Geometric Group Theory

Syllabus:

Review of group theory (basic notions and operations with groups, solvable and nilpotent groups), Groups with generators and relations: Generating sets,free groups, reduced words, presentation of a group, finitely presented groups.Products and extensions of groups, free products and amalgamated free products.
Groups and their Cayley graphs, trees and free groups, Review of groups actions, free actions, orbits and stabilisers, transitive actions. Free groups and actions on trees, The ping-pong lemma, The Tits alternative.
Quasi-isometry of metric spaces, quasi-isometry type of groups, quasi-geodesic and quasi-geodesic spaces, the ˇSvarc-Milnor Lemma with applications, quasi-isometry invariants.
Growth of groups, growth types, groups of polynomial growth, groups of uniform exponential growth.
ADDITIONAL TOPICS FROM: Hyperbolic groups, Ends and boundaries, Amenable groups.

Reference Texts:

(a) Clara Loh: Geometric Group Theory: An introduction.
(b) Pierre de la Harpe: Topics in Geometric Group Theory.
(c) John Meier: Groups, Graphs and Trees: An introduction to the geometry of infinite groups.
(d) M. Clay and D. Margalit: Office Hours with a geometric group theorist.

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

Information Theory

Syllabus: Introduction to Markov chains (if necessary). Shannons entropy, Gibbs inequality, Typical sequences of random vectors, Shannons theorem. Capacity-cost function and channel coding theorem. Rate distortion function and source coding theorem. Steins lemma and properties of information measures. Uniquely decipherable codes, Codes on trees, Krafts inequality, Krafts code, Huffmans code, Shannon-Fano-Elias code. Parsing codes and trees, Turnstalls code. Universal source coding, Empirical distributions, Kullback-Leibler divergence. Parsing entropy, Lempel-Ziv algorithm, Entropy equivalence. Mutual information and capacity of noisy channels.

ADDITIONAL TOPICS FROM: Stationary coding of finite alphabets, Ergodic theorem for binary alphabets and examples. Frequencies of finite blocks and Entropy theorem.

Reference Texts:

(a) T. M. Cover and J. A. Thomas. Elements of Information Theory.
(b) P. Bremaud. Discrete Probability Models and Methods.
(c) D. J. C. Mackay. Information Theory, Inference and Learning Algorithms.
(d) Robert J. McEliece. The Theory of Infomation and Coding.
(e) Paul C. Shields. The Ergodic Theory of Discrete Sample Paths.

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

Geometry

Syllabus:

AFFINE GEOMETRY: Affine spaces, mappings. Thales theorem, Pappus the orem, Desargues theorem. Convexity.

EUCLIDEAN GEOMETRY: Euclidean vector spaces, Euclidean affine spaces. Linear isometries and rigid motions. Spheres, Spherical triangles, Polyhedra and Eulers formula.

PROJECTIVE GEOMETRY: Projective spaces, Pappus and Desargues theorem. Projective duality, Projective transformations. Cross Ratio. Complex projective line and circular group

Reference Texts:

(a) M. Audin: Geometry
(b) R. Fenn: Geometry
(c) P. M. H. Wilson: Curved Spaces: From Classical Geometries to Elementary Differential Geometry

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

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

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.

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