Summer Math Course

Greetings;

Assuming a reasonable amount of interest, I'm going to offer an informal, graduate level course covering the topics below. Part of the idea for this course is to finally sit down and do all the math you wish you knew, starting from scratch and doing everything right. It's roughly speaking "math the way the mathematicians do it for non-mathematicians." There are no pre-requisites, but this will be a graduate level course which I think will be pretty challenging.

If you are interested in attending, please send me an email or just show up to the next meeting of the class.

Saul Youssef

Topics

1. Category theory | Categories and sets | homework | solutions
   • Definitions, distinguished morphisms, products and sums
   • Sets, equivalence relations, partial orders
   • Clustering and jet finding [*]
   • The Cantor-Bernstein theorem [nice to know, but not especially needed for the rest of the course]
   • Generalized Induction
2. Algebra | Group theory | homework | solutions
   • Groups, morphisms, subgroups, quotients, free groups
   • Even and odd permutations
   • Permutation groups, detecting mis-cabling [*]
   • Quaternions, SU(2), the SU(2)->SO(3) double covering [*]
   • Numbers, Rings and Fields basics
   • Group actions
3. Vector spaces | Vector spaces I | homework | Vector spaces II
   • Generating sub-spaces, equivalence relations, etc.
   • Basic theory including infinite dimensional spaces
   • Subspaces, quotients, duals, bases
   • Tensors, exterior algebra, determinants, inner product spaces, spectral theorems
   • Fitting polynomials to data [*]
   • Numerical Laplacian and Fourier transform [*]
4. Topology | homework | Topology
   • Definitions, categorical business
   • Compactness, connectedness, nets, limits
   • Rigid motion in R^n [*]
   • Differential calculus
   • Groebner bases[*]
   • Drift tube segment finding [*]
5. Analysis | Calculus...of variations, a tiny bit of analysis
   • Metric spaces, Cauchy sequences
   • Riemann integration [*]
   • Picard's theorem
   • Matrix exponential [*]
   • L_p spaces
6. Differential topology | Differential Topology
   • errata: In the last line of page 28, "I" should be the boundary of M.
   • Inverse and implicit function theorems, d^2f/dxdy=d^2f/dydx
   • Manifolds, differential forms, pullbacks, Hodge star, Stoke's theorem, vector bundles, Poincare lemma
   • Div, grad, curl, Laplacian on manifolds
   • Riemannian manifolds, Maxwell's equations, Hamiltonian vector fields, Relativistic motion [*]
7. Lie groups | Lie groups
   • Definitions, vector fields, Lie Algebra
   • Homogeneous spaces
   • SU(2), SO(3), SL(2,R)[*]
   • Campell-Baker-Hausdorff
   • Compact groups, integration and Haar measure
   • Peter Weyl theorem, Representation theory
8. Vector Bundles and Gauge theory

Reference materials

• Notes handed out during the course.
• Required: Mathematical Physics, by Robert Geroch.
• Some references:
  • Algebra, by Birkhoff and MacLane.
  • Algebra, by Michael Artin.
  • Linear Algebra done Right, by Sheldon Axler.
  • Partial Differential Equations, by Michael E. Taylor.
  • Differential Forms and Applications, by Manfredo P. do Carmo.
  • Introduction to Smooth Manifolds, by John M. Lee.
  • Representation Theory, by William Fulton and Joe Harris.
  • Lie Groups and Lie Algebras, by Roger Carter, Graeme Segal and Ian MacDonald.
  • Notes on Lattice Theory, by J.B. Nation.
  • Any online book by Allen Hatcher.