AFS-III - Annual Foundation School - III (2026)

Speakers and Syllabus

    • Algebra III [Field and Galois Theory]:

1. Definition and examples of fields and their extensions, algebraic and transcendental elements, degree of a field extension, algebraic extension.
2. Finite fields, function fields, transcendental extensions.
3. Constructions with ruler and compass, symbolic adjunction of roots.
4. Splitting field, algebraic closure, algebraically closed field.
5. Separable and inseparable extensions, normal extensions, fundamental theorem of Galois theory. Computation of Galois groups of field extensions.
6. Composite and simple extensions, cyclotomic extension, abelian extension.
7. Solvability by radicals. Cubic equations, symmetric functions, primitive elements, quartic equations, insolvability of quintic equations.

Reference books:
            1. Algebra by Michael Artin.
            2. Abstract Algebra by Dummit and Foote.
Speakers:

1.  Dr. Arjun Paul [AP] (IISER Kolkata) [4 Lectures]
2.  Dr. Md. Ali Zinna [Zin] (IISER Kolkata) [4 Lectures]
3.  Dr. Swarnendu Datta [SD] (IISER Kolkata) [4 Lectures]
4.  Prof. Mrinal Kanti Das [MKD] (ISI Kolkata) [4 Lectures]

    • Analysis III [Functional Analysis]:

1. Normed Linear Spaces: Definitions and examples; Banach spaces and Hilbert spaces; ℓp, C(K) (K compact, Hausdorff) and L^p; equivalence of norms in finite-dimensional spaces; Riesz lemma; non-compactness of the unit ball in infinite-dimensional normed spaces; quotient spaces and completeness.
2. Linear Operators and Dual Spaces: Bounded linear operators and continuity; linear functionals; dual space; examples of duals; natural embedding into the bidual.
3. Hahn–Banach Theorem and Convexity: Hahn–Banach extension theorem (real and complex); separation of convex sets; applications to existence of continuous linear functionals.
4. Completeness Theorems: Baire category theorem (statement); uniform boundedness principle; open mapping theorem; closed graph theorem.
5. Weak Topologies and Duality: Weak topology and weak convergence; weak* topology; Banach–Alaoglu theorem (statement).
6. Hilbert Spaces: Inner product spaces; Hilbert spaces; orthonormal sets and bases; orthogonal projections; Riesz representation theorem for Hilbert spaces; bounded linear operators on Hilbert spaces: self-adjoint and normal operators.
7. Compact Operators and Spectral Theory: Compact operators on Banach and Hilbert spaces; examples; spectrum of bounded linear operators; spectral theory of compact self-adjoint operators; Fredholm alternative (statement).

Reference book:

Introduction to Topology and Modern Analysis by Simmons and A course in functional analysis by Conway.

Speakers:
    1. Prof. Shibananda Biswas [Shi] (IISER Kolkata) [4 Lectures]
    2. Prof. Saugata Bandyopadhyay [Sau] (IISER Kolkata) [4 Lectures]
    3. Dr. Subrata Shyam Roy [SSR] (IISER Kolkata) [4 Lectures]
    4. Dr. Soumalya Joardar [SJ] (IISER Kolkata) [4 Lectures]

    • Topology III [Algebraic Topology]:

1. Basic Constructions. Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms.
2. Van Kampen’s Theorem. Free Products of Groups. The van Kampen Theorem. Applications to Cell Complexes.
3. Covering Spaces. Lifting Properties. The Classification of Covering Spaces. Construction of universal cover. Deck Transformations and Group Actions. Coverings of a wedge of two circles.
4. Simplicial and Singular Homology. Delta-Complexes. Simplicial Homology. Singular Homology. Homotopy Invariance. Exact Sequences and Excision.
5. Computations and Applications. Homology of surfaces. Euler characteristic. Mayer-Vietoris Sequences.
6. Additional Topics. Homology and Fundamental Group.

Reference books:

Algebraic Topology by Allen Hatcher

     Speakers:

     
    1. Prof. Pratulananda Das [PD] (Jadavpur University) [4 Lectures]
    2. Prof. Somnath Basu (IISER Kolkata) [Som] [4 Lectures]
    3. Dr. Subhabrata Das [Sub] (Presidency University) [4 Lectures]
    4. Dr. Kuldeep Saha [KS] (TCG CREST, Kolkata) [4 Lectures]

Time Table

 Time-table: There will be 2 lectures (1.5 hours each) and 2 tutorials (1 hour each) per day, six days per week. There will be 4 lectures on each topic per week. A (tentative) schedule is as follows:

Names of tutors & course associates:

• Lisa Mondal (LM)
• Samit Ghosh (SG)
• Sneha Sardar (SS)
• Soumyadeep Chakraborty (SC)
• Anubrotaa Bhuniya (AB)
• Sourasish Karmakar (SK)
• Mukilraj K (MK)
• Gourab Paul (GP)
• Arnab Chattopadhyay (AC)
• Bishal Bhunia (BB)
• Debanjit Mondal (DM)
• Jitender Sharma (JS)

[all from IISER K] and the speakers.