Annual Foundation School - Part III (2014)

Each resource person has taken four lectures of 90 minutes each covering one module during one week. This was supported by two tutorial sessions of two hours each. The final timetable is enclosed herewith.

Name and affiliation of resource persons:

• Diganta Bora IISER, Pune,
• S. S. Bhoosnurmath Karnatak University, Dharwar
• Ganesh Kadu Pune University, Pune
• S. A. Katre Pune University, Pune
• Dilip Patil IISc, Bangalore
• Manoj Keshari IITB, Mumbai
• Anant R. Shastri IITB, Mumbai
• Parvati Shastri University of Mumbai
• B. Subhash NIISER, Bhuvaneshwar
• B. N. Waphare Pune University, Pune

Course Associates

• Rajiv Garg IITB, Mumbai
• Dipankar Ghosh IITB, Mumbai
• Jai Laxmi IITB, Mumbai
• Ojas Sahasrabudhe IITB, Mumbai
• Prasant Singh IITB, Mumbai

• Field Theory: Ist week by Prof. S. A. Katre

1. Field Extensions and Examples. Algebraic and transcendental elements. Minimal polynomial. Degree of a field extension. Finite and infinite extensions. Simple extensions. Rationalising the denominator.
2. Transitivity of finite/algebraic extensions. Compositum of two fields. Algebraically closed fields. Uncountably many algebraically closed subfields of the field of complex numbers
3. Ruler and Compass constructions (in brief), Regular polygons. Squaring the circle etc.
4. Existence of a splitting field of a polynomial.
5. Symmetric polynomials. Newton’s theorem.
6. Fundamental Theorem of Algebra using symmetric polynomials. Luroth’s theorem (Covered by Prof. Avinash Sathaye in a special lecture)

• Field Theory: IInd week by Prof. B. N. Waphare
  In my lecture series we studied separable and inseparable extensions. An algebraic extension can be achieved in two stages: first, separable extension and then purely inseparable extension. The degree of a finite purely inseparable extension is a power of the characteristic of the field . The characterization of perfect fields and normal extensions are studied. The multiplicative group of a finite field is cyclic and determination of all subfields of a finite field are taken into consideration.

• Field Theory: IIIrd week by Prof. Parvati Shastri
  Lecture 1. Galois extensions, equivalent conditions, examples, Galois group of quadratic, bi-quadratic extensions and extensions like Q(³√2, ω), where ω is a primitive cube root of unity, Galois group of finite a extension of a finite field.
  Lecture 2. Cyclotomic fields, cyclotomic polynomials, Galois group of cyclotomic fields; Artin’s theorem on the fixed field of a finite automorphism group of a field.
  Lecture 3. Fundamental theorem of Galois Theory (FTGT), illustrations, Galois group of separable cubic polynomials, roots of unity in any field, cyclicity of radical extensions K( ⁿ√a), a ∈ K,  when the ground  field K contains distinct n^th roots of unity.
  Lecture 4. Applications of FTGT: Symmetric functions, symmetric functions are rational functions of elementary symmetric functions, discriminant of  polynomial, criterion for a Galois group to be subgroup of the alternating group, proof of Fundamental Theorem of Algebra via FTGT, constructibility of regular n-gons (Fermat primes), solution of inverse Galois problem for finite abelian groups.

• Field Theory: IVth week by Prof. Dilip Patil
  Starting with equivalent definitions of Galois extensions, in some examples Galois groups were calculated using groups actions. A characterisation of Cyclic Galois extensions using Hilbert’s theorem 90 was done. Roots of unity, Galois groups of Cyclotomic extensions and Galois groups of finite field extensions were discussed. At the end solvability of radicals of polynomials over arbitrary fields and their relations to Galois groups of their splitting fields was done. The last part was sketchy and not all complete proofs were given as lack of time and prerequisites of the participants. Many of them were fresh even in AFS III without attending AFS I or AFS II. I hope these comments are taken in a right spirit and used to improve further AFSs.

• Complex Analysis: Ist week by Dr. Diganta Borah

1.  Definition of complex numbers, topology of the complex plane, the Riemann sphere
2.  Complex differentiability, Cauchy-Riemann equations, complex versus Fr ́echet differentiability
3.  Power series, Abel’s lemma, uniform convergence of power series, differentiation of power series
4.  Exponential and trigonometric functions, branch of logarithm, branch of complex powers
5.  Conformal mappings, M ̈obius transformations

• Complex Analysis: II nd week by Dr. Ganesh Kadu
  Complex Integration (definition and some basic properties of contour integrals), Differentiation under integral sign (without proof), Existence of primitives, Primitive existence theorem, Cauchy-Goursat theorem, Cauchy’s theorem on convex region, Cauchy’s integral formula for derivatives, Morera’s theorem,Analyticity of complex differentiable functions, Taylor’s formula, Cauchy’s Estimate, Liouville’s theorem, Fundamental theorem of algebra, Gauss’ mean value theorem, maximum modulus theorem.

