MATH 669--TOPICS IN GEOMETRIC ANALYSIS--SPRING 2025--A. FREIRE

Syllabus (includes references)


Topics and references


Course log and notes:

Tu 1/21  Ricci-de Turck flow, Part I
Ref: [TOP] Ch 2, 4, 5

Th 1/23 Decomposition of Einstein equations/Initial data sets, constraint equations
Ref: Constraint equations notes

Tu 1/28 Ricci-de Turck flow, Part 2
Ref: [TOP], Ch. 5, [BRE], 2.2

Th 1/30 Intro to causality in Lorentzian manifolds.
Ref:[ON], Ch. 14

Tu 2/4 Evolution of curvature under RF, blowup criterion
Ref:[TOP], Ch. 3, 5

Th 2/6 Causality: global hyperbolicity, existence of timelike geodesics
Ref: [ON] Ch. 14

Tu 2/11 Higher derivative estimates (statement)/ reaction-diffusion eqn for curvature
Ref: [CK], p200-207; [TOP] p. 52-54/ [TOP] Prop 2.3.5 and 3.4.1; [BRE] p. 21-26

Th 2/13 Causal curves are locally Lip; convergence theorems via Ascoli/ Jacobi fields and timelike index theory
of geodesics/ Geodesics normal to a submanifold P, P-Jacobi fields, existence of focal points from curvature bounds
(Ref: [ON, Ch. 10]

Tu 2/18 Uhlenbeck's trick (NB)/Tangent and normal to convex sets
Ref: [BRE, Ch. 5]

Th 2/20 Causality theory: achronal hypersurfaces, Cauchy development
Theorem: the interior of the Cauchy development (of an achronal hypersurface) is globally hyperbolic

Tu 2/25 Hamilton's maximum principle (CF)
Ref: [BRE, Ch.5]

Th 2/27 Causality theory: spacelike hypersurfaces-Cauchy development is open, existence of maximizing normal
geodesics; edgeless, connected spacelike hypersurfaces in simply-connected spacetimes separate and are achronal
(Ref: [ON, Ch. 14])

Tu 3/4 Higher derivative estimates for Ricci flow: curvature tensor and others

Th 3/6 Future Cauchy hypersurfaces; Hawking's timelike incompleteness theorem

Tu 3/11: Hamilton's convergence theorem, part I: sectional curvature becomes constant at blowup time.
Curvature tensor and oscillation of sectional curvature

Th 3/13: Penrose incompleteness theorem, part I: geometry of null hypersurfaces (TI)

Tu 3/18, Th 3/20: SPRING BREAK

Tu 3/25: Hamilton's convergence theorem, part II: end of proof (GB)

Th 3/27 Penrose incompleteness theorem, part II: statement and proof.

Tu 4/1 Curvature pinching in dimension 3 [Bre, Ch.6/CLN, Ch.3, 6.7]

Th 4/3 Marginally outer-trapped surfaces (MOTS): stability and topology [Lee, 7.5]
Stability and topology of MOTS

Tu 4/8 Perelman's F-functional and W-entropy for Ricci flow [CLN 5.4, TOP Ch. 7]

Th 4/10 Stability/topology of MOTS (cont'd). Asympt flat initial data sets (def), total energy-momentum vector (def.)

Tu 4/15 Perelman's no-local-collapsing, applications [CLN, 5.4, 5.5/ TOP Ch.8] (CF)

Th 4/17 SPRING RECESS

Tu 4/22 Applications of non-collapsing: convergence of flows, limit ancient solution at a finite-time singularity

Th 4/24 Spacetime Penrose inequality: proof in spherically symmetric case [Lee, 7.6] (BW)
Computations in spherical symmetry

Tu 4/29 Singularity models at a finite-time singularity/ Hamilton-Ivey estimate in 3D

Th 5/1Harmonic functions proof of the PMT in 3D (based on [BKKS]

Tu 5/6 Singularity types (at finite time)/ Classification of Type I singularity models in 3D (ref: [Chow-Knopf, Section 9.4])