Course Code
Course Name & Syllabus
MA 4010
ANALYSIS OF FUNCTIONS OF A SINGLE VARIABLE
• Real number system : Field properties, ordered properties, completeness
axiom, Archimedean property, subsets of R, infimum, supremum, extended real
numbers. Finite, countable and uncountable sets, decimal expansion.
• Sequences of real numbers, Subsequences, Monotone sequences, Limit infimum, Limit
Supremum, Convergence of Sequences .
• Metric spaces, limits in metric spaces. Functions of single real variable, Limits of functions,
Continuity of functions, Uniform continuity, Continuity & compactness, Continuity and
connectedness, Monotonic functions, Limit at infinity.
• Differentiation, Properties of derivatives, Chain rule, Rolle's theorem, Mean-value
theorems, L'Hospital's rule, Derivatives of higher order, Taylor's theorem.
• Definition and existence of Riemann integral, properties, Differentiation and integration.
• Revision of Series, Sequences and Series of functions, Pointwise and uniform convergence,
Uniform convergence of continuous functions, Uniform convergence and differentiability,
Equicontinuity, Pointwise and uniform boundedness, Ascoli's theorem, Weierstrass
approximation theorem, Fourier series.

MA 4020
LINEAR ALGEBRA
• System of Linear Equations, Elementary Operations, Row-Reduced Echelon Matrices, Gaussian
Elimination.
• Vector Spaces, Subspaces, Direct Sums, Bases and Dimension, Linear Maps, Rank-Nullity
Theorem, The Matrix of a Linear Map, Invertibility.
• Metric spaces, limits in metric spaces. Functions of single real variable, Limits of functions,
Continuity of functions, Uniform continuity, Continuity & compactness, Continuity and
connectedness, Monotonic functions, Limit at infinity.
• Eigenvalues and Eigenvectors, Invariant Subspaces, Upper-Triangular Matrices, Diagonal
Matrices
• Inner Products, Norms, Orthonormal Bases, Gram-Schmidt process, Schur's theorem,
Orthogonal Projections and Minimization Problems, Linear Functionals and Adjoints.
• Self-Adjoint and Normal Operators, The Spectral Theorem for finite dimensional operators.
• Generalized Eigenvectors, The Characteristic Polynomial, Cayley-Hamilton Theorem, The
Minimal Polynomial, Jordan Form.

MA 4030
ORDINARY DIFFERENTIAL EQUATIONS
• Mathematical Models, Review of methods, First Order Equations, Existence, Uniqueness and
continuity theorems, separation and comparison theorems. Higher order equations, Solutions
in Power Series, Legendre equation, Bessel equation, generating functions, orthogonal
properties.
•System of differential equations, existence theorems, Homogeneous linear systems,
Nonhomogeneous linear systems, linear systems with constant coefficients.
• Two point boundary value problem, Green's functions, construction of Green's functions,
Sturm-Liouville problems, Eigen values and Eigen functions.
• Autonomous systems, Stability of linear systems with constant coefficients, Linear plane
autonomous systems.

MA 4060
COMPLEX ANALYSIS
• Spherical representation of extended complex plane, Analytic Functions, Harmonic
Conjugates, Elementary Functions, Cauchy Theorem and Integral Formula, Homotopic version .
•Linear fractional transformations, Power Series, Analytic Continuation and Taylor’s theorem,
Zeros of Analytic functions, Hurwitz Theorem, Maximum Modulus Theorem, Laurent’s
Theorem, Classification of singularities.
• Residue theorem and applications, Argument Principle, Theorem of Rouche, SchwarzChristoffel
Transformation.

