Syllabus:
For Prob III: All limit theorems used in this course will be stated in context with
applications. These can be proved in Probability III rigorously.
i) Sufficiency, Exponential family, Bayesian methods, Moment methods, Maximum
likelihood estimation.
ii) Criteria for estimators; UMVUE, Fisher Information.
iii) Multivariate normal distribution: Marginals, Conditionals; Distribution of linear
forms.
iv) Order statistics and their distributions.
v) Large sample theory: Consistency, asymptotic normality, asymptotic relative
efficiency.
vi) Elements of hypothesis testing; Neyman-Pearson Theory, UMP tests, Likelihood
ratio and related tests, Large sample tests.
vii) Confidence intervals.
Reference Texts:
(a) George Casella and Roger L Berger: Statistical Inference.
(b) Peter J Bickel and Kjell A Doksum: Mathematical Statistics.
(c) Erich L Lehmann and George Casella: Theory of Point Estimation.
(d) Erich L Lehmann and Joseph P Romano: Testing Statistical Hypotheses.
https://www.isibang.ac.in/~adean/infsys/database/Bmath/ISI.html
- Teacher: Mohan Delampady
Syllabus:
i) BASIC COUNTING TECHNIQUES: Double-counting, Averaging principle, Inclusion-
Exclusion principle. Euler indicator, Mobius function and inversion formula.
Recursions and generating functions.
ii) PIGEONHOLE PRINCIPLE: The Erdos-Szekeres theorem. Mantels theorem.
Turans theorem, Dirichlets theorem, Schurs theorem, Ramsey theory.
iii) GRAPHS: Eulers theorem and Hamilton Cycles. Spanning Trees. Cayleys
theorem and Spanning trees.
iv) SYSTEMS OF DISTINCT REPRESENTATIVES: Halls marriage theorem, Applications
to latin rectangles and doubly stochastic matrices, Konig-Egervary theorem,
Dilworths theorem, Sperners theorem.
v) FLOWS IN NETWORKS: Max-flow min-cut theorem, Ford-Fulkerson theorem,
Integrality theorem for max-flow.
vi) LATIN SQUARES AND COMBINATORIAL DESIGNS: Orthogonal Latin squares,
Existence theorems and finite projective planes. Block designs. Hadamard
designs, Incidence matrices. Steiner triple systems.
Reference Texts:
(a) S. Jukna: Extremal Combinatorics.
(b) J. H. van Lint & R. M. Wilson: A Course in Combinatorics.
(c) D. B. West: Introduction to Graph Theory.
(d) R. A. Beeler: How to Count: An Introduction to Combinatorics and Its Applications.
(e) H. J. Ryser: Combinatorial Mathematics.
https://www.isibang.ac.in/~adean/infsys/database/Bmath/DM1.html
- Teacher: Yogeshwaran D.
Syllabus:
i) Equivalence relations and partitions, Zorns lemma, Axiom of choice, Principle
of mathematical induction.
ii) Groups, subgroups, homomorphisms, Modular arithmetic, quotient groups, isomorphism
theorems.
iii) Groups acting on sets, Sylows theorems, Permutation groups,Semidirect products.
iv) Classification of groups of small order.
v) Notion of solvable groups and proof of simplicity of An (for n > 4).
vi) Matrix groups O(2); U(2); SL(2;R), Matrix exponential.
Reference Texts:
(a) M. Artin: Algebra.
(b) S. D. Dummit and M. R. Foote: Abstract Algebra.
(c) I. N. Herstein: Topics in Algebra.
(d) K. Hoffman and R. Kunze: Linear Algebra.
(e) J. A. Gallian: Contemporary Abstract Algebra.
https://www.isibang.ac.in/~adean/infsys/database/Bmath/GTh.html
- Teacher: Maneesh Thakur
Syllabus (Vector Calculus): Multiple
integrals,
Existence of the Riemann integral for sufficiently well-behaved
functions on rectangles, i.e. product of intervals. Multiple integrals
expressed as iterated simple integrals. Brief
treatment of multiple integrals on more general domains. Change of
variables and the Jacobian formula, illustrated with plenty of examples.
