Intro to Stochastic Processes

Syllabus:

DISCRETE-TIME MARTINGALES: Optional Stopping theorem, Martingale convergence theorem, Doobs inequality and convergence.
BRANCHING PROCESSES: Model definition. Connection with martingales. Probability of survival. Mean and variance of number of individuals.
DISCRETE-TIME MARKOV CHAINS: Classification of states, Stationary distribution, reversibility and convergence. Random walks and electrical networks. Collision and recurrence.
BASIC PROBABILISTIC INEQUALITIES AND APPLICATIONS: First and Second Moment methods. Applications to Longest increasing subsequences, Random k-Sat problem and connectivity threshold for Erdos-Renyi graphs. Chernoff bounds and Johnson-Lindenstrauss lemma.


https://www.isibang.ac.in/~adean/infsys/database/Bmath/SP.html

Computer Science IV

Syllabus: (Design and Analysis of Algorithms) Efficient algorithms for manipulating graphs and strings. Fast Fourier Transform. Models of computation, including Turing machines. Time and Space complexity. NP-complete problems and undecidable problems.

Reference Texts:

1. A. Aho, J. Hopcroft and J. Ullmann: Introduction to Algorithms and Data Structures
2. T. A. Standish: Data Structure Techniques
3. S. S. Skiena: The algorithm Design Manual
4. M. Sipser: Introduction to the Theory of Computation
5. J.E. Hopcroft and J. D. Ullmann: Introduction to Automata Theory, Languages and Computation
6. Y. I. Manin : A Course in Mathematical Logic

Analysis IV

Syllabus (Introduction to Function Spaces): Review of compact metric spaces. C([a; b]) as a complete metric space, the contraction mapping principle. Banachs contraction principle and its use in the proofs of Picars theorem, inverse and implicit function theorems. The Stone-Weierstrass theorem and Arzela-Ascoli theorem for C(X). Periodic functions, Elements of Fourier series - uniform convergence of Fourier series for well behaved functions and mean square convergence for square integrable functions.

Reference Texts:
1. T.M. Apostol: Mathematical Analysis.
2. T.M.Apostol: Calculus
3. S. Dineen: Multivariate Calculus and Geometry.

Intro to Algebraic Geometry

Syllabus

Prime ideals and primary decompositions, Ideals in polynomial rings, Hilbert basis theorem, Noether normalisation theorem, Hilbert's Nullstellensatz, Projective varieties, Algebraic curves, Bezout's theorem, Elementary dimension theory.

Reference Texts:

1. M. Atiyah and I.G. MacDonald: Commutative Algebra
2. J. Harris: Algebraic Geometry
3. I. Shafarervich: Basic Algebraic Geometry
4. W. Fulton: Algebraic curves
5. M. Ried: Undergraduate Commutative Algebra

https://www.isibang.ac.in/~adean/infsys/database/Bmath/AG.html

Analysis IV

Syllabus (Introduction to Function Spaces): Review of compact metric spaces. C([a; b]) as a complete metric space, the contraction mapping principle. Banachs contraction principle and its use in the proofs of Picars theorem, inverse and implicit function theorems. The Stone-Weierstrass theorem and Arzela-Ascoli theorem for C(X). Periodic functions, Elements of Fourier series - uniform convergence of Fourier series for well behaved functions and mean square convergence for square integrable functions.

Reference Texts:
1. T.M. Apostol: Mathematical Analysis.
2. T.M.Apostol: Calculus
3. S. Dineen: Multivariate Calculus and Geometry.

https://www.isibang.ac.in/~adean/infsys/database/Bmath/A4.html


Intro to Differential Topology

Syllabus

Manifolds. Inverse function theorem and immersions, submersions, transversality, homotopy and stability, Sards theorem and Morse functions, Embedding manifolds in Euclidean space, manifolds with boundary , intersection theory mod 2, winding numbers and Jordan- Brouwer separation theorem, Borsuk- Ulam fixed point theorem.

Reference Texts:

1. V. Guillemin and Pollack: Differential Topology (Chapters I, II and Appendix 1, 2).
2. J. Milnor: Topology from a differential viewpoint.

https://www.isibang.ac.in/~adean/infsys/database/Bmath/DiffTop.html

Differential Equations

Syllabus: Ordinary differential equations - first order equations, Picards theorem (existence and uniqueness of solution to first order ordinary differential equation), Second order linear equations - second order linear differential equations with constant coefficients, Systems of first order differentialequations, Equations with regular singular points, Introduction to power series and power series solutions, Special ordinary differential equations arising in physics and some special functions (eg. Bessels functions, Legendre polynomials, Gamma functions). Partial differential equations - elements of partial differential equations and the three equations of physics i.e. Laplace, Wave and the Heat equations, at least in 2 - dimensions. Lagranges method of solvingfirst order quasi linear equations.

Reference Texts:
1. G.F. Simmons:Differential Equations
2. R. Haberman: Elementary applied partial differential equations
3. R. Dennemeyer: Introduction to partial differential equations and boundary value problems

https://www.isibang.ac.in/~adean/infsys/database/Bmath/DiffEq.html

Physics IV

Syllabus:(Modern Physics and Quantum Mechanics): Special theory of Relativity: Michelson-Morley Experiment, Einstein's Postulates, Lorentz Transformations, length contraction, time dilation, velocity transformations, equivalence of mass and energy. Black body Radiation, Planck's Law, Dual nature of Electromagnetic Radiation, Photoelectric Effect, Compton effect, Matter waves, Wave-particle duality, Davisson-Germer experiement, Bohr's theory of hydrogen spectra, concept of quantum numbers, Frank-Hertz experiment, Radioactivity, X-ray spectra (Mosley's law), Basic assumptions of Quantum Mechanics, Wave packets, Uncertainty principle, Schrodinger's equation and its solution for harmonic oscillator, spread of Gaussian wave packets with time.

https://www.isibang.ac.in/~adean/infsys/database/Bmath/Phy4.html

Computer Science III

Syllabus: (Data Structures) Fundamental algorithms and data structures for implementation. Techniques for solving problems by programming. Linked lists, stacks, queues, directed graphs. Trees: representations, traversals. Searching (hashing, binary search trees, multiway trees). Garbage collection, memory management. Internal and external sorting.

Reference Texts: Data Structures Using C and C++ 2 Edition, by Yedidyah Langsam, Aaron M. Tenenbaum, Moshe J. Augenstein, PHI 2009

https://www.isibang.ac.in/~adean/infsys/database/Bmath/CS3.html