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)