The concept of sigma-algebra, Borel subsets of real line, Construction of Lebesgue and Lebesgue-Stieltjes measures on the real line following outer measure.
Abstract measure theory: definition and examples of measure space, measurable functions, Lebesgue integration, convergence theorems (Fatouâs Lemma, Monotone convergence and dominated convergence theorem).
Caratheodory extension theorem, completion of measure spaces.
Product measures and Fubiniâs theorem.
L^p-spaces, Riesz-Fischer Theorem, approximation by step functions and continuous functions.
Absolute continuity, Hahn-Jordan decomposition, Radon-Nikodym theorem, Lebesgue decomposition theorem. Functions of bounded variation.
Complex measures.
Review of sequences and series of functions including power series, Complex differentiation and Cauchy-Riemann equation, Cauchyâs theorem and Cauchyâs integral formula, Power series expansion of holomorphic function, zeroes of holomorphic functions, Maximum Modulus Principle, Liouvilleâs Theorem, Moreraâs Theorem.
Complex logarithm and winding number, Singularities, Meromorphic functions, Casorati Weierstrass theorem, Riemann sphere, Laurent series, Residue Theorem and applications to evaluation of definite integrals, Open Mapping Theorem, Roucheâs Theorem.
Conformal maps, Schwarz lemma, Linear fractional transformations, automorphisms of a disc, Introduction to Gamma function.
Equicontinuity and Arzela-Ascoli Theorem, Normal family, Montelâs theorem and Riemann mapping theorem.
Quick review of sequences and series of functions, equicontinuity, Arzela-Ascoli theorem.
Normed linear spaces and Banach spaces. Bounded linear operators. Dual of a normed linear space. Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, closed graph theorem. Computing the dual of some well-known Banach spaces. Weak and weak-star topologies, Banach-Alaoglu Theorem. The double dual. L^p-spaces and their duality, Weierstrass and Stone-Weierstrass Theorems.
Hilbert spaces, adjoint operators, self-adjoint and normal operators, spectrum, spectral radius, analysis of the spectrum of a compact operator on a Banach space, spectral theorem for compact self-adjoint operators on Hilbert spaces.
Basics of complex measures and statement of the Riesz representation theorem for the space C(X) for a compact Hausdorff space X. Sketch of proof of Riesz representation theorem (if time permits).
Metric Topology of R^n. Topology induced by L^p norms (p = 1, 2, infinity) on R^n and their equivalence. Continuous functions on Rn. Separation properties. Compact subsets of R^n. Path-connectivity. Topological properties of subgroups like GLn(R), GLn(C), O(n), Hilbert-Schmidt norm and operator norm on Mn(R). Sequence and series in Mn(R). Exponential of a matrix.
Differentiation and integration of functions on R^n. Partial derivatives of realvalued functions on R^n. Differentiability of maps from R^m to R^n and the derivative as a linear map. Jacobian theorem. Chain Rule. Mean value theorem. Higher derivatives and Schwarz theorem, Taylor expansions in several variables. Inverse function theorem and implicit function theorems. Local maxima and minima, Lagrange multiplier method.
Vector fields on R^n. Integration of vector fields and flows. Picardâs Theorem.
Riemann integration of bounded real-valued functions on rectangles (product of intervals). Existence of the Riemann integral for sufficiently well-behaved functions. Iterated integral and Fubiniâs theorem. Brief treatment of multiple integrals on more general domains. Change of variable and the Jacobian formula.
Differential forms on R^n. Wedge product of forms. Pullback of differential forms. Exterior differentiation of forms. Integration of compactly supported n-forms on R^n. Change of variable formula revisited. Integration of k-forms along singular k-chains in R^n. Stokesâ theorem on chains. [Special emphasis on curves and surfaces in R^2 and R^3. Line integrals, Surface integrals. Gradient, Curl and Divergence operations, Greenâs theorem and Gaussâs (Divergence) theorem.
Basics of ODE: (local as well as global) existence and uniqueness results, Picard iteration, Gronwallâs inequality, solving some first order and second order equations.
Introduction to PDE: order of a PDE, classification of PDEs into linear, semilinear, quasi-linear, and fully nonlinear equations, Examples of equations from Physics, Geometry, etc. The notion of well-posed PDEs.
First order PDEs: Method of characteristics, existence and uniqueness results of the Cauchy problem for quasilinear and fully nonlinear equations. [9 lectures]
Second order linear PDEs in two independent variables: classification into hyperbolic, parabolic and elliptic equations, canonical forms.
Laplace equation: Definition of Harmonic functions. Mean-value property, Strong Maximum principle for harmonic functions, Liouvilleâs theorem, smoothness of harmonic functions, Poissonâs formula. Harmonic functions in rectangles, cubes, circles, wedges, annuli.
Heat equation: Fundamental solution of heat equation, Duhamelâs principle, weak and strong maximum principles, smoothness of solutions of heat equation, ill-posedness of backward heat equation.
Wave equation: well-posedness of initial and boundary value problem in 1D and dâAlembert formula. method of descent in 2D and 3D. Duhamelâs principle, domain of dependence, range of influence, finite speed of propagation.
Boundary problems: Separation of variables, Dirichlet, Neumann and Robin conditions. The method of seperation of variables for Laplace, Heat and Wave equations.
Fourier Series on T:
a) Dirichlet problem for the unit disc and origin of Fourier series, continuity of translation on L^p(T) and elementary convolution inequalities, approximate identity.
b) Fourier series and its elementary properties, completeness of trigonometric polynomials and Riemann-Lebesgue lemma, Uniqueness of Fourier coefficients... and the Fourier inversion, Plancherel theorem, Weylâs equidistribution theorem.
c) Dirichlet kernel and pointwise convergence of Fourier series for Lipschitz continuous functions, Riemannâs localization principle, existence of a continuous function with divergent Fourier series.
d) Cesaro and Abel summability, Poisson integral and solution of Dirichlet problem for appropriate function classes.
e) Marcinkiewicz interpolation theorem, Youngâs inequality and Hausdorff-Young inequality, norm convergence of Fourier series for L^p, 1 < p < 1.
