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 incompletness theorem