Annual Foundation School -III (2018) - Kozhikode

Speakers and Syllabus 

First Week

Topology:
What is topology in general and algebraic topology in particular, went on describe the central problem, motivated the study of homotopy classes of maps, introduced the notion of Homotopy lifting property and Homotopy extension property.

Second lecture: The fundamental group, and calculated it for the circle, all the while emphesisng on the properties of the exponential map and how these properties themselves are going to be axiomatized in the notion of covering spaces..

Third lecture:  Various basic notions such as contractibility, relative homotopy, deformation retracts and various examples. An equivalent condition for cofibrations was established. The last lecture was completely divoted to categories and functors. Gave a large number of examples and was able to just define the notion of covariant and contravariant functors with some examples. The  students were given soft copy of my book on algebraic topology and were encouraged to go through the first chapter.

Second Week Topics:
Homotopic maps induce maps differing by an isomorphism at the level of fundamental groups; equivalent conditions for a map from a sphere to be nullhomotopic; application to existence of positive eigenvalue for a positive square matrix, fundamental theorem of Algebra; retractions versus deformation retractions; eg of a retraction that is not a deformation retraction; homotopy equivalences; eg a homotopy equivalence that is not a deformation retraction; idea of covering space and Galois theory of coverings Homological Algebra Basics: Short exact and long exact sequences, complexes and homology as a measure of non-exactness; zeroth homology module and reduced zeroth homology module and connectedness

Third Week Topics:
Singular Chain Complex and Homology groups; functoriality; chain homotopies; homotopic maps induce the same map on homology; use of the Prism Operator to show how a topological homotopy induces an algebraic chain homotopy and implies homotopy invariance of homology. Contractibility, asphericality and acyclicity; eg of aspherical but not contractible space; universal property of abelianisation; first homology is the abelianised first fundamental group; the Moebius strip can be deformed to the median circle but not to its boundary.

Fourth Week Topics:
Relative homology, fundamental lemma of homology theory, relative homology exact sequence, splitting of short exact sequences and applications for relative homology with respect to path components and retractions, Excision theorem and applications to computing the homology of spheres, applications to Brouwer Fixed Point, Borsuk-Ulam, Ham-Sandwich (Bisection) Hairy Ball theorems, existence and non-existence of vector fields on spheres, introduction to the Jordan-Brouwer Separation and Domain Invariance theorems.
Tutorials were conducted and in general the students found problem-solving somewhat hard, as they had not encountered these kind of topics before.

1. Review of definition and examples of fields, characteristic, field extensions and examples.
2. Algebraic and transcendental Extensions, finite extensions, degree of a field extension, transitivity of degree and algebraicity.
3. Adjoining roots, idea of splitting field, existence of finite fields. Some idea of isomorphisms between fields over a given field.
4. Application to classical Greek geometry: Ruler and compass construction, constructible numbers, necessary condition for construction  of  regular  n-gons. These topics spill over the first 7 sections of Artin’s book, Second edition,  Chapter  15

• Lecture 1, June 10: Motivational introduction to Galois theory without getting into details and a quick recap of first week's material covered by Prof. Parvati Shastri.
• Lecture 2, June 11: Algebraic independence, transcendence base, purely transcendental extensions. Lecture 3, June 12: Little bit on function fields. Also indicated an elementary proof of Luroth's theorem.
• Lecture 4, June 13: Algebraically closed fields, existence and uniqueness of algebraic closure, mostly only what is covered in Artin's book but indicated the proof in generality due to Emil Artin (but without all the details).

1. Group actions, many examples of group actions in algebra, geometry and topology
2. The fundamental theorem of Galois theory, with proof. 3. Discriminant of a polynomial and using it to determine whether the Galois group is a subgroup of the alternating group,
3. Splitting fields and Galois groups of quadratic and cubic extensions,
4. Cyclotomic extensions: roots of prime order in depth and other case partially. Cyclotomic polynomials were not defined.

Analysis

Week 1

• Lecture 1 Preliminaries from topology and measure theory
• Lecture 2 Normed Linear Spaces, Riesz lemma, Finite dimensional normed linear spaces, Definition of Hilbert spaces, Cauchy-Schwarz inequality and some consequences
• Lecture 3 Gram-schmidt Orthogonalization, Bessels inequality, Parsevals relation, Existence of orthonormal bases
• Lecture 4 Projection Theorem, Riesz Representation Theorem, classification of Hilbert spaces up to unitary equivalence

Week 2

• Lecture 1 Introduction to Fourier transform in the generality of locally compact abelian groups Isomorphism between
• L² (S¹) and l₂(Z)
• Lecture 2 Analytic formulation of Hahn-Banach theorem
• Lecture 3 Geometric formulation of Hahn Banach theorem, positive linear functionals on C(X) as application of Hahn-Banach theorem
• Lecture 4 Banach Alaoglu theorem and existence of Haar measure on compact groups. Exercises covered Existence of conditional expectation, Isoperimetric in-equality, Lax-Milgram lemma, Holder’s inequality, Minkowski’s inequality, completeness of l_p and computation of their duals.

Lectures given under AFS III on Analysis during July 18 – 21, 2018.

• Topic: Baire's Theorem and Applications
• July 18: Lecture 1: Baire's theorem, Banach-Steinhaus theorem and ramifications. Application to Fourier series: there exists a dense set of
• functions in the space of $2\pi$ periodic continuous functions on $[\pi,\pi]$ for each of which the Fourier series diverges on a dense set of points.
• July 19: Lecture 2: Open mapping and closed graph theorems and ramifications.
• July 20: lecture 3: Annihilators, adjoints. Orthogonality relation between the kernel and range of a continuous linear map and its adjoint.
• July 21: Lecture 4: Characterization of surjective maps. Equivalence of the grand theorems of functional analysis in a Hilbert space. A simple proof of the BanachSteinhaus theorem without the use of Baire's theorem.
• Book: Functional Analysis, by S. Kesavan, TRIM 52.

• Lecture 1: Banach algebra structure of continuous functions on a compact Hausdorff space and Bounded operators on a Hilbert space. Weak topologies for operators. Definition of an abstract Banach algebra. Regular and singular elements in a Banach algebra.
• Lecture 2: Topological divisors of zero, Spectrum and spectral radius
• Lecture 3: The structure of commutative Banach algebras, Gelfand mapping, Spectral radius formula.
• Lecture 4: Involutive Banach algebras, C*-algebras. Gelfand Naimark theorem. Spectral theorem for normal operators and Continuous functional Calculus.
• The book Topology and Modern Analysis by G. F. Simmons was used as a reference book for the course.