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


Bundles, connections:

Characteristic classes:

Splitting principle:

Mathai-Quillen representative:

Symplectic geometry:

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)