MA 4070
ELEMENTS OF GROUPS AND RINGS
•Binary operation and its properties, Definition of Groups, Examples and basic properties.
Subgroups, Coset of a subgroup, Lagrange’s theorem. Cyclic groups. Normal subgroups,
Quotient group. Homomorphisms, Isomorphism theorems. Permutation groups, Cayley’s
theorems. Direct and semidirect product of groups. Group actions and Sylow theorems.
•Definition of Rings, Examples and basic properties, Zero divisors, Integral domains, Fields,
Characteristic of a ring, Quotient field of an integral domain. Subrings, Ideals, Quotient rings,
Isomorphism theorems. Ring of polynomials. Prime, Irreducible elements and their properties,
Unique Factorization Domains, Principal Ideal Domains, and Euclidean domains. Prime ideal,
Maximal ideal, Prime avoidance theorem, Chinese remainder theorem.

MA 4080
PARTIAL DIFFERENTIAL EQUATIONS
• First order partial differential equations : Surfaces and Curves, Classification of 1st order p.d.e.
Classification of solutions-Pfaffian differential equations - Quasi-linear equations, Lagrange's
method-compatible systems-Charpit's method- Jacobi's method-Integral surfaces passing through
a given curve- method of characteristics for quasi-linear and non-linear p.d.e., Monge cone,
characteristic strip.
• Second order partial differential equations :Origin of second order p.d.e's - classification of second
order p.d.e's. Wave equation - D'Alemberts' solution - vibrations of a finite string - existence and
uniqueness of solution - Riemann method. Laplace equation - boundary value problems,
Uniqueness and continuity theorems - Dirichlet problem for a circle - Dirichlet problem for a circular
annulus - Neumann problem for a circle - Theory of Green's function for Laplace equation. Heat
equation - Heat conduction problem for an infinite rod - Heat conduction in a finite rod - existence
and uniqueness of the solution.

MA 4090
ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES
• Functions of several-variables, Directional derivative, Partial derivative, Total derivative, Jacobian,
Chain rule and Mean-value theorems, Interchange of the order of differentiation, Higher
derivatives, Taylor's theorem, Inverse mapping theorem, Implicit function theorem, Extremum
problems, Extremum problems with constraints, Lagrange's multiplier method.
• Multiple integrals, Properties of integrals, Existence of integrals, iterated integrals, change of
variables. Curl, Gradient, div, Laplacian cylindrical and spherical coordinate, line integrals, surface
integrals, Theorem of Green, Gauss and Stokes.

MA 5010
COMBINATORICS AND GRAPH THEORY
• Basic counting : Bijections, Counting objects with repetitions, de Bruijn-Erdos theorem, Listing
combinatorial objects.
• Permutations : Combinatorial representation of a permutation, Descents and Eulerian
polynomial, Tree representation for permutations.
• Inclusion-Exclusion principle : Use of Rook polynomial, Some arithmetic and Mobius functions.
• Parity : Parity in Graph theory, Eulerian circuits in graphs, digraphs and de Bruijn circuits,Hypercubes and Gray codes, Parity of a permutation, Quadratic reciprocity.
• • Pigeonhole principle : Ramsey theorem, The infinite case.
• Geometry : Regular polytopes and tessellations of plane, triangulations and Sperner’s lemma.
• Recurrence relations : Fibonacci recurrence relation, Linear homogeneous recurrence relations with constant coefficients, Case of repeated roots, Difference tables and sums of polynomials, Other types of recurrence relations.

MA 5020
FUNCTIONAL ANALYSIS
• Normed linear spaces. Non-compactness of the unit ball in infinite dimensional normed linear
spaces. Product and quotient spaces. Banach spaces, Hilbert spaces.
• Permutations : Combinatorial representation of a permutation, Descents and Eulerian
polynomial, Tree representation for permutations.
• Linear maps. Boundedness and continuity. Linear isometries, linear functionals. Examples.
• Hahn-Banach extension theorem, applications. Banach-Steinhaus theorem, closed graph
theorem, open mapping theorem and bounded inverse theorem, Spectrum of a bounded
operator.
• Gram-Schmidt orthogonalization. Bessel’s inequality, Riesz-Fisher theorem. Orthonormal
basis, Parseval’s identity, Projection, orthogonal decomposition. Bounded linear functionals on
Hilbert spaces.

