Annual Foundation School -III (2016) - IISER Thiruvananthapuram - Speakers and Syllabus

1. Field Theory

Module I
Speaker: Parvati Shastri(Mumbai University)

1. Field extension and examples. Algebraic and transcendental elements, minimal polynomial. Degree of a field extension, finite and infinite extensions. Simple extensions.
2. Transitivity of finite/algebraic extensions. Compositum of two fields.
3. Ruler and compass constructions. Characterization of constructible numbers via square root towers of fields. Impossibility of squaring the circle, trisection of angles and duplication of cubes by ruler and compass
4. Gauss’ criterion of constructible regular polygons. (Wantzel’s characterization of constructible regular p-polygons. Richmond’s construction of a regular pentagon).
5. Examples of symmetric polynomials. The fundamental theorem of symmetric polynomials. Newton’s identities for power sum symmetric polynomials.
6. 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

Module II
Speaker: Tony Puthenpurakal(IIT Bombay)

1. Splitting fields and algebraic closures; existence and isomorphisms..
2. Criterion for multiple roots of polynomials in terms of derivatives. Characterization of perfect fields of positive characteristic.
3. Separable and inseparable extensions. Transitivity of separable extensions,Roots of an irreducible polynomial have equal multiplicity. Separable degree.
4. 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.
5. Factorization of polynomials over finite fields. Primitive element theorem.Finite separable extensions have a primitive element.
6. Normal extensions and their examples. Characterization of normal extensions in terms of embeddings and splitting field.

Module III
Speaker
: Viji Thomas(IISER TVM)

1. Galois extensions. Galois groups of finite extensions od finite fields and quadratic extensions. Galois groups of biquadratic extensions. Galois groups of a separable cubic polynomials. Fundamental Theorem of Galois theory (FTGT).
2. 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.
3. Fundamental theorem of algebra via FTGT. Gauss’ criterion for constructible regular polygons. Symmetric rational functions. Galois group of some binomials.
4. Roots of unity in a field. Galois group of x roots of unity. Irreducibility of the cyclotomic polynomial Φn(x) over Q. A recursive formula for the cyclotomic polynomial.
5. Discriminant of Φp(x). Subfields of Q(ζp). Kronecker-Weber theorem for quadratic extensions of Q. Algorithms for construction of primitive elements of subfields of Q(ζp). Subfields of Q(ζ7), Q(ζ13) and Q(ζ17).
6. Infinitude of primes p ≡ 1(mod n). Inverse Galois problem for finite abelian groups. Structure of some cyclic extensions. n − a over a field having n-th

Module IV
Speaker : Viji Thomas(IISER TVM)

1. Cyclic extensions of degree p over fields with characteristics p. Solvable groups. Simplicity of An.
2. Galois group of composite extensions Galois closure of a separable field extension.
3. Radical extensions. Solvability by radicals and solvable Galois groups. A quintic polynomial which is not solvable by radicals.
4. Cardano’s method for roots of cubic equations. Lagrange’s method for roots of quartic equations. Ferrari’s method for roots of quartic equations.
5. Galois group as a group of permutations. Irreducibility and transitivity.Galois groups of quartics.
6. 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

 References

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

2.Complex Analysis

1. Quick review of algebra and topology of complex plane sequences and series, uniform convergence, 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.
Speaker: Siddhartha Bhattacharya(TIFR Mumbai)

2. Line Integrals basic properties, differentiation under integral sign, Primitive existence theorem, Cauchy-Goursat theorem (statements and 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.
Speaker: Siddhartha Bhattacharya(TIFR Mumbai)

3. Homotopy and Homology Version of Cauchy’s theorem, Inverse function theorem, Rouche’s theorem. Schwarz’s lemma.
Speaker: A. R. Shastri(IIT Bombay)

4. 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.
Speaker: Joseph Mathew(Retired Professor from ISI Kolkata)

3. Algebraic Topology

1. Statements 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. Computation of fundamental group of the circle and applications.
Speaker: A. R. Shastri(IIT Bombay)

2. CW-Complexes and 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 sub division and simplicial approximation theorem. Applications: cellular Approximation theorem, Brouwer’s invariance of domains etc.
Speaker : K. Ramesh (ISI Bangalore)

3. Covering spaces and Fundamental groups lifting properties,relation with fundamental group. Classification of covering spaces(proof of existence may be skipped), Computation of Fundamental groups: simpler cases of Van-Kampen theorem. Effect of attaching n-cells
Speaker: B. Subhash(IISER Tirupati)

4. 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. Computations and applications: Separation theorems, Invariance of Domain. Euler characteristic
Speaker: A. R. Shastri(IIT Bombay)

References

1. A. Hatcher: Algebraic Topology, Cambridge University Press.
2. C. R. F Maunder: Algebraic Topology, Van Nostrand Reinhold Company, London.
3. E. H. Spanier: Algebraic Topology, Tata McGraw-Hill.
4. Anant R. Shastri: Basic Algebraic Topology, CRC Press, Taylor and Francis group, 2013.

Time Table
Week - One

Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
20June FT1 T CA1 L TFT1 T TCA1
21June AT1 T FT2 L TAT1 T TRT2
22June CA2 T AT2 L TCA2 T TAT2
23June FT3 T CA3 L TFT3 T TCA3
24June AT3 T FT4 L TAT3 T TFT4
25June CA4 T AT4 L TCA4 T TAT4

 Week - Two

Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
27June FT5 T CA5 L TFT5 T TCA5
28June AT5 T FT6 L TAT5 T TFT6
29June CA6 T AT6 L TCA6 T TAT6
30June FT7 T CA7 L TFT7 T TCA7
01July AT7 T FT8 L TAT7 T TFT8
02July CA8 T AT8 L TCA8 T TAT8

 Week - Three

Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
04July FT9 T CA9 L TFT9 T TCA9
05July AT9 T FT10 L TAT9 T TFT10
06July CA10 T AT10 L TCA10 T TAT10
07July FT11 T CA11 L TFT11 T TCA11
08July AT11 T FT12 L TAT11 T TFT12
09July CA12 T AT12 L TCA12 T TAT12

 Week - Four

Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
11July FT13 T CA13 L TFT13 T TCA13
12July AT13 T FT14 L TAT13 T TFT14
13July CA14 T AT14 L TCA14 T TAT14
14July FT15 T CA15 L TFT15 T TCA15
15July AT15 T FT16 L TAT15 T TFT16
16July CA16 T AT16 L TCA16 T TAT16

 • FTn : nth lecture in field theory

• TFTn: nth tutorial in field theory

• CAn : nth lecture in complex analysis

• TCAn: nth tutorial in complex analysis

• ATn : nth lecture in algebraic topology

• TATn: nth tutorial in algebraic topology

• T : Tea

• L : Lunch