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