TEW Analysis & Algebra (2021)

School Convener(s)

Lecture 1: Continuous functions, Types of discontinuity through examples, limit at infinity, infinite limits, limit of functions.
Lecture 2: Sandwich theorems for sequences and functions, nested intervals, Intermediate Value Theorem (IVT).
Lecture 3: Examples: log and exp functions, uniform continuity, Lipschitz continuous functions.
Lecture 4: Topological preliminaries, converse of Intermediate Value Theorem is not true (example), certain consequences, Theorem of Darboux using both intermediate and mean value theorems.
Lecture 5: Differentiation, Mean Value Theorem, Rolle’s theorem, monotonic property.
Lecture 6: Extended Mean Value Theorem EMVT), Some applications of EMVT, Taylor’s theorem.

Lecture 1. Basics of Complex Numbers, Geometry of Complex Numbers, The exponential function, Complex functions as mappings, The function arg(z), many to one functions, branches of arg(z), principle branch of arg(z), The complex logarithmic function log(z), complex powers.
Lecture 2. Brief introduction to multivariable differential calculus, directional derivative, total derivative, Jacobian, Complex function differentiation, holomorphicity and C-R equations.
Lecture 3. Power series, its derivative of power series, definition of analyticity of a complex function.
Lecture 4. Integration along the curves, Goursat's Theorem, Cauchy theorem for disc.
Lecture 5. Keyhole Contour, Cauchy's Integral Formula, Cauchy's estimate, Holomorphicity implies Analyticity, Liouville's theorem for entire functions, Fundamental theorem of algebra, Identity theorem, Analytic continuation and Morera's theorem.
Lecture 6. Holomorphicity of the limit of the sequence of holomorphic functions, Zeros and Poles of complex functions, Residue of a pole, Riemann's theorem on removable singularity, Essential Singularity. Then without proof, we discussed Casorati-Weierstrass theorem, Argument principle, Open mapping theorem and Maximum modulus principle.

Lecture 1: Explained my pedagogical strategies. Collected a few topics of interest from the participants. Gave a quick introduction to quantifiers and its importance in teaching algebra. Well ordering principle, induction principle and their equivalence. Explained the ring, vector space and module faces of L(V,V) (set of all linear maps from V to V, where V is a vector space over a field)
Lecture 2: Axiomatic approach based introduction to ring theory. Definition and examples of Rings.
Lecture 3: normal subgroups and Quotient groups, Ideals and Quotient Rings. Polynomial rings
Lecture 4: Commutative Ring structure of polynomials with coefficients in field Division algorithm for polynomials over Fields analogous to the division algorithm for integers. Correspondence theorem of subrings of ring and quotient ring
Lecture 5: Primality and irreducibility, Quadratic residues modulo p, Lecture 6:C[0,1] - A commutative ring with identity and not an integral domain. Maximal ideals of C[0,1].