Expository Articles by Prof. S. Kumaresan

 These are the articles (mainly expository in nature) written by me. They were used by me either as hand-outs in some of my courses or assigned as seminar material to students. While most of them are original, a few of them are my versions of others' work disseminated in a form suitable for my students.

If you use them in any way, please send me your comments. kumaresa at gmail dot com.
 
All the articles given below are in pdf format. To download, please (left)click on the title of the article. To open pdf files, you may have to use pdf readers like xpdf or Acrobat reader which you can freely download from the internet.

• Set Theory
1. Exercises on Inverse Images
2. Finite Sets
3. Knaster Fixed Point Theorem
4. Partial Order
5. Partial Order, Total Order and Well-ordering
6. Uncountability of R
7. Axiom of Choice and its Equivalents

• Algebra
1. Insolvability of Quintic
2. Isometries of Rn and Sylvester Criterion for Positive Definite Matrices
3. Burnside Theorem on Matrix Algebras and Its Application to Group Theory
4. Classification of Real Division Algebras
5. Outline of a Course in Field Theory
6. Fundamental Theorem of Algebra via Linear Algebra
7. Isometries of Rn
8. Jordan Canonical Form
9. Jordan Canonical Form
10. Row Rank of a Matrix equals its Column Rank
11. A Problem Course in Module Theory
12. A Problem Course in Module Theory - Exercises
13. Sign of a Permutation
14. Polar Decomposition, Singular Values, Spectral Theorem
15. How to work with quotient rings?
16. Finite Dimensional Spectral Theory
17. Structure of Linear Maps
18. Sylow Theorem
19. Sylvester Criterion for Positive Definiteness

• Analysis
1. Arzela-Ascoli Theorem and Applications
2. Picard's Little Theorem
3. Bishop-Stone-Weierstrass Theorem
4. Review of Differential Calculus
5. Completeness of R
6. Change of Variable Formula for Multiple Integrals
7. LUB Property of Ordered Fields
8. A Diagnostic Quiz in Complex Analysis
9. Contraction Principle and its Applications
10. Existence of Continuous Logarithms
11. A Continuous but Nowhere Differentiable Function
12. Decimal, Binary and Ternary Expansions
13. Solution of Dirichlet Problem - Perron's Method
14. Double Series
15. The Fundamental Theorem of Algebra
16. Fubini Theorem
17. Fundamental Theorem of Calculus in HK-Integral
18. Implicit Function Theorem for Banach Spaces
19. Integers and Polynomials
20. Lebesgue Integral without Measure Theory
21. Similarity of Matrices via Module Theory
22. Compact Subsets of Lp Spaces
23. The Role of LUB Axiom in Real Analysis
24. Minkowski Inequality
25. Miscellaneous Results in Analysis
26. A Conceptual Introduction to Several Variable Calculus
27. Applications of Nested Interval Theorem
28. Nested Interval Theorem and Its Applications
29. The Role of LUB Axiom in Real Analysis
30. Construction of real numbers
31. Existence and Uniqueness of Solutions of ODE
32. Little Picard Theorem
33. Topics in Measure Theory
34. An Outline of Real Analysis
35. Riesz-Markov Representation Theorem
36. Properties equivalent to the LUB property
37. Stokes Theorem | A simple and intuitive proof
38. Stokes Theorem ? An Outline of a Proof
39. Stone-Weierstrass Theorem
40. Uniform Convergence
41. Every Real Sequence has a Monotone Subsequence
42. Weierstrass Approximation Theorem

• Topology
1. Every compact metric space is the continuous image of the cantor set
2. Simple and Direct Proofs of Some Results in the Theory of Compact Spaces
3. Discrete Subgroups of R^n
4. Lebesgue Covering Lemma
5. Vector Fields on Spheres and Brouwer's Fixed Point Theorem
6. Characterization of Compact Metrizable Spaces via Metrics and Continuous Functions
7. Existence of Smooth Functions and Partition of Unity
8. Subspace Topology
9. Topological Groups | via Problems
10. Generating Topologies| A Unified View of Subspace, Product and Quotient Topologies
11. Tychonoff?s Theorem
12. Noncontractibility of the Circle
13. Connected and Path Connected spaces
14. Continuity of Roots of Polynomials
15. R^m is homeomorphic to R^n iff m = n.
16. Any compact hypersurface in Rn is a level surface
17. Maps into the Punctured Plane
18. Existence of Continuous Functions
19. Winding Numbers of Loops

• Geometry
1. Problem Set In Differential Geometry
2. Frenet-Serret Formulas for Space Curves
3. Implicit Function Theorem
4. Inverse Mapping Theorem
5. Gauss Golden Theorem - Isometry - Examples
6. The Method of Lagrange Multipliers - A Geometric Treatment
7. How to draw the graphs of functions on R2?
8. Geometric Applications of Sturm Comparison Theorem
9. Theory of Surfaces|A Rapid Introduction

• Harmonic Analysis
1. Plancherel Theorem and Fourier Inversion Theorem