Part 1 Real Analysis.
Point
Set Topology, Banach and Hilbert Space, Measure Theory, Fourier Series and
Transforms, Distribution Theory, Locally Convex Spaces, Basics of
Probability Theory, Hausdorff Measure and Dimension.
Selected topics include Bernstein Polynomials, Pointwise Converge of Fourier Series, L^{p} spaces, Brownian Motion, Measures on Polish Spaces, Haar Measure, Convexity, Alexandroff–Hausdorff and BanachMazur theorems, Knaster–Kuratowski Fan, Krein Milman Theorem, moment problems, fixed point theorems including existence of invariant subspaces for compact operators, Hermite expansions, NyquistShannon sampling theorem, Riesz products, fundamental solutions of classical linear PDEs and the MalgrangeEhrenpreis theorem, law of large numbers, central limit theorem, law of the iterated logarithm, Poisson processes, Markov chains, Carathéodory construction, inductive limits and ordinary distributions. 

Part
2a Basic Complex Analysis. Cauchy Integral
Theorem, Consequences of the Cauchy Integral Theorem (including
holomorphic iff analytic, Local Behavior, PhragménLindelöf, Reflection
Principle, Calculation of Integrals), Montel, Vitali and Hurwitz’s
Theorems, Fractional Linear Transformations, Conformal Maps, Zeros and
Product Formulae, Elliptic Functions, Global Analytic Functions,
Picard’s Theorem. Selected topics include Goursat Argument, Ultimate and Ultra Cauchy Integral Formulas, Runge’s Theorem, complex interpolation, Marty’s Theorem, continued fraction analysis of real numbers, Riemann mapping theorem, Uniformization theorem (modulo results from Part 3), Mittag Leffler and Weirstrass product theorems, finite order and Hadamard product formula, Gamma function, EulerMaclaurin Series and Stirling’s formula to all orders, Jensen’s formula and Blaschke products, Weierstrass and Jacobi elliptic functions, Jacobi theta functions, PaleyWiener theorems, Hartog’s phenomenon, Poincaré’s theorem that in higher complex dimensions, the ball and polydisk are not conformally equivalent. 

Part
2b Advanced Complex Analysis. Conformal metric
methods, topics in analytic number theory, Fuchsian ODEs and
associated special functions, asymptotic methods, univalent functions,
Nevanlinna theory. Selected topics include Poincaré metric, AhlforsRobinson proof of Picard’s theorem, Bergmann kernel, Painlevé’s conformal mapping theorem, Jacobi 2 and 4squares theorems, Dirichlet series, Dirichlet’s prime progression theorem, zeta function, prime number theorem, hypergeometric, Bessel and Airy functions, Hankel and Sommerfeld contours, Laplace’s method, stationary phase, steepest descent, WKB, Koebe function, Loewner evolution and introduction to SLE, Nevanlinna’s First and Second Main theorems. 

Part
3 Harmonic Analysis. Maximal functions and
pointwise limits, harmonic functions and potential theory, phase space
analysis, H^{p} spaces, more inequalities. Selected topics include HardyLittlewood maximal function, von Neumann and Birkhoff ergodic theorems, Weyl equidistribution, ergodicity of Gauss (continued fraction) map, ergodicity of geodesic flow on certain Riemann surfaces, Kingman subadditive ergodic theorem, Ruelle–Oseledec theorem, martingale convergence theorem, subharmonic functions, Perron’s method, spherical harmonics, Frostman’s theorem, KelloggEvans theorem, potential theory on Riemann surfaces, pseudodifferential operators, coherent states, wavelets, BMO, real interpolation and Marcinkiewicz theorem, HardyLittlewoodSobolev inequalities, Sobolev spaces, CalderónZygmund method, CalderónVaillancourt estimates, Hypercontractive and LogSobolev estimates, Lieb Thirring and CLR bounds, TomasStein theorem. 

Part
4 Operator Theory. Eigenvalue Perturbation
Theory, Operator Basics, Compact Operators, Orthogonal Polynomials,
Spectral Theory, Banach Algebras, Unbounded SelfAdjoint Operators. Selected topics include analytic functional calculus, polar decomposition, HilbertSchmidt and RieszSchauder theorems, Ringrose structure theorems, trace ideals, trace and determinant, Lidskii’s theorem, index theory for Fredholm operators, OPRL, OPUC, BochnerBrenke theorem, Chebyshev polnomials, spectral measures, spectral multiplicity theory, trace class perturbations and Krein spectral shift, Gel’fand transform, Gel’fandNaimark theorems, almost periodic functions, Gel’fandRaikov and PeterWeyl theorems, Fourier analysis on LCA groups, Wiener and Ingham tauberian theorems and the prime number theorem, Spectral and Stone’s theorem for unbounded selfadjoint operators, von Neumann theory of selfadjoint extensions, quadratic forms, BirmanKreinVershik theory of self adjoint extensions, Kato’s inequality, BeurlingDeny theorems, moment problems, BirmanSchwinger principle. 