Resources

• Terence Tao’s Notes
• Keith Conrad’s Notes
• AMS Open Math Notes
• Topics of Interest:
  • Lectures on the Combinatorics of Free Probability
  • Free Probability and Random Matrices
  • A Guided Tour of the Connes Embedding Problem
  • Introduction to Random Matrices
  • Strong Convergence: A Short Survey
  • An Introduction to Hyperlinear and Sofic Groups
  • Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture
  • Free Entropy
  • A Survey of Measured Group Theory
  • Around the Orbit Equivalence Theory of the Free Group, Cost, and \(\ell^2\)-Betti Numbers
  • A Brief Introduction to Sofic Entropy Theory
• Personal Notes:
  • Graduate Classes:
    • Differential Topology
    • Algebra I
    • Algebra II
  • Qualifier Prep:
    • Real Analysis
    • Complex Analysis
  • Algebra:
    • Conjugation and the Sylow Theorems
    • Algebraic Geometry (partial work-through)
  • Analysis:
    • Amenability in Discrete Groups
    • Von Neumann Algebras (all various levels of incomplete):
      • Structure of von Neumann Algebras
      • Projections in von Neumann Algebras
      • Completely Positive Maps
      • Standard Representations and Modular Theory
      • Injective implies Hyperfinite
      • Product-Type Constructions
    • Measure Theory/Real Analysis:
      • Inequalities and the $L_p$-Spaces
      • Egorov’s Theorem and Lusin’s Theorem
      • Three Convergence Theorems
      • The Lebesgue Measure
      • Signed Measures and the Lebesgue–Radon–Nikodym Theorem
    • Functional Analysis/General Operator Algebras:
      • Extreme Points, the Krein–Milman Theorem, and Applications
      • Compact and Fredholm Operators
      • Functional Calculus in Banach and $C^{\ast}$-Algebras
      • Spectral Theory for Normal Operators
      • Positive Elements and Ideals in $C^{\ast}$-Algebras
      • States and Representations
      • Fixed-Point Theorems
      • Generalized Limits
    • Miscellaneous:
      • Functional Analysis Exercises
      • Folland Exercise Workthrough
      • Quantum Theory for Mathematicians
      • Banach Algebras and Operator Theory
  • Topology:
    • Urysohn’s Lemma
    • Compactness in Topological Spaces
• Undergraduate Notes:
  • Real Analysis II
  • Real Analysis
  • Complex Analysis
  • Partial Differential Equations
  • Ordinary Differential Equations
  • Multivariable Calculus
  • Mathematical Methods of Physics II
  • Mathematical Methods of Physics I
  • Advanced Linear Algebra
  • Algebra II
  • Set Theory and Foundations of Mathematics