# Annual Foundation School -III (2018) - Kozhikode

 Venue: Kerala School of Mathematics,  Kozhikode, Kerala Date: 2nd, Jul 2018 to 28th, Jul 2018

 Convener(s) Name Prof. M Manickam Prof. Venkata Balaji T E Dr. A.K. Vijayarajan Mailing Address DirectorKerala School of MathematicsKunnamangalam, P.OKozhikode – 673571, Kerala Assistant ProfessorIIT Madras Associate Professor Kerala School of Mathematics Kozhikode Kerala

 Speakers and Syllabus

Resource Persons and Tutorial Assistants

• Topology
• AS – Anant Shastri, TEVB – T. E. Venkata Balaji
• Algebra
• PS – Parvati Shastri, UKA – U.K. Anandavardhanan, MK – Manoj Kummini
• Analysis
• PSC – Partha Sarathi Chakraborty, SK – S. Kesavan, VR - Vijayarajan

Tutorial Assistants

 Sr. Name / Subject Affiliation Duration 1 Mr. Subhajit Chanda / Algebra Research Scholar, Dept of Maths, IIT-Madras Last two weeks 2 Ms. Pratiksha Shingavekar / Algebra Research Scholar, Dept of Maths, IIT-Madras Last two weeks 3 Mr.Repana Devendra /Analysis and Topology Dept of Maths, IIT-Madras Analysis Second and third weeks, and Topology for the third week 4 Mr. Syamkrishnan M. S. / Functional Analysis Research Scholar, KSOM First, Second and Fourth weeks 5 Ms.Bandi Prasuna / Algebra and Topology PhD Scholar, School of Mathematics TIFR First two weeks 6 Mr. Jayakumar Ravindran / Algebra and Topology JRF Mathematics, IMSc, Chennai Algebra for the first two weeks and Topology for all 4 weeks 7 Mr. Md Hasan Ali Biswas /Functional Analysis Dept of Maths, IIT-Madras 3rd and 4th week

Syllabus Covered by the Resource Persons

 Anant Shastri 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. Venkata Balaji T E 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. Parvati Shastri Algebra: Review of definition and examples of fields, characteristic, field extensions and examples. Algebraic and transcendental Extensions, finite extensions, degree of a field extension, transitivity of degree and algebraicity. Adjoining roots, idea of splitting field, existence of finite fields. Some idea of isomorphisms between fields over a given field. 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 U K Anandavardhanan 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). Manoj Kummini Group actions, many examples of group actions in algebra, geometry and topology 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, Splitting fields and Galois groups of quadratic and cubic extensions, Cyclotomic extensions: roots of prime order in depth and other case partially. Cyclotomic polynomials were not defined. Parthasarathy Chakraborty 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 lp and computation of their duals. S. Kesavan 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. A. K. Vijayarajan 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.

Time Table

 Day Date Lecture 1(9.30– 11.00) Tea(11.00– 11.30) Lecture 2(11.30– 1.00) Lunch(1.00– 2.30) Tutorial 1(2.30– 3.30) Tea(3.30- 4.00) Tutorial 2(4.00- 5.00) Snacks5.00 Mon 02-07-2018 AS Tea  Break PS LunchBreak AS Tea Break PS Tues 03-07-2018 PSC AS PSC AS Wed 04-07-2018 PS PSC PS PSC Thu 05-07-2018 AS PS AS PS Fri 06-07-2018 PSC AS PSC AS Sat 07-07-2018 PS PSC PS PSC SUNDAY : HOLIDAY Mon 09-07-2018 UKA Tea Break PSC LunchBreak UKA Tea Break PSC Tues 10-07-2018 TEVB UKA TEVB UKA Wed 11-07-2018 PSC TEVB PSC TEVB Thu 12-07-2018 UKA PSC UKA PSC Fri 13-07-2018 TEVB UKA TEVB UKA Sat 14-07-2018 PSC TEVB PSC TEVB SUNDAY : HOLIDAY Mon 16-07-2018 KH Tea Break TEVB LunchBreak KH Tea Break TEVB Tues 17-07-2018 SK KH SK KH Wed 18-07-2018 TEVB SK TEVB SK Thu 19-07-2018 KH TEVB KH TEVB Fri 20-07-2018 SK KH SK KH Sat 21-07-2018 TEVB SK TEVB SK SUNDAY : HOLIDAY Mon 23-07-2018 MK Tea Break TEVB LunchBreak MK Tea Break TEVB Tues 24-07-2018 VR MK VR MK Wed 25-07-2018 TEVB VR TEVB VR Thu 26-07-2018 MK TEVB MK TEVB Fri 27-07-2018 VR MK VR MK Sat 28-07-2018 TEVB VR TEVB VR

 Actual Participants

The KSOM is located at a distance of about 12Kms from the Kozhikode city centre (train/bus Stations) situated off the Kozhikode-Medical College-Kunnamangalam bus route. If you choose to travel from the city to KSOM by bus, catch one from the Palayam Bus Station (also known as the Old Bus Stand) going in the above route (there is a bus in about every 10 minutes) and get down at a bus stop called Mundikkalthazham’ which is a few stops after the more prominent Medical College Bus Stop. The KSOM is by the side of a bye lane leading to Peringolam from this bus stop and at a distance of about 2Kms. Alternatively you can take a bus from the Kunnamangalam Bus Station to Peringolam’ and walk up a hill to reach KSOM.