This Annual Foundation School (AFS-III) is aimed at first and second year Ph.D. students. The main objective is to bring up students with diverse backgrounds to a desirable level and help them acquire basic knowledge in Algebra, Analysis and Topology. Knowledge of the material covered in AFS-I and AFS-II in Group Theory, Real Analysis and Point-set Topology will be assumed.
Participants will attend morning lectures in the topics of Ring Theory, Functional Analysis and Differential Topology. The afternoons will feature tutorials on these topics. A key goal of the program is the development of problem-solving ability and a mastery of examples and counter examples.
The School will draw expertise from reputed institutions such as, Harish Chandra Research Institute, Shiv Nadar University, IISc, Bangalore, IIT Gandhinagar and University of Hyderabad.
Research students who have previously attended AFS-I and/or AFS-II will be given preference in the selection. Otherwise, the recommending faculties should make it clear that the candidate has the required mathematical background.The participants will be provided accommodation on the campus of HRI, Allahabad.
Syllabus
Course Length of each module : 4 Lectures of 1.5 hours each and 4 Tutorials of 1 hour each
1. ALGEBRA (GALOIS THEORY) | |||
Instructor | Affiliation | Dates | Module |
Satyanarayana Reddy | Shiv Nadar University, Delhi. | 29 June to 4 July 2015 | Module LA1:Field extension and examples. Algebraic and transcendental elements, minimal polynomial.Degree of a field extension, finite and infinite extensions. Simple extensions. Transitivityof finite/algebraic extensions. Compositum of two felds. Ruler and compass constructions.Characterization of constructible numbers via square root towers of fields. Impossibility ofsquaring the circle, trisection of angles and duplication of cubes by ruler and compass. Gauss’criterion of constructible regular polygons. (Wantzel’s characterization of constructible regular p-polygons. Richmond’s construction of a regular pentagon. Examples of symmetric polynomials. The fundamental theorem of symmetric polynomials. Newton’s identities for power sum symmetric polynomials. Discriminant in terms of power-sum symmetric polynomials.Discriminant of a cubic. Existence of a splitting field of a polynomial. Fundamental theorem of algebra via symmetric polynomials. |
D.S. Ramana | HRI, Allahabad | 06 July to 11 July 2015 | Module LA2 :Splitting fields and algebraic closures; existence and isomorphisms. Criterion for multiple roots of polynomials in terms of derivatives. Characterization of perfect fields of positive characteristic. Separable and inseparable extensions. Transitivity of separable extensions. Roots of an irreducible polynomial have equal multiplicity. Separable degree. Finite fields: existence and uniqueness, algebraic closure. Finite subgroup of the multiplicative group of a field is cyclic. Gauss’ formula for the number of monic irreducible polynomials of a given degree over a finite field. Factorization of polynomials over finite fields. Primitive element theorem. Finite separable extensions have a primitive element. Normal extensions and their examples. Characterization of normal extensions in terms of embeddings and splitting fields. |
Gyan Prakash | HRI, Allahabad | 13 July to 18 July 2015 | Module LA3 :Galois extensions. Galois groups of finite extensions of finite fields and quadratic extensions. Galois groups of biquadratic extensions. Galois groups of a separable cubic polynomials. Fundamental Theorem of Galois theory (FTGT). Artin’s Theorem about fixed field of a finite group of automorphisms. Behavior of Galois group under isomorphisms. Normal subgroups of the Galois groups and their fixed fields. Fundamental theorem of algebra via FTGT. Gauss’ criterion for constructible regular polygons. Symmetric rational functions. Galois group of some binomials. Roots of unity in a field. Galois group of x Irreducibility of the cyclotomic polynomial Φn(x) over Q. A recursive formula for Φn(x). Discriminant of Φp(x). Subfields of Q(ζp). Kronecker-Weber Theorem for quadratic extensions of Q. Algorithm for construction of primitive elements of subfields of Q(ζp). Subfields of Q(ζ7),Q(ζ13) and Q(ζ17). Infinitude of primes p ≡ 1 (mod n). Inverse Galois problem for finite abelian groups. Structure of some cyclic extensions. |
C. S. Dalawat | HRI, Allahabad. | 20 July to 25 July 2015 | Module LA4 :Cyclic extensions of degree p over fields with characteristics p. Solvable groups. Simplicity of An. Galois group of composite extensions Galois closure of a separable field extension. Radical extensions. Solvability by radicals and solvable Galois groups. A quintic polynomial which is not solvable by radicals. Cardano’s method for roots of cubic equations. Lagrange’s method for roots of quartic equations. Ferrari’s method for roots of quartic equations. Galois group as a group of permutations. Irreducibility and transitivity. Galois groups of quartics. The norm and the trace function. Multiplicative form of Hilbert’s Theorem 90. Cyclic extensions of degree n. Additive version of Hilbert’s 90. Cyclic extensions of prime degree: Artin-Schreier Theorem. |
