MATH 70330, Intermediate Geometry and Topology
Notre Dame, Fall 2021
Schedule: Monday, Wednesday 12:30-1:45, Pasquerilla Center 102
Exercise sessions: Friday 4:00-5:00, Hurley 258
References.
Bundles, connections:
- Husemoller "Fibre bundles," link
- Cohen "Bundles, homotopy and manifolds," link
- Hatcher "Vector bundles and K-theory," link
- Steenrod "The topology of fibre bundles"
- Ehresmann "Les connexions infinitésimales dans un espace fibré
differentiable," link
(in French)
Characteristic classes:
- Milnor, Stasheff, "Characteristic classes," link
- Bott, Tu, "Differential forms in algebraic topology," link
- Bott, "On the Chern-Weil homomorphism and the continuous
cohomology of Lie groups," link
Splitting principle:
- May, "A note on the splitting principle," link
- Bott, Tu, p.273
Mathai-Quillen representative:
- Getzler, "The Thom class of Mathai and Quillen and probability
theory," link
Symplectic geometry:
- Cannas da Silva, "Lectures on symplectic geometry," link
- Lecture notes on Moser's trick and Darboux theorem, link
- Cannas da Silva, "Symplectic toric manifolds," link
Lecture notes:
Characteristic classes
8/23 (fiber
bundles, vector bundles)
8/25 (vector bundles,
principal bundles)
8/30 (principal
bundles, Ehresmann connections, connections in vector bundles)
9/1 (connections in
vector bundles and in principal bundles)
9/6 (Stiefel-Whitney
classes)
9/8 (Stiefel-Whitney
numbers, unoriented cobordism, classifying map)
9/13 (cohomology of
the Grassmannian)
9/15 (Thom
isomorphism, Euler class, Chern classes)
9/20 (Chern and
Pontrjagin classes)
9/22 (Pontrjagin
classes, classifying G-bundles)
9/27 (classifying
G-bundles, Milnor's join construction, group cohomology)
9/29 (Chern-Weil
homomorphism)
10/4 (Chern-Weil
continued: Chern, Pontrjagin and Euler classes)
10/6 (equivariant cohomology: Borel, Weil and Cartan models -- talk
by Xiyan and Guoran)
10/11 (simplicial construction for EG and BG -- talk by Lorenzo)
10/13 (Milnor's exotic 7-spheres -- talk by Jiayi)
10/15
(extra lecture: Mathai-Quillen representative for Thom and Euler
classes)
Symplectic geometry
10/25 (symplectic
linear algebra, symplectic manifolds)
10/27 (symplectic
volume, Lagrangian submanifolds in a cotangent bundle)
11/1 (generating
function for a symplectomorphism, Moser's trick)
11/3 (Darboux theorem,
Weinstein's Lagrangian neighborhood theorem)
11/8 (Weinstein's
tubular neighborhood theorem, applications, Hamiltonian vector
fields)
11/10 (contact manifolds -- talk by Jinxuan)
11/15 (symplectic vs.
hamiltonian vector fields, integrable systems)
11/17 (classical Chern-Simons theory -- talk by Cory and Justin)
11/22 (Hamiltonian
group actions, moment maps)
11/29 (symplectic
quotients)
12/1 (convexity
theorem, classification of toric manifolds)
12/6 (symplectic toric
manifolds, their homology, symplectic blow-up)
Exercise sheets:
8/27 (bundles)
9/3
(connections); updated
version
9/10
(cohomology of projective spaces, Stiefel-Whitney numbers)
9/17 (Thom
isomorphism in de Rham cohomology)
9/24 (Chern
character)
10/1
(Chern-Weil)
10/8 (Berezin
integral, Levi-Civita connection, Mathai-Quillen representative of
the Thom class)
10/29
(symplectic linear algebra, symplectic manifolds)
11/5
(coisotropic reduction, canonical relations, generating functions,
Legendre transform)
11/12
(Hamiltonian vector fields)
11/19
(coisotropic reduction - geometric setting, coadjoint orbits)
12/3 (moment
maps, Lie algebra cohomology)