Annual Foundation School - Part III - Speakers and Syllabus

 1. Field theory

 Speakers

 1.2 Syllabus

1. Algebraic and separable extensions, splitting field of polynomials, primitive element theorem, normal extensions;

2. Fundamental theorem of Galois theory (FTGT), applications to fundamental theorem of algebra, symmetric functions, solutions of cubic and quartic polynomial equations via FTGT;

3. Galois group of X n − a, solvable extensions, cyclotomic extensions, constructible regular polygons, inverse Galois problem for finite abelian extensions; and

4. Hilbert’s theorem 90 and structure of cyclic extensions, Dedekind’s reduction mod p technique, and construction of polynomials with Galois group Sn and An .

2. Complex analysis

2.1  Speakers

2.2  Syllabus

Numbers in parentheses indicate sections from reference [4] below.
1. Quick review of algebra and topology of complex plane, sequences and series, uniform convergenc Weierstrass M -test (1.1–1.7), complex differentiability, basic properties, analytic functions, power series, Abel’s theorem, examples (2.1–2.4), Cauchy-Riemann equations,Cauchy derivative versus Fr´chet derivative, geometric interpretation of holomorphy, formale differentiation (3.4, 3.5), Mobius Transformation and the Riemann sphere. (3.6, 3.7);

2. Line integrals, basic properties, differentiation under integral sign (4.1), primitive existence theorem, Cauchy-Goursat theorem (statements and sketch of the proof only), Cauchy’s theorem on a convex domain (4.2, 4.3), Cauchy’s integral formula, Taylor’s theorem, Liouville,maximum modulus principle, (4.5–4.7), zeros of holomorphic functions, identity theorem, open mapping theorem and isolated singularities (5.1–5.2), Laurent series and residues (5.3–
5.4), winding number and argument principle (5.5, 5.6);

3. Schwartz lemma, inverse function theorem, Rouche’s theorem (7.1, 7.2), convergence of sequences of holomorphic and meromorphic functions, theorems of Weierstrass and Hurwitz(8.1, 8.2), Runge’s approximation theorem (8.6, 7.4), homology form of Cauchy’s theorem,Mittag-Leffler theorem (8.6, 7.4), infinite products Weierstrass’s theorems on products (8.6);and

4. Simple connectivity, homotopy version of Cauchy’s theorem (7.3), harmonic functions, mean value property, maximum principle, etc., Schwartz reflection principle (4.8), Harnack’s principle, subharmonic functions (9.2, 9.3), Dirichlet’s problem, Perron’s solution (9.4), Green’s function and a proof of Riemann mapping theorem (9.6, 8.11), multiply connected domains
(9.7).

References

1. J. B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics,11, Springer-Verlag, 1973.

2. T. W. Gamelin, Complex analysis, Undergraduate Texts in Mathematics, Springer-Verlag,2001.

3. R. Remmert, Theory of complex functions, Graduate Texts in Mathematics, 122, Springer-Verlag, 1980.

4. A. R. Shastri, Basic complex analysis of one variable, MacMillan Publishers India Ltd., 2011.

3. Differential geometry

3.1 Speakers

3.2 Syllabus

Numbers in parentheses indicate sections from reference [2]. It is assumed that the participants have been exposed to differential manifolds and Stokes’ theorem, which is likely if they have attended AFS I.

1. Quick review of differential forms on manifolds, exterior differentiation, Lie derivatives and Cartan’s formula (2.2), Frobenius theorem (2.3), the Maurer-Cartan form of a Lie group (2.4),vector bundles on Manifolds, various operations (5.1), geodesics and parallel translation ofvectors, covariant derivatives, curvature (5.2); and

2. Smooth triangulation of smooth manifolds, smooth singular chain complex (3.1), integration and Stokes theorem (3.2). de Rham cohomology, Poincare lemma (3.3). Cech cohomology; proof of de Rham theorem (3.4) (see also [1, Section V.9] for a quick proof), applications of de Rham theorem, Hopf invariant, mapping degree, linking number, etc. (3.5);

3. Riemannian metric on manifolds, Hodge star operator (4.1), harmonic forms (4.2), Hodge theorem, outline of a proof of Hodge decomposition (4.3), applications of Hodge theorem, Poincare duality, Euler characteristic, intersection number, etc. (4.4); and

4. Connections on vector bundles, curvature (5.3), invariant polynomials, Pontrjagin classes, Levi-Civita connection (5.4), Chern classes, relation with Pontrjagin classes (5.5), Orientation on vector bundles, Euler class (5.6), applications: Gauss-Bonnet, characteristic classes of complex projective spaces, characteristic numbers (5.7).

References

1. G. Bredon, Topology and geometry, Graduate Texts in Mathematics, 139, Springer-Verlag,1993.

2. S. Morita, Geometry of differential forms, Translations of Mathematical Monographs, 201, Iwanami Series in Modern Mathematics, American Mathematical Society, 2001.