Resources

• Topics of Interest:
  • Sofic and Hyperlinear Groups:
    • An Introduction to Hyperlinear and Sofic Groups, Pestov and Kwiatowska
    • Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture, Capraro and Lupini
    • A Brief Introduction to Sofic Entropy Theory, Bowen
    • On Sofic Groups
  • Measured Group Theory and Cost:
    • A Survey of Measured Group Theory
    • Around the Orbit Equivalence Theory of the Free Group, Cost, and $\ell^2$-Betti Numbers
    • Invariants of Orbit Equivalence Relations and Baumslag–Solitar Groups
  • Free Probability/Entropy:
    • Lectures on the Combinatorics of Free Probability
    • A Guided Tour of the Connes Embedding Problem
    • Introduction to Random Matrices
    • Strong Convergence: A Short Survey
    • Free Entropy
    • Random Matrices and Free Probability Lecture Notes
• Neat Links:
  • Slick Proof of the Mazur–Ulam Theorem
  • Notes on Thompson’s Groups
  • Open Problems in Operator Algebras
• Class Notes:
  • Differential Topology
  • Algebra I
  • Algebra II
  • Project on Free Entropy
• Qualifier Prep:
  • Real Analysis
  • Complex Analysis
  • Algebra
• Amenability in Discrete Groups
• Terence Tao’s Notes
• Keith Conrad’s Notes
• AMS Open Math Notes
• Jesse Peterson’s Notes
• Personal Notes:
  • Algebra:
    • Conjugation and the Sylow Theorems
    • Algebraic Geometry (partial work-through)
  • 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
    • The Disintegration 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 Analysis Notes:
    • Functional Analysis Exercises
    • Folland Exercise Workthrough
    • Quantum Theory for Mathematicians
    • Banach Algebras and Operator Theory
  • Topology:
    • Urysohn’s Lemma
    • Compactness in Topological Spaces
• Undergraduate Class 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