MA 5030
MEASURE AND INTEGRATION
• Sigma-algebra of measurable sets. Completion of a measure. Lebesgue Measure and its
properties. Non-measurable sets.
• Measurable functions and their properties. Integration and Convergence theorems. Lebesgue
integral, Functions of bounded variation and absolutely continuous functions. Fundamental
Theorem of Calculus for Lebesgue Integrals.
• Product measure spaces, Fubini's theorem.
• Lp spaces, duals of Lp spaces. Riesz Representation Theorem for C([a,b]).

MA 5040
TOPOLOGY
• Definition of Topologies in terms of open sets, neighborhood system, closed sets and closure
operations and their equivalence, points of accumulation, interior, exterior and boundary
points.
• Base and subbase of a topology, subspace, product space, quotient space, continuous, open
and closed maps, homeomorphism convergence of sequence and nets
• Separation axioms, Urysohn’s Lemma, Tietze extension theorem, separability.
• Compactness, local compactness, sequential and countable compactness, Tychonoff’s
theorem, Lindelof space. One point compactification.
• Connectedness and local connectedness.
• Urysohn's metrization theorem.

MA 5050
MATHEMATICAL METHODS
• Integral Transforms : Laplace transforms: Definitions - properties - Laplace transforms of some elementary functions - Convolution Theorem - Inverse Laplace transformation - Applications.
• Fourier Transforms : Definitions - Properties - Fourier transforms of some elementary functions- Convolution theorems - Fourier transform as a limit of Fourier Series - Applications to PDE.
• Integral Equations :Volterra Integral Equations: Basic concepts - Relationship between Linear differential equations and Volterra integral equations - Resolvent Kernel of Volterra Integral equation - Solution of Integral equations by Resolvent Kernel - The Method of successive approximations - Convolution type equations, solution of integral differential equations with the aid of Laplace transformation.
• Fredholm Integral equations : : Fredholm equations of the second kind, Fundamentals - Iterated Kernels, Constructing the resolvent Kernel with the aid of iterated Kernels - Integral equations
with degenerate Kernels - Characteristic numbers and eigen functions, solution of
homogeneous integral equations with degenerate Kernel - non homogeneous symmetric
equations - Fredholm alternative.
• Calculus of Variations : Extrema of Functionals: The variation of a functional and its properties
- Euler's equation - Field of extremals - sufficient conditions for the Extremum of a Functional
conditional Extremum Moving boundary problems - Discontinuous problems - one sided
variations - Ritz method.

MA 5060
NUMERICAL ANALYSIS
•Floating point representation of numbers, floating point arithmetic, errors, propagation of
error.
•Solution of nonlinear equations: Iterative methods, Fixed point iteration method, convergence
of fixed point iteration, Newton-Raphson method, complex roots and Muller’s method.
• Interpolation : Existence and uniqueness of interpolating polynomial, error of interpolation -
interpolation of equally and unequally spaced data - Inverse interpolation - Hermite
interpolation
• Approximation : Uniform approximation by polynomials, data fitting, Least square, uniform and
Chebyshev approximations
• Solution of linear systems : Direct and iterative methods, ill-conditioned systems, Eigen values
and eigen vectors: Power and Jacobi methods.
• Integration : Newton-cotes closed type methods; particular cases, error analysis - Romberg
integration, Gaussian quadrature; Legendre, Chebyshev formulae.
• Solution of Ordinary differential equations : Initial value problems: Single step methods; Taylor’s, Euler method, modified Euler method, Runge-Kutta methods, error analysis.

