Syllabus:
i) Majorization and doubly stochastic matrices. Matrix Decomposition Theorems (Polar, QR, LR, SVD etc.) and their applications. Perturbation Theory. ii) Nonnegative matrices and their applications. Wavelets and the Fast Fourier Transform. Basic ideas of matrix computations.
Suggested Texts :
(a) R. Bhatia, Matrix Analysis, GTM (169), Springer (Indian reprint 2004).
https://www.isibang.ac.in/~adean/infsys/database/MMath/E28ALA.html
- Teacher: B V Rajarama Bhat
- Teacher: Chaitanya G K
Syllabus:
i) Basics of ODE: (local as well as global) existence and uniqueness results, Picard iteration, Gronwalls inequality, solving some first order and second order equations. [7 lectures]
ii) 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 [1 lecture]
iii) 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. [1 lecture]
iv) Laplace equation: Definition of Harmonic functions. Mean-value property, Strong Maximum principle for harmonic functions, Liouvilles theorem, smoothness of harmonic functions, Poissons formula. Harmonic functions in rectangles, cubes, circles, wedges, annuli. [9 lectures]
v) Heat equation: Fundamental solution of heat equation, Duhamels principle, weak and strong maximum principles, smoothness of solutions of heat equation, ill-posedness of backward heat equation. [9 lectures]
vi) Wave equation: well-posedness of initial and boundary value problem in 1D and dAlembert formula. method of descent in 2D and 3D. Duhamels principle, domain of dependence, range of influence, finite speed of propagation. [8 lectures]
vii) Boundary problems: Separation of variables, Dirichlet, Neumann and Robin conditions. The method of seperation of variables for Laplace, Heat and Wave equations. [2 lectures]
Suggested Texts :
(a) Evans, L. C. Partial Differential Equations, AMS, 2010.
(b) Han, Q. A Basic Course in Partial Differential Equations, AMS, 2011.
(c) McOwen, R. Partial Differential Equations: Methods and Applications, Pearson, 2002.
(d) Pinchover, Y. and Rubinstein, J. An Introduction to Partial Differential Equations, Cambridge, 2005.
(e) Fritz John Partial Differential Equations, Springer.
(f) Partial Differential Equations: An introduction by Walter Strauss.
https://www.isibang.ac.in/~adean/infsys/database/MMath/C13PDE.html
- Teacher: Gobinda Sau
- Teacher: Renjith Thazhathethil
Syllabus:
i) Review of discrete probability, First and Second Moment methods, Chernoff bounds and some applications.
ii) Percolation on lattices: Phase-transition phenomena, subcritical and supercritical phases, Uniqueness.
iii) Random graphs: Phase transition, Influences, Russos formula and Sharp thresholds. Noise Sensitivity and Stability.
iv) Introduction to Markov chains and Martingales. Branching processes. Random walks and electrical networks, Uniform spanning trees.
Suggested Texts :
(a) C. Garban and J. Steif: Noise Sensitivity of Boolean Functions and Percolation.
(b) N. Lanchier: Stochastic Modelling.
(c) Sebastien Roch: Modern Discrete Probability: A toolkit. (Notes).
(d) R. Lyons and Y. Peres: Probability on trees and networks.
(e) M. Barlow: Random walks and heat kernel on Graphs.
https://www.isibang.ac.in/~adean/infsys/database/MMath/E19TDP.html
- Teacher: Yogeshwaran D.
- Teacher: Subhajit Ghosh