Swayam Chube

Notes and Blurbs

Notes

These were taken as a course was going on (liveTeX’d if you will) and I never bothered correcting mistakes, so read them with caution. The notes on Analytic NT were not taken as part of a course and (hopefully) contain few mistakes.

  • Topology PDF
  • Measure Theory PDF
  • Complex Analysis PDF
  • Galois Theory PDF
  • Commutative Algebra PDF
  • Analytic Number Theory PDF
  • Functional Analysis PDF

Blurbs and Expositions

Commutative/Non-commutative Algebra

  • The Quillen-Suslin Theorem PDF
  • PIDs that are not EDs PDF
  • The Jacobson Radical PDF
  • On \(\displaystyle\prod_{i = 1}^\infty\mathbb{Z}\) PDF
  • Rings of Continuous Functions PDF (Most importantly, contains a discussion of the Krull dimension of \(C(X)\) for a compact Hausdorff space \(X\).)
  • Flat, Projective, and Injective Modules PDF (This is a work in progress, and is not very well organized. Contains stuff about the interplay of flat and projective modules. Also contains a section on injective hulls and a nice application to Cantor-Schroeder-Bernstein for injective modules. There’s a section on the Eakin-Nagata theorem in there too for some reason.)

Group Theory

  • Group Transfer PDF
  • Subnormality in Group Theory PDF (The main goal of this was to write a self-contained exposition of Horosevskii’s theorem. It also contains useful results on Sylow subgroups, Hall subgroups, nilpotent, and solvable groups.)

Lie Groups/Lie Algebras

  • Theorems of Levi and Ado-Iwasawa PDF
  • The Poincaré-Birkhoff-Witt Theorem PDF (Also contains a section on epimorphisms of Lie algebras.)

Analysis

  • The Picard Theorems PDF
  • Lebesgue Differentiation PDF
  • The Inverse Galois Problem over \(\mathbb{C}(t)\) PDF