• Complex Analysis: IIIrd week by Prof. A. R. Shastri

1. Zeros of holomorphic functions, identity Theorem, open mapping theorem,
2.  Isolated Singularities, Riemann’s removable singularity, Poles and essential singularities. Laurent series, residues.
3.   Winding number, argument principle, Logarithmic residue theorem, Homotopy version of Cauchy’s theorem.
4.  Homology version of Cauchy’s theorem, extension of residue theorems. Statement and sketch of proof of Riemann mapping theorem.

• Complex Analysis: IVth Week by Prof. S. S. Bhoosnurmath
  Mean Value property, maximum principle, Schwarz’s reflection principle, Harmonic functions, Subhar-monic functions, Dirichlet’s problem, Green’s functions, Outline of a proof of Riemann mapping theorem. Comments to the Coordinator: From the students’ point of view and also from my own view
  point, material on harmonic functions from your book was very good and was extremely useful. Students’ response was very good. Some Participants felt that during 4th week some topics were much above their head. Can you help the students during the next AFS-III.

• Topology: I st week by Prof. Manoj Keshri

                Four lectures of 90 minutes each.

Lecture 1. I introduced some examples asking them whether they are homeomorphic, and introduced homotopy of two maps, problem whether in a triagle, if two maps are given, can one find the third one so that the diagram commutes.

Lecture 2. I defined the cofibration, retraction, (strong) deformation retraction etc and relations between them ware done in tutorial.

Lecture 3. I introduced path homotopy, definition of π1(X) etc.

Lecture 4 was π₁(S¹) = Z (only statement), Van-Kampen theorem (statement) and some applications of V-K, e.g. π₁ of torus, figure eight, R³ \ (x − axis ) etc.; Brouwer fixed point theorem was also done.

• Topology: IInd week by Prof. A. R. Shastri

Lecture 1. Basic problem in Topology/algebraic topology: The workspace for algebraic topologists viz. CW-complexes. (By this time I realised that most of the students are very week in topology.) Many examples and only statements of basic topological properties of CW-complexes.

Lecture 2. Simplicial complexes, lots of examples, simplicial maps, barycentric subdivision.

Lecture 3. Simplicial approximation.

Lecture 4. Applications to Sperner lemma and Brouwer’s fixed point theorem. Brouwer’s invariance of domain (with proof of weaker version only, viz, Rⁿ and R^m, m =! n are not homeomorphic).

• Topology: IIIrd week by Dr. B. Subhash

My lectures started recalling some relevant material from the first week: The fundamental group of the circle S^l was computed highlighting the lifting properties of the exponential map exp : R → S¹ In the second lecture, covering spaces were introduced and ample examples were discussed. Path and Homotopy lifting properties of the covering projection was discussed. The statement that covering projections are fibrations was proved.In the third lecture, the lifting problem was discussed with respect to the covering projection and necessary and sufficient conditions obtained for the same. The deck transformation groups were introduced, the relation of the deck transformation group with the fundamental group was discussed. Normal coverings were introduced and equivalent conditions for a covering to be normal were discussed.In the fourth lecture, the equivalence of coverings were introduced and the classification for coverings was taken up, leading to the one one correspondence between the equivalence classes of coverings and conjugacy classes of subgroups of the fundamental group for connected, locally path connected, semi locally simply connected spaces.

• Topology: IVth week by Prof. A. R. Shastri

Lecture 1. Categories and functors, Chain complexes, homology and Snake lemma.

Lecture 2 Singular chain complex and statement of axioms: proof of the long exact sequence of a pair.

Lecture 3. Computation of homology using Mayer-Vietoris. Suspension theorem. Homology of spheres.

Lecture 4. Application to Brouwer’s fixed point theorem, invariance of domain and separation theorem etc.

Click here for tutorials and time table

Instructions to candidates:

1. All further inquiries/communications, regarding the school should be addressed to Prof. S. A. Katre (sakatre@gmail.com).
2. Selected candidates should communicate the confirmation of  their participation on or before 2nd April 2014, along with a copy of their onward journey ticket where-ever applicable, failing which their selection will be cancelled automatically and some wait-listed candidate will be offered the chance.
3. If you are a selected candidate  and  decide NOT to attend the school for any reason what-so-ever, at any stage even after after confirmation, please communicate this immediately to the organizers so that some wait-listed candidates may  benefit.
4. Wat-listed candidates may therefore book their tickets so as not miss such a chance merely because they would not get a train-ticket.
5. All the selected candidates should visit the following website immediately:
   www.math.iitb.ac.in/~ars/afsIIIPune2014.html

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