MA 5070
MODULES AND FIELDS
•Review of Rings, Modules, Free modules, Cartesian products and direct sums of modules, quotient
modules, Simple and semisimple modules, isomorphism theorems. Modules over principal ideal
domains and applications. Noetherian and Artinian rings/Modules, Hilbert basis theorem. JordanHolder
theorem. Projective/Injective modules.
•Field extensions. Algebraic/transcendental elements, Algebraic extensions. Finite fields, Cyclotomic
fields. Splitting field of a polynomial. Algebraic closure of a field, Uniqueness. Normal, separable,
purely inseparable extensions. Primitive elements, simple extensions. Fundamental theorem of
Galois theory. Solvability by radicals - Solutions of cubic and quartic polynomials, Insolvability of
quintic and higher degree polynomials. Geometric constructions.

MA 5080
ADVANCED PROGRAMING
•Mathematical background, Model - What to Analyze,
Abstract Data Types (ADT’s), The List ADT, The Queue ADT, The Stack ADT,
Preliminaries, Binary Trees, The Search Tree ADT, Binary Search Trees, AVL Tree,
Preliminaries, Insertion Sort, Shell Sort, Merge Sort, Quick Sort,
Definitions, Topological Sort and Minimal Spanning Tree.

MA 5090
SETS LOGICS AND BOOLEAN ALGEBRA
• Sets and Relations : Types of relations, Peano Axioms and Mathematical Induction, Cardinality, Recursion.
• Boolean Algebra : Partially Ordered Sets, Lattices, Subalgebras, Direct Product, Homomorphisms,
Boolean Functions, Representation and Minimization of Boolean functions.
• Mathematical Logic :Connectives, Normal Forms, Theory of Inference for the Statement Calculus.

MA 5100
INTRODUCTION TO ALGEBRAIC TOPOLOGY
• Homotopy, Fundamental group, The Fundamental group of the circle, Retractions and fixed
points, Application to the Fundamental Theorem of Algebra, The Borsuk-Ulam theorem, Homotopy
equivalence and Deformation retractions, Fundamental group of a product of spaces, and
Fundamental group the torus, Sphere, and the real projective n-space.
• Free Products of Groups, The Van Kampen Theorem, Fundamental Group of a Wedge of Circles,
Definition and construction of Cell Complexes, Application to Van Kampen Theorem to Cell
Complexes, Statement of the Classification Theorem for Surfaces, Fundamental groups of the
closed orientable surface of genus g
• Introduction to Covering spaces, Universal Cover and its existence, Unique Lifting Property, Galois
Correspondence of covering spaces and their Fundamental Groups, Representing Covering Spaces
by Permutations - Deck Transformations, Group Actions, Covering Space Actions, Normal or
Regular Covering Spaces.

MA 5110
FOURIER ANALYSIS AND APPLICATIONS
•Definition, Examples, Uniqueness of Fourier series, Convolution, Cesaro summability and
Abel summability of Fourier series, Mean square convergence of Fourier series, A continuous
function with divergent Fourier series. Some applications of Fourier series, The isoperimetric
inequality, Weyl's equidistribution theorem.
• Fourier transform on the real line and basic properties, The Schwartz space, Approximate
identity using Gaussian kernel, Solution of heat equation, Fourier inversion formula, L^2-
theory .
• Some basic theorems of Fourier Analysis, Poisson summation formula, Heisenberg
uncertainty principle, Hardy's theorem, Paley-Wiener theorem, Wiener's theorem, Shannon
sampling theorem.
• The class of test functions, Distributions, Convergence, differentiation and convolution of
distributions, Tempered distributions, Fourier transform of a tempered distribution.

MA 5120
NUMERICAL LINEAR ALGEBRA
•Gaussian elimination and its variants. Sensitivity of system of linear systems. QR factorization and
The least squares. The singular value decomposition. Computing Eigenvalues and Eigenvectors.
Iterative methods for linear systems.