Fourier transform on R^d:
a) Elementary properties of the Fourier transform involving translation, dilation, rotation, decay and smoothness, Riemann Lebesgue lemma, Fourier transform of Gaussian and the Poisson kernel, the Fourier inversion. Schwartz class functions and its image under Fourier transform.
b) Fourier transform of L^2-functions and the Plancherel theorem, Hausdorff-Young inequality, Paley-Wiener theorem, Poisson summation formula.
c) Tempered distribution and its Fourier transform, computation of some distributional Fourier transform.
d) Weak L^p spaces, Method of maximal function, Lebesgue differentiation theorem, almost everywhere convergence of Poisson integrals.
e) Nontangential convergence of Poisson integral and characterization of Poisson integral of L^p functions (If time permits).
Brief introduction to topological vector spaces (TVS) and locally convex TVS.
Linear Operators. Uniform Boundedness Principle. Geometric Hahn-Banach theorem and applications (Markov-Kakutani fixed point theorem, Haar Measure on locally compact abelian groups, Liapounovs theorem). Extreme points and Krein-Milman theorem.
In addition, one of the following topics:
(a) Geometry of Banach spaces: vector measures, Radon-Nikodym Property and geometric equivalents. Choquet theory. Weak compactness and Eberlein- Smulian Theorem. Schauder Basis.
(b) Banach algebras, spectral radius, maximal ideal space, Gelfand transform.
(c) Unbounded operators, Domains, Graphs, Adjoints, spectral theorem.
Definition and examples of Banach and C^* algebras, Gelfand theory for abelian Banach and C^* algebras, Spectral theorem for bounded normal operators on a Hilbert spaces.
General noncommutative C^* algebras: basic properties, Positive elements and states, GNS theorem, ideals and quotients of C^* algebras, C^* algebra of compact operators.
Definition of examples of von Neumann algebras, the double commutant theorem and Kaplansky density theorem, abelian von Neumann algebras and the L^1-functional calculus.
Time permitting: discussion on some concrete examples like AF algebras, group C^* algebras for discrete countable groups, Cuntz algebras, noncommutative tori etc.
Review of Matrices and Graphs. Incidence matrix, Adjacency matrix and Laplace matrix of a graph. The Laplace operator on graphs. Random walks on graphs. Dirichlet Problem. Spectral properties. Eigenvalues and mixing time. Cheegerâs inequality. Eigenvalues on infinite graphs. Estimates of the heat kernel. Carne- Varopoulous bound.
Finite dimensional quantum probability spaces: Events, observables and states. Pure and mixed states. Tensor products, notions of separability and entanglement for states. Distance measures for density matrices. Bellâs inequality, Dynamics in finite dimensional quantum probability spaces. Basic notions of spectral integration. Self-adjoint operators viewed as observables.
Completely positive maps, Stinespringâs theorem, Choi-Kraus representations. State preserving maps and quantum channels. Majorization. Generalized measurements (POVMs) and Naimarkâs theorem.
Revision of Measure theory: probability spaces, distributions, random variables, standard random variable examples, expected value, inequalities (Holder, Cauchy-Schwarz, Jensen, Markov, Chebyshev) , convergence notions(convergence in probability and almost sure, Lp), application of DCT, MCT, Fatou with examples, Revision of Fubiniâs theorem.
Independence,sum of random variables, constructing independent random variables, weak law of large numbers, Borel-Cantelli lemmas, First and Second Moment methods, Chernoff bounds and some applications.
Strong law of large number, Kolmogorov 0 â 1 law. Convergence of random series. Kolmogorovâs three series theorem.
Weak convergence, tightness, characteristic functions with examples, Central limit theorem (iid sequence and triangular array).
Random walks Finite Length: Stopping times and games, The ruin problem, Reflection Principle, Arc Sine law for last visit to origin
Random walks infinite Length: Higher Dimension, Recurrence versus transience.
B3-Probability III and M2 are slightly different to this but close enough. So M2 (when offered) or B3-Probability III (when offered) instructor can modify things a little.
This is the same as (M.Math elective) and the student can take it along with other students when offered.
Review of Gaussian random variables. Slepian and Sudakov-Fernique inequalities. Variance bounds, Poincare inequality, Isoperimetric inequality, Log-sobolev inequality, Concentration and transport inequalities. Maxima of Gaussian processes. Majorizing measures and Generic chaining. Excursion probabilities. Hypercontractivity.
Geometry of Gaussian random fields. Stein's method. Introduction to Malliavin calculus.
Absolute continuity and singularity of measures. Hahn-Jordon decomposition, Radon-Nikodym Theorem, Lebesgue decomposition. Conditional expectation - Definition and Properties. Regular conditional probability, proper RCP, and regular conditional distribution.
Discrete parameter martingales, sub-and super-martingales. Doob's Maximal Inequality, Upcrossing inequality, martingale convergence theorem, Lp inequality, uniformly integrable martingales, reverse martingales, Levy's upward and downward theorems. Stopping times, Doob's optional sampling theorem. Discrete martingale transform, Doob's Decomposition Theorem.
Finite and infinite exchangeable sequence of random variables, de Finetti's Theorem. SLLN for U-Statistics for exchangeable data.
Introduction to continuous parameter martingales: definition, examples, and basic properties.
(If time permits) Martingale Central Limit Theorem and applications.