Signatures in topology, algebra, & dynamics (Math 2720)

Here are some handwritten notes from the course. Use the topic schedule below if you want to search for something specific in them.

Course summary

This is a graduate topics course centered around the topic of “signatures”, which are numerical invariants associated to quadratic forms, manifolds, knots, and more. While many things are signatures, many more things are not, and we will use the theme of signatures as an opportunity to discuss as many interesting topics as possible. This may include (depending on time and interest)

• signatures of manifolds, cobordism, Hirzebruch signature theorem
• the E8 manifold, the Poincare homology sphere, exotic spheres
• knots, Seifert surfaces, concordance, fibered knots
• conjugacy of symplectic matrices, the symplectic group, Maslov index
• Novikov additivity, signature cocycle for surface bundles
• SL(2,ℤ), signature defect, connection to modular forms
• Sturm’s theorem (counting real roots of a polynomial)

Textbook

Ghys-Ranicki, Signatures in algebra, topology, and dynamics

Topic schedule

• Week 1
  • Thursday 1/27: no class. The course starts with the ICERM program on Braids.
• Week 2 (quadratic forms)
  • Tuesday 2/1: real quadratic forms and signature, course preview
  • Thursday 2/3: rational quadratic forms and p-signatures, E8 lattice
• Week 3
  • Tuesday 2/8: lattices and isospectral tori, strong/weak Hasse principle
  • Thursday 2/10: quadratic forms and hyperbolic manifolds
• Week 4
  • Tuesday 2/15: quadratic forms on Z^2, positive definite forms and the mass formula
  • Thursday 2/17: indefinite forms on Z^2, general indefinite unimodular forms
• Week 5
  • Tuesday 2/22: no class (university holiday)
  • Thursday 2/23: no class (math travel)
• Week 6
  • Tuesday 3/1: intersection forms of manifolds, representing homology by submanifolds, plumbing
  • Thursday 3/3: the E8 manifold, representing homology by submanifolds
• Week 7
  • Tuesday 3/8: signatures and cobordism
  • Thursday 3/10: no class (math travel)
• Week 8
  • Tuesday 3/15: Rokhlin’s theorem, K3 manifold, Pontryagin-Thom construction
  • Thursday 3/17: no class (serving on an NSF panel)

• Week 9: ICERM workshop

• Week 10: spring break
  • Tuesday 3/29: spring break
  • Thursday 3/31: spring break
• Week 11
  • Tuesday 4/5: signature of knots, concordance, Seifert surfaces
  • Thursday 4/7: knot genus, knot signature is well-defined, knot signature canonically
• Week 12
  • Tuesday 4/12: more knot signatures, classification of symplectic matrices, signature additivity
  • Thursday 4/14: Maslov index, Novikov additivity
• Week 13
  • Tuesday 4/19: Wall non-additivity, symplectic group cocycles
  • Thursday 4/21: Meyer signature cocycle, topology of the Lagrangian Grassmannian

• Week 14: ICERM workshop

• Week 15
  • Tuesday 5/3: Fixed points of surface diffeomorphism, Arnold conjecture
  • Thursday 5/5: Conley-Zehnder index, Floer theory, Arnold conjecture (lecture by Kenny Blakey)