MA 5130
FOURIER ANALYSIS AND APPLICATIONS
• Regualr Languages : Finite Automata, Non-determinism, Regular Expressions, Nonregular
Languages.
• Context-Free Languages : Context-free Grammars, Pushdown Automata, Non-context-free
Languages.
• The Church-Turing Thesis : Turing Machines and Variants.
• Decidability : Decidable Languages, The Halting Problem.
•Reducibility : Undecidable Problems, Example, Mapping Reducibility.
•Time Complexity : Measuring Complexity, The classes of P and NP.

MA 5140
MATHEMATICAL INTRODUCTION TO ELLIPTIC CURVES
•Plane curves, Bezout’s theorem, Basic Theory of Elliptic Curves. Reduction modulo p, Torsion
points. Elliptic curves over the complex numbers, Lattices and bases, Doubly periodic functions.
Heights, Mordell-Weil theorem, rank of E(Q), Neron-Tate pairing, Nagell-Lutz Theorem, Elliptic
curves over finite fields and local fields, Elliptic Curves and it’s relation with modular forms.

MA 5150
ALGEBRAIC NUMBER THEORY
•Localisation, Integral ring extensions, Dedekind domains, discrete valuation rings, unique
factorisation of ideals, ideal class groups, finiteness of class number, some class number
computations, valuations and completions of number fields, Hensel's lemma, norm, trace,
discriminant, different, Ramification theory of p-adic fields, Decomposition groups, Inertia
groups, cyclotomic fields, Gauss sums, quadratic reciprocity, geometry of numbers, Ostrowski's
theorem, Dirichlet’s unit theorem.

MA 5160
AN INTRODUCTION TO MODULAR FORMS
•Modular group, congruence subgroups, modular forms, examples, Eisenstein series, lattice
functions, Some number theoretic applications, space of modular functions, expansions at infinity,
zeroes and poles using contour integrals, Hecke operators, Theta functions, Atkin-Lehner theory,
Petersson inner product, Eigenforms, L-functions and some properties, relation between Modular
forms and Elliptic curves.

MA 5170
BASIC INTRODUCTION TO ALGEBRAIC GEOMETRY
• Algebraic curves in the plane, Singular points and tangent lines, local rings, intersection
multiplicities, Bezout's theorem for plane curves, Max Noether's theorem and some of its
applications. Affine spaces, Projective spaces, Affine and projective varieties, coordinate rings,
morphisms and rational maps, local ring of a point, function fields, dimension of a variety, Zariski's
main theorem.

MA 5180
ADVANCED MEASURE THEORY
• Revision on Radon-Nikodym Theorem, Radon-Nikodym derivative and their applications.
• Complex measure and its various properties, Complex analogue of Radon-Nikodym.
• Theorem. Dual of C0(X), the space of all complex valued continuous functions vanishing at
infinity on a locally compact Hausdorff X.
• A revision on the spaces Lp(\mu) for a \sigma finite measure \mu. Dual of Lp(\mu). Dense
subclasses of Lp(\mu).
• Modes of convergence: pointwise convergence, convergence in measure, convergence almost
uniformly. Egoroff's Theorem.
• Fundamental Theorem of Calculus for Lebesgue Integrals. Derivative of an integral.
• Derivative of a measure: The Lebesgue Differentiation Theorem. Functions of Bounded
Variation and Rectifiable curves in the plane. Absolutely continuous functions.

MA 5190
ADVANCED PARTIAL DIFFERENTIAL EQUATIONS
• Review : Quasi-linear PDE, Cauchy problem, higher order PDE, classification, wave equation, heat
equation, Laplace equation.
• Introduction to non-linear waves : 1-D linear equation, basic non-linear equations, expansion
wave, centered expansion wave, breaking and examples. Shock waves, discontinuous shocks,
equal area rule, asymptotic behavior, shock structure, Burgers equation, Thomas equation.
• Second order systems : The equations of shallow water theory, method of characteristics, waves
on a sloping beach, linear and nonlinear theory, conservation equations and boundary value
problems,, exact solutions for certain nonlinear equations.

