 comments

Terence Tao wrote 1 year ago: In the previous lecture, we studied high curvature regions of Ricci flows on some time interval , an … more →

Tags: math.DG, math.AT, Surgery, canonical neighbourhoods, standard solution, horns

285G, Lecture 18: The structure of high-curvature regions of Ricci flow — 1 comment

Terence Tao wrote 1 year ago: Having characterised the structure of -solutions, we now use them to describe the structure of high … more →

Tags: math.DG, canonical neighbourhoods, concentration and compactness, geometric limits, necks, Ricci flows, soul theorem

285G, Lecture 17: The structure of κ-solutions — 2 comments

Terence Tao wrote 1 year ago: Having classified all asymptotic gradient shrinking solitons in three and fewer dimensions in the pr … more →

Tags: math.DG, compactness and contradiction, geometric limits, gradient shrinking solitons, kappa-solutions, necks

285G, Lecture 16: Classification of asymptotic gradient shrinking solitons — 1 comment

Terence Tao wrote 1 year ago: In the previous lecture, we showed that every -solution generated at least one asymptotic gradient s … more →

Tags: math.DG, gradient shrinking solitons, geometric limits, Cheeger-Gromoll splitting theorem, Toponogov theory, Hamilton's rounding theorem

285G, Lecture 15: Geometric limits of Ricci flows, and asymptotic gradient shrinking solitons — 13 comments

Terence Tao wrote 1 year ago: We now begin using the theory established in the last two lectures to rigorously extract an asymptot … more →

Tags: math.AP, math.CA, math.DG, Distributions, geometric limits, gradient shrinking solitons, kappa-solutions, parabolic regularity

285G, Lecture 14: Stationary points of Perelman entropy or reduced volume are gradient shrinking solitons — 4 comments

Terence Tao wrote 1 year ago: We continue our study of -solutions. In the previous lecture we primarily exploited the non-negative … more →

Tags: math.DG, gradient shrinking solitons, Perelman entropy, reduced volume, Ricci flow

285G, Lecture 13: Li-Yau-Hamilton Harnack inequalities and κ-solutions — 6 comments

Terence Tao wrote 1 year ago: We now turn to the theory of parabolic Harnack inequalities, which control the variation over space … more →

Tags: math.AP, Harnack inequalities, kappa-solutions, Ricci flow, strong maximum principle

285G, Lecture 12: High curvature regions of Ricci flow and κ-solutions — 9 comments

Terence Tao wrote 1 year ago: In previous lectures, we have established (modulo some technical details) two significant components … more →

Tags: math.DG, canonical neighbourhoods, compactness and contradiction, gradient shrinking solitons, kappa-solutions

285G, Lecture 11: κ-noncollapsing via Perelman reduced volume — 9 comments

Terence Tao wrote 1 year ago: Having established the monotonicity of the Perelman reduced volume in the previous lecture (after fi … more →

Tags: math.DG, Entropy, non-collapsing, reduced volume, Ricci flow

285G, Lecture 10: Variation of L-geodesics, and monotonicity of Perelman reduced volume — 12 comments

Terence Tao wrote 1 year ago: Having completed a heuristic derivation of the monotonicity of Perelman reduced volume (Conjecture 1 … more →

Tags: math.DG, Bishop-Gromov comparison inequality, geodesics, Perelman reduced length, Perelman reduced volume, variation formulae

285G, Lecture 9: Comparison geometry, the high-dimensional limit, and Perelman reduced volume — 19 comments

Terence Tao wrote 1 year ago: We now turn to Perelman’s second scale-invariant monotone quantity for Ricci flow, now known a … more →

Tags: math.DG, Bishop-Gromov inequality, high-dimensional limit, Perelman reduced length, Perelman reduced volume

285G, Lecture 8: Ricci flow as a gradient flow, log-Sobolev inequalities, and Perelman entropy — 9 comments

Terence Tao wrote 1 year ago: It is well known that the heat equation (1) on a compact Riemannian manifold (M,g) (with metric g st … more →

Tags: math.AP, math.CA, math.DG, gradient flow, least eigenvalue, log-Sobolev inequality, Nash entropy, non-collapsing, Perelman entropy

285G, Lecture 7: Rescaling of Ricci flows and κ-noncollapsing — 13 comments

Terence Tao wrote 1 year ago: We now set aside our discussion of the finite time extinction results for Ricci flow with surgery (T … more →

Tags: math.AP, math.DG, coercivity, comparison geometry, Criticality, noncollapsing, scale invariance

285G, Lecture 6: Finite time extinction of the third homotopy group, II — 4 comments

Terence Tao wrote 1 year ago: In this lecture we discuss Perelman’s original approach to finite time extinction of the third … more →

Tags: math.AP, math.DG, curvature, curve shortening flow, minimal disk, Perelman width, ramps, Ricci flow

285G, Lecture 5: Finite time extinction of the third homotopy group, I — 11 comments

Terence Tao wrote 1 year ago: In the previous lecture, we saw that Ricci flow with surgery ensures that the second homotopy group … more →

Tags: math.DG, math.AT, Hurewicz theorem, homotopy sphere, homotopy group, Colding-Minicozzi, minimal surface, min-max functional

285G, Lecture 4: Finite time extinction of the second homotopy group — 13 comments

Terence Tao wrote 1 year ago: Returning (perhaps anticlimactically) to the subject of the Poincaré conjecture, recall from Lecture … more →

Tags: math.DG, math.AT, Ricci flow, second fundamental group, Gauss curvature, minimal surfaces, Sacks-Uhlenbeck theory, monotonicity formula

285G, Lecture 3: The maximum principle, and the pinching phenomenon — 25 comments

Terence Tao wrote 1 year ago: We now begin the study of (smooth) solutions to the Ricci flow equation , (1) particularly for compa … more →

Tags: math.AP, math.DG, convexity, maximum principle, pinching phenomenon, Ricci flow, Riemann curvature, tensor bundles

285G, Lecture 2: The Ricci flow approach to the Poincaré conjecture — 15 comments

Terence Tao wrote 1 year ago: In order to motivate the lengthy and detailed analysis of Ricci flow that will occupy the rest of th … more →

Tags: math.GT, math.DG, math.AT, Poincaré conjecture, 3-manifolds, spherical space form, Surgery, geometrization conjecture

285G, Lecture 1: Flows on Riemannian manifolds — 36 comments

Terence Tao wrote 1 year ago: In the first lecture, we introduce flows on Riemannian manifolds , which are recipes for describing … more →

Tags: math.AP, math.DG, de Turck trick, diffeomorphisms, Ricci flow, variation formulae

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