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PMATH 950

Topics in Analysis

Graduate

Winter 2027 · Interplay between Geometric and Functional Inequalities

Topics

  • Convex geometry and functional inequalities
  • Symmetrization
  • Optimal transport
  • Decompositions of the identity
  • Variational arguments
  • Integration by parts
  • Tensorization
  • Heat-flow arguments
Winter 2025 · Topics in Analysis - Analytic methods in convex geometry

Topics

  • Convex sets in Euclidean space
  • Analytic, combinatorial and algebraic methods in convex geometry
  • Isoperimetric inequalities
  • Concentration phenomena
  • Uniqueness problems

Readings

  • Lectures on Convex Geometry
    D. Hug and W. Weil · recommended
  • Asymptotic Geometric Analysis
    S. Artstein-Avidan, A. Giannopoulos and V. D. Milman · Parts I and II · recommended
  • An Elementary Introduction to Modern Convex Geometry
    K. Ball · recommended
  • A Course in Convexity
    A. Barvinok · recommended
  • Mostly self-contained instructor course notes.
    unspecified
Winter 2023 · Quantum Representation Theory

Topics

  • Classical compact groups and representation theory
  • C*-algebras, states and GNS construction
  • Compact quantum groups, Haar integrals and Hopf algebras
  • Unitary representations, tensor products and complete reducibility
  • Schur–Weyl orthogonality and quantum Peter–Weyl theory
  • Tensor categories and Tannaka–Krein reconstruction
  • Drinfeld–Jimbo quantum groups
  • Possible quantum-group actions, amenability and subfactor applications

Readings

  • Compact Quantum Groups and Their Representation Categories
    S. Neshveyev and L. Tuset · recommended
    Société Mathématique de France, 2014.
  • An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond
    T. Timmerman · recommended
    EMS Press, 2008; author spelling as given in outline.
  • Compact Quantum Groups and Their Combinatorics
    A. Freslon · recommended
    Cambridge, 2022 as listed; outline describes it as in press and provides a PDF to students.
  • No required textbook; self-contained lectures sample material from the listed optional books.
    unspecified
Winter 2022 · Operator Algebras

Topics

  • Kadison’s similarity problem
  • Completely bounded and positive maps
  • Amenability and nuclearity
  • Reductive and strongly reductive operator algebras
  • Gifford’s total reduction property
  • Invariant and almost-invariant subspaces

Readings

  • C*-algebras and Finite-Dimensional Approximations
    N. Brown and N. Ozawa · recommended
  • Completely Bounded Maps and Dilations
    V. Paulsen · recommended
  • Similarity Problems and Completely Bounded Maps
    G. Pisier · recommended
  • Instructor notes and their bibliography.
    unspecified
  • J. E. Gifford, Operator Algebras with a Reduction Property (PhD thesis).
    unspecified
  • A. Popov and A. Tcaciuc, Every Operator Has Almost-Invariant Subspaces.
    unspecified
Fall 2021 · Dynamics of Groups

Topics

  • Topological and measurable group actions
  • Dynamical approaches to Ramsey theory and additive number theory
  • Approximation properties of groups
  • Random walks and Poisson boundaries
  • Strongly amenable and Choquet–Deny groups
Winter 2020 · Fractal Geometry

Topics

  • Hausdorff measure and dimensions
  • Box and Assouad dimensions
  • Iterated function systems
  • Self-similar sets and measures
  • Local dimension and multifractal analysis
  • Bernoulli convolutions
  • Laplacians on fractals
  • Besicovitch and Kakeya sets

Readings

  • Fractal Geometry: Mathematical Foundations
    K. Falconer · recommended
    Title as listed in source; other books by Falconer are also accepted.
  • Some material may come from recent research papers.
    unspecified
Fall 2020 · The tensor theory of Banach spaces, operator spaces, and operator systems

Topics

  • Tensor norms on Banach spaces
  • Grothendieck’s tensor theory
  • Tensor theories of operator spaces and operator systems
  • Completely contractive and completely positive maps
  • Tensor characterizations of operator algebras
  • Similarity and factorization degree
  • Tensor products of C*-algebras
Winter 2019 · Group C* Algebras

Topics

  • Group C*-algebras and group von Neumann algebras
  • Representation theory and basic structure theory
  • Approximation and regularity properties
  • Dynamical systems and crossed products
  • Advanced structure theory
Sources