MA 6040
FUZZY LOGIC CONNECTIVES AND THEIR APPLICATIONS
• Fuzzy Logic Connectives : Classes and their generation process, Algebraic and analytical
properties, related conjunctions.
• Fuzzy implications : Classes and their generation process, Algebraic and analytical properties.
Fuzzy Measures and Integrals: An Introduction.
• Applications : Including but not limited to :Approximate Reasoning, Clustering and Data Analysis,
Image Processing

MA 6050
WAVELETS AND APPLICATIONS
• Fourier transform - Continuous wavelet transform, frames - Multiresolution analysis, discrete
wavelets, - Spline, orthogonal and biorthogonal wavelets - Applications in Image processing,
Numerical analysis

MA 6060
REDUNDANT AND SPARSE REPRESENTATION THEORY
•Redundant representations, Orthogonal, nonorthogonal and frame type bases, Sparsity,
Coherence, Uncertainty Principle , L1 minimization, Probabilistic and deterministic approaches,
Convex and iterative methods, Applications in analog-to-digital conversion, Nyquist sampling
theory, Low-rank matrix recovery, Dictionary design, Recent develop.

MA 6070
APPROXIMATION THEORY
•The Theorems of Weierstrass, Bernstein, Fejer, and Korovkin, Stone's Approximation Theorem and
the Stone-Weierstrass Theorem, Some applications, Best approximation in normed spaces: some
basic notions and results, Degree of uniform approximation by algebraic and trigonometric
polynomials - Modulus of continuity and modulii of smoothness - Jackson's theorems - Bernstein's
inequality for trigonometric polynomials - Inverse theorems for uniform trigonometric
approximation, Bernstein and Markov inequalities for algebraic polynomials, Characterizations of
best uniform approximants - Theorems of Collatz and Schewdt, Collatz and Kolmogorov - Haar
systems and the Haar-Kolmogorov Theorems - Chebyshev's Alternation Theorem and some
applications.

MA 6080
MEASURE THEORETIC PROBABILITY
•Classical Probability and Preliminaries : Discrete Probability, Conditional Probability, Expectation,
Theorems on Bernoulli Trials. Basic definitions of algebraic structures, few facts about Banach
Spaces; Measure Theory: Sigma Algebra, Measurable functions, Positive and Vector valued
measures, Total Variation of a measure, Spaces of measures, Lebesgue Measure on R, Completion,
Caratheodory’s theorem.
• Lebesgue Integration : Abstract Integral, Convergence theorems of
Lebesgue and Levi, Fatou’s Lemma, Radon-Nikodym Theorem, Modes of convergence of
measurable functions; Product Spaces: Finite Products, Fubini’s Theorem, Infinite Products,
Kolmogorov’s Extension Theorem; Independence: Random Variables, Distributions, Independent
Random Variables, Weak and Strong Law of Large Numbers, Applications.

MA 6090
OPERATOR THEORY
•Operators on Hilbert spaces: Basics of Hilbert spaces; Bounded linear operators, Adjoint of
operators between Hilbert spaces; Self-adjoint, normal and unitary operators; Numerical range
and numerical radius; compact operators, Hilbert Schmidt operators. Spectral results for Hilbert
space operators: Eigen spectrum, approximate eigen spectrum; Spectrum and resolvent; Spectral
radius formula; Spectral mapping theorem; Riesz-Schauder theory; Spectral results for normal,
self-adjoint and unitary operators; Functions of self-adjoint operators.
Spectral representation of
operators: Spectral theorem and singular value representation for compact self-adjoint operators;
Spectral theorem for self-adjoint operators. Unbounded Operators: Basics of unbounded closed
Operators in Hilbert spaces, Cayley transform, Spectral theorem for unbounded self-adjoint
operators.

