TEW Algebra, Discrete Mathematics, Complex Analysis (2017)

Algebra:  by Sankaran Viswanath (10.5 hours)

Topics covered

• Group actions on sets: orbits and stabilizers
• Burnside’s Lemma
• The class equation and applications
• Applications of group actions to Group Theory: the Sylow Theorems
• Applications of group actions to Combinatorics: counting necklace configurations, weighted version of Burnside’s Lemma, simple instances of Polya’s enumeration theorem.
• Application of group actions to elementary number theory: orbits for the action of the automorphism group of a cyclic group and an identity involving d(n) (the number of divisors of n ) and φ(n) (Euler’s Totient function).

Complex Analysis: by Parameswaran Sankaran (8.5 hours)

Topics covered

• Holomorphic functions, Cauchy-Riemann equations, power series.
• Local properties of holomorphic functions, singularities, meromorphic functions, Laurent series.
• Complex integration, residue calculus, Rouché’s theorem.
• Infinite product, Weierstrass’ theorem; growth of an entire function, Hadarmard’s theorem.

The emphasis was on explaining the concepts with the help of examples rather than on proving theorems.

Complex Analysis:  by K. Srinivas (2.5 hours)
The Riemann zeta function was introduced. It was shown that

• The Riemann zeta function ζ(s), s = σ + it , is analytic in the half-plane σ > 1
• ζ(s) admits analytic continuation in σ > 0
• by means of functional equation, ζ(s) extends to a meromorphic function in the entire complex
• place with a simple pole at s = 1 with residue 1
• ζ(s) has real zeros at s = −2, −4, · · ·
• The following topics were briefly discussed:
• The complex zeros of ζ(s) , Riemann Hypothesis, and Hardy’s theorem.
• prime number theorem.

Discrete Mathematics:  by K. N. Raghavan (11.5 hours)

Topics covered

• Colourings and exact colourings of graphs; the chromatic polynomial; acyclic orientations and statement of Stanley’s theorem that the chromatic polynomial evaluated at −1 gives, up to sign,the number of acyclic orientations.
• Definition of heaps of pieces, enumeration of heaps.
• Generating functions of heaps, the inversion lemma.
• proof of Stanley’s theorem using the inversion lemma.
• calculations with generating functions: Fibonacci sequence; Fibonacci polynomials; Dyck paths.
• Matching polynomials of graphs; statement about all their roots being real.

The emphasis was more on examples and calculations rather than on theory.

 Time Table

How to reach

The Chennai neé Madras, and to IMSc, the Institute of Mathematical Sciences. Here is some information on getting to the Institute. Address and phone numbers are

THE INSTITUTE OF MATHEMATICAL SCIENCES
C I T CAMPUS, TARAMANI, CHENNAI 600 113
Telephone Numbers

Here are some maps to give you a rough idea of the layout of Madras and the area around the Institute.

• A simplified schematic map of Madras, showing the major roads and Hotels and marking the area in which the CIT Campus is located.
• A postscript version of this figure if you want to download it to print at your end.
• A detailed Madras map.
• A gzipped postscript version of this figure
• A schenatic map of the location of IMSc within CIT Campus and its environs
• Detailed map of the location of IMSc within CIT Campus and other landmarks in the Greater Adyar area.

The Institute of Mathematical Sciences is located in a campus-like area variously known as the ``CIT Campus'' or the ``CPT Campus''. Many people in Madras may not have heard of the Institute, but almost everyone would have heard of the M.G.R. Film City and Tidel Park which are further down in the same campus area. IMSc has two complexes. One houses the main Institute building, the library building and the new building. The other has the students' hostel and Guest House. These two complexes are across the road from each other.