# Prelim/GRE Level

## References and Study Material

• L. Ahlfors, ‘Complex Analysis’ (McGraw-Hill, 1973).
• V. Arnold, ‘Ordinary Differential Equations’ (MIT Press)
• M. Artin, ‘Algebra’ (Prentice Hall, 1991)
• F. Kirwan, ‘Complex Algebraic Curves’, London Mathematical Series Student Texts 23 (Cambridge, 1992).
• W. Massey, ‘A Basic Course in Algebraic Topology’ Graduate Texts in Mathematics Vol. 127 (Springer-Verlag, 1991).
• J. Milnor, ‘Topology from the Differentiable Viewpoint
• S. Ross, ‘A First Course in Probability’ (5th edition, Prentice-Hall, 1997).
• W. Rudin, ‘Real and Complex Analysis
• J-P. Serre, ‘A course in Arithmetic
• J-P. Serre, ‘Linear Representations of Finite Groups

## Linear Algebra

Finite dimensional vector spaces (over R) and linear maps between them – subspaces, quotient spaces, dimension, bases, matrix representations. Positive definite inner products, orthonormal bases, extensions of orthonormal subsets. Eigenvalues and eigenvectors for automorphisms. Characteristic polynomial.

References: M. Artin, ‘Algebra’ (Prentice Hall, 1991), Chapters 1,3,4K. Hoffman and R. Kunze, ‘Linear Algebra’, Chapters 1-6, (Prentice-Hall, 1971).

## Abstract Algebra

Definitions of groups, rings, fields, and modules over a ring. Homomorphisms of these objects. Subgroups, normal subgroups, quotient groups. Cyclic groups, finite abelian groups (structure theorem). Ideals, prime and maximal and their quotients — basic examples Z, *****k*[X], rings of algebraic integers. Field extensions, splitting fields of polynomials, normal extensions.

References: M. Artin, ‘Algebra’, Chapters 2, 10, 11, 12, 13, 14I. Herstein, ‘Topics in Algebra’ (Blaisdell Publishers, 1964).

## Point-set Topology

Open and closed sets, continuous functions. Connectedness, compactness, Hausdorff, normality. Metric spaces, Rn. Heine-Borel theorem.

Reference: J. Munkres, ‘Topology, A First Course’, Part I (Prentice-Hall).

## Calculus

Differential of a smoothing mapping between open subsets in Euclidean spaces. Matrix of partial derivatives. Inverse and implicit functions. Multivariable Riemann integration.

References: W. Rudin, ‘Principles of Mathematical Analysis’ (McGraw-Hill, 1964)A. Browder, ‘Mathematical Analysis: An Introduction’ (Springer, 1996).

## Complex Analysis

Definition of holomorphic functions, Cauchy integral formula, power series representations of holomorphic functions, radius of convergence, meromorphic functions, residues.

Reference: L. Ahlfors, ‘Complex Analysis’, (McGraw-Hill, 1973), Chapters 1- 5.Real Analysis: A thorough working knowledge of advanced calculus, at the level of the books of W. Rudin or A. Browder as listed under Calculus.