Name of the Speakers with their affiliation.

No. of Lectures/Hrs

Detailed Syllabus

Speakers

1) Anupam K. Singh (IISER, Pune)

2) Ganesh Kadu (S. P. Pune Univ., Pune)

3) Vijay Patankar (JNU, New Delhi)

4) ParvatiShastri (Mumbai Univ.)

Associate Teachers:

1) Dilpreet Kaur (IISER, Pune)
2) JatinMajithia (COEP, Pune)

6 Hrs
each

Algebra I (Linear Algebra and Group Theory)
[Artin] Chapters 6,7,8,9 (1^st Edition) or Chapters 7,8,9,10 (2^nd Edition)

1. [Artin] Michael Artin, Algebra, Pearson, 1991, Prentice-Hall of India, New-Delhi 2003.

Topics:

(i) Operations of a group on itself, the class equation of the icosahedral group, operations on subsets, the Sylow theorems, the groups of order 12, computations on the symmetric group, the free group, generators and relations, the Todd-Coxeter Algorithm.

(ii) Bilinear forms, symmetric forms: orthogonality, the geometry associated to a positive form, Hermitian forms, the spectral theorem, conics and quadrics, the spectral theorem for normal operators, skew-symmetric forms, summary of results in matrix notation.

(iii) Linear groups, the classical linear groups, the special unitary group SU2, the orthogonal representation of SU2, the special linear group SL2(R), one-parameter subgroups, the Lie algebra, translations in a group, simple groups.

(iv) Group representations, G-invariant forms and unitary representations, compact groups, G-invariant subspaces and irreducible representations, compact groups, G-invariant subspaces and irreducible representations, characters, permutation representations and the regular representation, the representations of the icosahedral group, one-dimensional representations, Shur’s Lemma, and proof of the orthogonality relations.

Speakers

1) Chandrasheel Bhagwat (IISER, Pune)

2) Rohit Holkar

3) Shantanu Dey

4) A. R. Shastri (IIT, Mumbai)

Associate Teachers:

1) Krishna Masalkar (Garware College, Pune)
2) PratulGadagkar (Modern College, Pune)
3) Divakaran (IMSc)

Analysis I (Complex Analysis)

[Stein and Shakarchi] Chapters 2,3,4.

1. [Stein and Shakarchi] Elias M. Stein and Rami Shakarchi Complex Analysis, Princeton Lectures in Analysis-II, 2003.

Topics:

Cauchy’s Theorem and Its Applications, Meromorphic Functions and the Logarithm, The Fourier Transform

Speaker

1) Sandeep Singh (IITB, Mumbai)

2) Amit Kuber (IITK, Kanpur)

3) Priyavrat Deshpande (CMI)

4) Dheeraj Kulkarni (IISER, Bhopal)

Associate Teachers:

1) Nanasaheb Phatangare (Fergusson College, Pune)
2) Diwakaran (IMSc)

Topology I (Point Set Topology):

[Simmons] Chapters 1-7 and Quotient spaces: [Armstrong] Chapter 4 ‘Identification Spaces’.

1. [Simmons] G. F. Simmons, Introduction to Topology and Modern Analysis, McGrawHill, 1983.

2. [Armstrong] M. A. Armstrong, Basic Topology, Springer International Edition.

Topics:

Sets and functions, metric spaces, topological spaces, quotient spaces, compactness, separation, connectedness, approximation, Weierstrass approximation theorem, Stone-Weierstrass theorem.

Day

Lecture 1

(9.30–11.00)

Tea

(11.00–11.30)

Lecturer 2

(11.30–1.00)

Lunch

(1.00–2.30)

Tutorial 1

(2.30–3.30)

Tea

(3.30-4.00)

Tutorial 2

(4.00-5.00)

Snacks

5.00

Mon

Algebra-L1

Analysis-L1

Algebra-T1

Algebra-T2

Tues

Topology-L1

Algebra-L2

Analysis-T1

Analysis-T2

Wed

Analysis-L2

Topology-L2

Topology-T1

Topology-T2

Thu

Algebra-L3

Analysis-L3

Algebra-T3

Fri

Topology-L3

Algebra-L4

Analysis-T3

Analysis-T4

Sat

Analysis-L4

Topology-L4

Topology-T4

Topology-T4