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Algebraic Number Theory

Syllabus: Number fields and number rings, prime decomposition in number rings, Dedekind domains, definition of the ideal class group, Galois theory applied to prime decomposition and Hilbert ramification theory, Gauss reciprocity law, Cyclotomic fields and their ring of integers as an example, the finiteness of the ideal class group, Dirichlet Unit theorem.

Reference Texts:
1. D. Marcus: Number fields
2. G. J. Janusz: Algebraic Number Theory

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

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Syllabus:

(Note: A priori knowledge of Commutative Algebra is desirable.)
Algebraic numbers and algebraic integers; Brief review of integral extensions; Norm, trace and discriminant; Existence of integral basis. Dedekind domains, ideal class group. Minkowsky theory, finiteness of class group. Dirichlet unit theorem. Factoring of prime ideals on extensions, fundamental identity; Quadratic number fields (computation of class numbers, prime decomposition, Pells equa tions). Hilberts ramification theory (decomposition and inertia groups); Cyclo tomic fields. Valuations, completions, local fields

Suggested Texts :
(a) G.J. Janusz: Algebraic Number Fields, (chapter 1-4), AMS (1996).
(b) D.A. Marcus: Number Fields, Springer-Verlag (1977).
(c) J. Neukirch: Algebraic Number Theory, Springer (1999).
(d) P. Ribenboim: Classical Theory of Algebraic Numbers, Springer Science and Business Media (2001).
(e) J. Esmonde and M. Ram Murty: Problems in Algebraic Number Theory, Springer (Indian reprint 2006).
(f) TIFR pamphlet on Algebraic Number Theory.

https://www.isibang.ac.in/~adean/infsys/database/MMath/E24ANT.html

Introduction 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.

Reference Texts:

(a) N. Lanchier: Stochastic Modelling.
(b) W. Feller: Introduction to Probability: Theory and Applications - Vol. I and II..
(c) L. Levine, Y. Peres and E. Wilmer: Markov chains and mixing times.
(d) Sheldon Ross: Probability Models.
(e) Santosh S. Venkatesh: Theory of Probability - Explorations and Applications.
(f) R. Meester: A Natural Introduction to Probability Theory.
(g) S. R. Athreya and V. S. Sunder: Measure and Probability.
heart Sebastien Roch: Modern Discrete Probability: A toolkit. (Notes).

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

Design and Analysis of Algorithms

Syllabus: Basics of Algortihm Analysis: Models of computation, asymptotic order of growth, algorithm analysis, time and space complexity, average and worst case analysis, lower bounds.
Algorithm design techniques: Greedy algorithms, Divide and conquer, dynamic pro- gramming, amortization, randomization.
Complexity classes: Problem classes P, NP, PSPACE; reducibility, NP-hard and NP complete problems. Approximation algorithms for some NP-hard problems.

Reference Texts:

(a) T. H.Cormen, C.E.Leiserson, R.L.Rivest, C. Stein: Introduction to Algorithms
(b) J. Kleinberg and E. Tardos: Algorithm Design
(c) R. Sedgewick and P. Flajolet: Introduction to the Analysis of Algorithms
(d) R. Sedgewick: Algorithms
(e) A. Aho, J. Hopcroft and J. Ullmann: Introduction to Algorithms and Data Structures
(f) M. Sipser: Introduction to the Theory of Computation
(g) S. S. Skiena: The algorithm Design Manual

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

Probability III

Syllabus:

- Sigma-algebras, axioms of probability, pi - lamda theorem (proof can be skipped), uniqueness of extension for probability measures. Examples of countable probability spaces, Borel sigma-algebra on the real line and standard probability distributions on the real line.
- Construction of Lebesgue measure (statement alone). Random variables and examples. Push-forward of a probability measure (sketch of proof) . Borel probability measures on Euclidean spaces as push-forward of Lebesgue measure (statement alone); Cumulative distribution function and properties.
- General definition of expectation and properties. Change of variables. Review of conditional distribution and conditional expectation, General definition, Examples.
- Limit theorems: Monotone Convergence Theorem (MCT) (without proof), Fatous Lemma, Dominated Convergence Theorem (DCT), Bounded Convergence Theorem (BCT), Cauchy-Schwartz, Jensen and Chebyshev inequalities.
- Different modes of convergence and their relations, Weak Law of large numbers, First and Second Borel-Cantelli Lemmas, Strong Law of large numbers (proof under finite variance).
- Characteristic functions, properties, Inversion formula and Levy continuity theorem (statements only), CLT in i.i.d. finite variance case. Slutskys Theorem.
- Introduction to Finite Markov chains - Definition. Random mapping representation. Examples. Irreducibility and aperiodicity. Stationary distribution and reversibility. Random walks on graphs.

Reference Texts:

(a) N. Lanchier: Stochastic Modelling.
(b) W. Feller: Introduction to Probability: Theory and Applications - Vol. I and II..
(c) J. Pitman: Probability.
(d) Sheldon Ross: Probability Models.
(e) Santosh S. Venkatesh: Theory of Probability - Explorations and Applications.
(f) R. Meester: A Natural Introduction to Probability Theory.
(g) S. R. Athreya and V. S. Sunder: Measure and Probability

https://www.isibang.ac.in/~adean/infsys/database/Bmath/PT3.html