MATH 661- MODERN TOPOLOGY-FALL 2024

Syllabus

Presentation topics

Course log

8/20  Tu   Hodge laplacian, self-adjointness/ Hodge existence theorem, decomposition theorem

8/22  Th Applications to de Rham cohomology/ ellipticity of Hodge laplacian (start)

8/27  Tu interior product: two properties/symbols of composition and adjoints/symbol of Hodge Laplacian
              regularity of weak solutions: from open sets to manifolds

8/29 Th L2 Sobolev spaces, elliptic estimates and compactness/geometric expressions for d and d*

9/3   Tu Weitzenböck formula: statement and proof/curvature term in the case of 1-forms
             A lemma on the codifferential

9/5   Th First applications of the Bochner argument: vanishing and estimation results for harmonic 1-forms/
        Introduction to Clifford algebras.
Clifford algebras and the Clifford bundle

Reference: Pierre Bérard, From Vanishing Theorems to estimating theorems: The Bochner technique revisited.
Bull. AMS vol. 19 (2), Oct. 1988, p.371-406.

Possible presentation topic: Theorem III in section B of this paper (estimation), following the proof given in Appendix II(a).

9/10 Tu Clifford bundles of a Riemannian manifold: Levi-Civita connection, its curvature operator
Inner product on Cl(V,g) and the Adjoint action

9/12 Th Cl(X)-modules and compatible connections/Dirac operators/ general Weitzenböck formula

9/17 Tu Weitzenböck formula for the Dirac operator on Cl(X)/Curvature operator and the curvature term for p-forms (Gallot-Meyer's theorem)

9/19 Th Cl(X)-modules: definition/ Existence of compatible connections on a general Cl(X) module: direct proof.
Compatible connections on Cl(X)-modules
(Ref: Booß-Bavnek and Wojciechowski, Elliptic Boundary value problems for Dirac operators (Birkhäuser 1993)--p.10

9/24 Tu  The Spin_n group in Cl_n^+; complex representations induced from Cl_n; definition of spin manifold

9/26 Th The spinor bundle of a spin manifold/review of connection 1-forms and curvature 2-forms (with values in a Lie algebra) associated to a local`frame'
(local section of a G-principal bundle); compatibility conditions/ induced connection on an associated vector bundle/the Lie algebra of Spin_n and the differential of the
standard covering map to SO_n.

10/1 Tu Curvature formula for the spinor bundle/Lichnerowicz formula and application

10/3 Th Lecture postponed

10/8  FALL BREAK

10/10 Th Lichnerowicz formula with boundary/ Killing connection on spinor bundle: modified Lichnerowicz formula, complete manifolds with Killing spinors.

10/15 Tu (two lectures) Harmonic functions on complete manifolds: two classic theorems of S-T Yau.
Yau's theorem on positive harmonic functions in nonnegative Ricci curvature

10/17 Th smooth support functions for the distance function, with Laplacian estimate/ First variation formula for scalar curvature/ Linearization of scalar curvature and the formal adjoint operator/ case of nonempty boundary: the mass integrand.

Linearization of scalar curvature and first variation of total scalar curvature

10/22, 10/24: lectures postponed (due to conference)

10/29 Tu Static potentials and the mass I

10/31 Th Static potentials and the mass II

11/5 Tu ELECTION DAY (no class)

11/7 Th Static potentials and the mass III

11/12  Tu Lecture I: geometric best constants in Sobolev ineqs, etc./ Lecture II: proof of PMT via minimal hypersurfaces: analytic aspects, weighted Sobolev spaces

11/14 Th Lecture I: positive isotropic curvature, Hodge and Bochner on manifolds with boundary/ Lecture II: min sfce proof of PMT: geometric/topological part

11/19 Tu: spinor proof of PMT (outline)

11/21Th: first presentation: From vanishing theorems to estimating theorems via the Bochner technique (George, Bryan)
Notes on P. Bérard's paper (in progress)

11/26 Tu: second presentation: Second betti number under PIC and 2-convexity (Jeff, Caden)
Notes on Bochner with boundary and PIC    (in progress)

11/28: THANKSGIVING

12/ 3 Tu (last day): third presentation: Bandwidth under PIC, 2-convexity and Betti number hypotheses (Isuru, Tariq)