Inverse and implicit functions theorems
(without proofs). More advanced topics in the calculus of one and
several variables curves in R2 and R3 . Line integrals,
Surfaces in R3, Surface integrals, Divergence,
Gradient and Curl operations, Green's, Strokes' and Gauss' (Divergence)
theorems. Sequence of functions - pointwise versus uniform convergence
for a function defined on an interval of
R, integration of a limit of a sequence of functions. The
Weierstrass's theorem about uniform approximation of a continuous
function by a sequence of polynomials on a closed
bounded interval.
- Teacher: B Rajeev
Syllabus:
Introduction to Statistics with examples of its use; Descriptive
statistics; Graphical representation of data: Histogram, Stem-leaf
diagram, Box-plot; Exploratory statistical
analysis with a statistical package; Basic distributions, properties;
Model fitting and model checking: Basics of estimation, method of
moments, Basics of testing, interval estimation;
Distribution theory for transformations of random vectors; Sampling
distributions based on normal populations: t, x2 and Fx distributions.
Bivariate data, covariance, correlation and
least squares.
References Texts:
1. Lambert H. Koopmans: An introduction to contemporary statistics
2. David S Moore and William I Notz: Statistics Concepts and
Controversies
3. David S Moore, George P McCabe and Bruce Craig: Introduction to the Practice of Statistics
4. Larry Wasserman: All of Statistics. A Concise Course in Statistical
Inference
5. John A. Rice: Mathematical Statistics and Data Analysis.
- Teacher: Rituparna Sen
Syllabus: Systems of linear equations:
Gaussian elimination, LU, QR decompositions, Singular values, SVD, Inner
product spaces, Projections onto subspaces, Perron-Frobenius,
Fundamental theorem of LP, The simplex algorithm,
Duality and applications, LP and Game theory.
References Texts:
1. Harry Dym: Linear algebra in action, AMS Publications 2011
2. A. R. Rao & Bhimasankaram: Linear Algebra
3. Strang, Gilbert Linear algebra and its applications. Second edition.
Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London,
1980.
4. C. R. Rao: Linear statistical inference
5. H. Karloff: Linear programming
6. S-C Fang & S.Puthenpura: Linear optimization and extensions
- Teacher: Soumyashant Nayak
Syllabus: (Rings and Modules) Ring
homomorphisms, quotient rings,
adjunction of elements. Polynomial rings. Chinese remainder theorem and
applications. Factorisation in a ring. Irreducible and prime elements,
Euclidean domains, Principal Ideal
Domains, Unique Factorisation Domains. Field of fractions, Gauss's
lemma. Noetherian rings, Hilbert basis theorem. Finitely generated
modules over a PID and their representation matrices.
Structure theorem for finitely generated abelian groups. Rational form
and Jordan form of a matrix.
- Teacher: Jishnu Biswas
Syllabus (Thermal physics and Optics):
Kinetic theory of Gases; Ideal Gas equation; Maxwells Laws for
distribution of molecular speeds. Introduction to Statistical mechanics;
Specification of state of many particle system; Reversibility and
Irreversibility; Behavior of density of states; Heat and Work;
Macrostates and Microstates; Quasi static processes;
State function; Exact and Inexact differentials; First Law of
Thermodynamics and its applications; Isothermal and Adiabatic changes;
reversible, irreversible, cyclic processes; Second
law of thermodynamics; Carnots cycle. Absolute scale of temp; Entropy;
Joule Thomson effect; Phase Transitions; Maxwells relations; Connection
of classical thermodynamics with statistical
mechanics; statistical interpretation of entropy.; Third law of
Thermodynamics.
Optics: Light as a scalar wave; superposition of waves and interference; Youngs double slit
experiment; Newtons rings; Thin films; Diffraction; Polarization of light;transverse nature of light waves.
References Texts:
1. F.Reif: Statistical and Thermal physics
2. Kittel and Kroemer: Thermal Physics
3. Zeemansky and Dittman: Heat and Thermodynamics
4. Jenkins and White: Optics
- Teacher: Chittaranjan Hens