1.Linear Algebra and Group Theory:

* We will cover most of the material found in Chapters 6, 7, 8, and 9 in Michael Artin’s book Algebra, Pearson, 1991.

Detailed Syllabus:
1. Chapter 6: The operations of a group on itself-The class equation of the Icosahedral group Operations on subsets-The Sylow theorems-The groups of order 12-Computation in the symmetric group-The free group-Generators and relations-The Todd-Coxeter algorithm
2. Chapter 7: Definition of bilinear form-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
3. Chapter 8: 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 Translation in a group-Simple groups
4. Chapter 9: Definition of a group representation-G-Variant forms and Unitary representations-Compact groups-G-invariant subspaces and irreducible representations-Characters-Permutation representations and the regular representations-The representations of the icosahedral group- One dimensional representations-Schur’s lemma and proof of the orthogonality relations- Rep-resentation of the group SU₂

Speakers:

• Dr. Neeraj Kumar, Associate Professor, IIT Hyderabad, Telangana.
• Dr. Anirban Mukhopadhyay, Professor, Institute of Mathematical Sciences, Chennai.
• Dr. C G Karthick Babu, Assistant Professor, IIT Goa, Goa.
• Dr. Vivek Sadhu, Associate Professor, IISER Bhopal.

Course Associates: Sureshkumar Mariappan (IMSc Chennai), Sivasankar (IIT Madras), Santosh Kumar Nayak (IISc Bangalore), Souvik Mandal (IIT Madras).

References

1. Michael Artin: Algebra, first edition, Prentice Hall of India, 1999.
2. S. Lang: Algebra, Revised 3rd Edition, Springer.
3. N. Jacobson: Basic Algebra, Vol.1 and Vol.2, Dover,2011.

2.Complex Analysis :

* We will cover most of the material found in Chapters 2, 3, and 4 of Stein and Shakarchi’s book Complex Analysis, Princeton Lectures in Analysis-II, 2003.

Detailed syllabus:
1. Chapter 2: Goursat’s theorem, Local existence of primitives and Cauchy’s theorem in a disc-Evaluation of some integrals, Cauchy’s integral formula-Cauchy’s integral formulas-Morera’s theorem, Sequences of holomorphic functions, Holomorphic functions defined in terms of
integrals-Schwarz reflection principle-Runge’s approximation theorem
2. Chapter 3: Zeros and poles-The residue formula, Examples-Singularities and meromorphic functions-The argument principle and applications -Homotopies and simply connected domains- The complex logarithm-Fourier series and harmonic functions
3. Chapter 4: The Fourier Transform- The class F -Action of the Fourier transform on F -Paley- Wiener theorem, Exercises, Problems

Speakers:
• Dr. Sushmita Venugopalan, Reader, Institute of Mathematical Sciences, Chennai.
• Dr. Samya kumar ray, Reader, Institute of Mathematical Sciences, Chennai.
• Dr. Sairam Kali Raj, Associate Professor, IIT Ropar, Punjab.
• Dr. D. Venku Naidu, Professor, IIT Hyderabad, Telangana.

Course Associates: Shubham R. Bais (IISc Bangalore), Ramesh Mete (IIT Bombay), , Ravitheja Vangala (IISc Bangalore).

References
1. Elias Stein and Rami Shakarchi: Complex Analysis, Princeton Lectures in Analysis-II, 2003.
2. W. Rudin: Real and Complex Analysis, 3rd Edition, Tata McGraw-Hill Higher Education, New Delhi, 1987.

3.Topology :

* We will cover most of the material found in Chapters 1-7 of Simmons’s book listed below in the references and quotient spaces from Chapter 4 of Armstrong’s book listed below.
Detailed syllabus:
1. Chapter 1 – Sets and Functions
2. Chapter 2 – Metric spaces: Convergence, completeness, Baire’s theorem, space of continuous mappings.
3. Chapter 3 – Topological spaces: Bases, weak topologies, function algebras.
4. Chapter 4 – Compactness: Product spaces, Tychonoff’s theorem, and Ascoli’s theorem.
5. Chapter 5 – Separation: T1, T2, T3, T4-spaces, and Urysohn imbedding theorem.
6. Chapter 6 – Connectedness: Components, Totally disconnected and Locally connected spaces.
7. Chapter 7 – Approximation: The Weierstrass approximation theorem and Stone-Weierstrass theorem
8. Chapter 4 (Armstrong) – Identification spaces: Mobius strips, Topological groups, and orbit spaces.

Speakers:
• Dr. B. Subhash, Assistant Professor, IISER Tirupati, Andhra Pradesh.
• Dr. Ramesh Kasilingam, Assistant Professor, IIT Madras, Chennai.
Course Associates: Sumanta Das (IIT Bombay), Himanshu Chandrakar (IMSc), Ajay Raj (Comenius University, Slovakia)

References
1. G.F. Simmons: Introduction to Topology and Modern Analysis, McGraw-Hill, 1983.
2. M. A. Armstrong: Basic Topology, Springer.
3. J. R. Munkres: Topology, A First Course, PHI, New Delhi, 1987.