2. COMPLEX ANALYSIS | |||
Instructor | Affiliation | Dates | Module |
R. Thangadurai | HRI Allhabad | 29 June to 4 July 2015 | Module LB1 :Quick review of algebra and topology of complex plane, sequences and series, uniform con- vergence, Weierstrass M-test. Complex differentiability, basic properties, analytic functions: power series, Abel’s theorem, examples, Cauchy-Riemann equations: Cauchy derivative versus Frechet derivative. Geometric interpretation of holomorphy, formal differentiation, Mobius transformation and the Riemann sphere. |
S. D. Adhikari | HRI Allhabad | 06 July to 11 July 2015 | Module LB2 :Line integrals, basic properties, differentiation under integral sign. Primitive existence theorem, Cauchy-Goursat theorem (statementsand sketch of the proof only), Cauchy’s theorem on a convex domain theorem. Cauchy’s integral formula, Taylor’s theorem, Liouville, Maximum modulus principle. Zeros of Holomorphic functions, identity theorem, open mapping theorem. Isolated singularities, Laurent series and residues. Winding number and argument principle. |
B. Ramakrishnan | HRI, Allahabad | 13 July to 18 July 2015 | Module LB3 :Homotopy and Homology versions of Cauchy’s theorem, Inverse function theorem, Rouche’stheorem. Schwarz’s lemma. |
R.R. Simha (tentative) |
20 July to 25 July 2015 | Module LB4 : Harmonic functions: Mean Value property maximum principle etc. Schwarz reflection principle. Harnack’s Principle, Subharmonic functions, Dirichlet’s problem, Perron’s solution, Green’s function and an outline of proof of Riemann Mapping theorem. | |
3. ALGEBRAIC TOPOLOGY | |||
Instructor | Affiliation | Dates | Module |
Umesh V Dubey | IISc, Bangalore. | 29 June to 4 July 2015 | Module LC1 :Statement of basic problems in Algebraic Topology: extension problems and lifting problems; homotopy, relative homotopy, deformation, contraction, retracts etc. Typical constructions: Adjunction spaces, Mapping cones, Mapping cylinder, Smash-product, reduced cones reduced suspension etc. Categories and Functors. Definition and examples. Equivalence of functors, adjoint functors, examples. |
Sanjay Amrutiya | IIT, Gandhinagar. | 06 July - 11 July, 2015 |
Module LC2 :Covering Spaces and Fundamental group, lifting properties, relation with fundamental group. Classification of covering spaces (proof of existence may be skipped), Computation of Funda- mental groups: simpler cases of Van-Kampen theorem. Effect of attaching n-cells. (4.6). |
Archana S Morye | University of Hyderabad, Hyderabad | 13 July - 18 July, 2015 | Module LC3 :CW-complexes Simplicial complexes: basic topological properties of CW complexes. Products of CW complexes (especially the CW-structure on X × [0, 1]). Homotopy theoretic properties of CW complexes. Abstract simplicial complexes and geometric realization, barycentric subdivision and simplicial approximation theorem. Applications: cellular Approximation theorem, Brouwer’s invariance of domain etc. |
N. Raghavendra | HRI, Allahabad. | 20 July - 25 July, 2015. | Module LC4 :Singular and Simplicial Homology: Chain complexes, exact sequences of complexes, snake lemma, four lemma and five lemma, homology long exact sequence. Axioms for homology, construction of singular chain complex, verification of axioms (except homotopy axiom and excision axiom). Simplicial and singular simplicial homologies. Statement of equivalence of all these homologies. |
3. Tentative Time Table
Date | 9:30 - 11:00 | 11:30 -13:00 | 14:30 -15:30 | 16:00-17:00 |
29 June | LA1 | LB1 | TA1 | TA1 |
30 June | LC1 | LA1 | TB1 | TB1 |
01 July | LB1 | LC1 | TC1 | TC1 |
02 July | LA1 | LB1 | TA1 | TA1 |
03 July | LC1 | LA1 | TB1 | TB1 |
04 July | LB1 | LC1 | TC1 | TC1 |
06 July | LA2 | LB2 | TA2 | TA2 |
07 July | LC2 | LA2 | TB2 | TB2 |
08 July | LB2 | LC2 | TC2 | TC2 |
09 July | LA2 | LB2 | TA2 | TA2 |
10 July | LC2 | LA2 | TB2 | TB2 |
11 July | LB2 | LC2 | TC2 | TC2 |
13 July | LA3 | LB3 | TA3 | TA3 |
14 July | LC3 | LA3 | TB3 | TB3 |
15 July | LB3 | LC3 | TC3 | TC3 |
16 July | LA3 | LB3 | TA3 | TA3 |
17 July | LC3 | LA3 | TB3 | TB3 |
18 July | LB3 | LC3 | TC3 | TC3 |
20 July | LA4 | LB4 | TA4 | TA4 |
21 July | LC4 | LA4 | TB4 | TB4 |
22 July | LB4 | LC4 | TC4 | TC4 |
23 July | LA4 | LB4 | TA4 | TA4 |
24 July | LC4 | LA4 | TB4 | TB4 |
25 July | LB4 | LC4 | TC4 | TC4 |
LA, LB, LC stand for the modules on Galois Theory, Complex Analysis and Algebraic Topology respectively. TA, TB, TC are the corresponding tutorial sessions.
05 July, 12 July and 19 July 2015 are Sundays which are holidays for the programme.
Tea Breaks.— 11:00 to 11:30 and 15:30 to 16:00.
Lunch Break.— 13:00 to 14:30