MA 6100
MATHEMATICS BEHIND MACHINE LEARNING
•Data Representation : Eigenvalues - Eigenvectors - PCA - SVD - Fischer Discriminant; Functionals -
Hilbert Spaces - Riesz Representation Theorem - Kernel Trick - Kernel PCA - Kernel SVM; Norm
Minimization - LLE - Sparse Representation Theory - Dimensionality Reduction.
• Supervised Learning : Convex Optimisation - Primal-Dual Transformations - Karush-Kuhn-Tucker
Conditions - SVM; Probability and Measures - Types of Convergences - Statistical Learning Theory
- VC dimension and Capacity - Some bounds.
• Unsupervised Learning : Expectation Maximization - EM-based Clustering - C-means clustering -
Fuzzy CM clustering; Operator Theory - Decomposition of Operators and Subspaces - Subspace
Clustering.

MA 6110
CONVEX FUNCTIONS AND THEIR APPLICATIONS
•Basic properties of convex functions; Convex functions on a normed linear spaces; Various notions
of differentiability of a convex function on a normed linear space; Monotone operators, Asplund
spaces and Radon Nikodym property; A smooth variational principle and more on Asplund spaces.

MA 6120
AN INTRODUCTION TO OPERATOR ALGEBRAS
•Banach Algebras: Banach Algebras & invertible group; spectrum; multiplicative linear functionals;
Gelfand transform & applications; maximal ideal spaces; Non-unital Banach Algebras.
C*-algebras: C*-algebras; commutative C*-algebras; the spectral theorem and applications; polar
decomposition; positive linear functional and states; The GNS Construction; non unital C*-algebras
von Neumann Algebras: Topologies on B(H); Existence of projections; the Double Commutant
Theorem; the Kaplansky density theorem; the Borel functional calculus; Abelian von Neumann
algebras; the Lα functional Calculus; equivalence projections; Type decompositions

MA 6130
BANACH SPACE THEORY
•Basic properties of Banach spaces; Classical Banach spaces and their various properties; Linear
operators in Banach spaces; Schauder bases; Convexity and smoothness.

MA 6140
COMPRESSIVE SENSING
•Nyquist Sampling Theorem, Under-determined linear systems, Classical solution techniques, l0, l1
and l2 norm minimization problems, Theoretical guarantees for sparse recovery, Greedy
and Convex optimization techniques, Dictionary Learning, Applications in Signal Processing.

MA 6150
DISCRETE DYNAMICAL SYSTEMS
•Phase portraits, Topology of the Real numbers, periodic points and stable sets, Sarkovskii's
theorem, Families of dynamical systems, bifurcation, The logistic function, Cantor sets and chaos,
topological conjugacy. period-doubling cascade. Symbolic dynamics. Newton's method. Complex
dynamics, quadratic family, Julia sets, Mandelbrot set.

MA 6160
BANACH ALGEBRAS
•Banach algebras : Definition, homomorphism, spectrum, basic properties of spectra, GelfandMazur
theorem, spectral mapping theorem, group of invertible elements. Commutative Banach
algebras and Gelfand theory: Ideals, maximal ideals and homomorphism, semi-simple Banach
algebra, Gelfand topology, Gelfand transform, involutions.
Banach*-algebras, Gelfand-Naimark theorem, applications to non-commutative Banach algebras.
A characterization of Banach * - algebras.

MA 6190
TRANSCENDENTAL NUMBER THEORY
•Irrational Numbers: Decimal representation of real numbers, repeating decimals and rational
numbers, irrationality of k-th root of an integer, irrationality of e, π, irrationality of various
trigonometric functions at rational arguments, irrationality of ζ(3).
Transcendental Numbers: Liouville’s construction of transcendental numbers, transcendence of e
and π, Lindemann’s theorem on algebraic independence of exponentials of algebraic numbers and
its corollaries, Gelfond - Schneider theorem on transcendence of algebraic exponents of algebraic
numbers and its corollaries, linear forms in logarithms - Baker’s theorem with application to the
Catalan’s conjecture, Mahler’s construction of transcendental numbers. applications to non-commutative Banach algebras.
A characterization of Banach * - algebras.