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NPTEL

Introduction to Algebraic Topology (Part-II)

NPTEL and Indian Institute of Technology Bombay via Swayam

Overview

As stated above, this is PG level course in Mathematics, which requires basic knowledge of Linear algebra, Point set topology, and group theory. This course is central to many areas in modern mathematics. The subject itself saw tremendous growth during 1950 and currently has attained a matured status.The syllabus I have chosen is common to MA5102 at IIT Bombay and AFS-III program of National Centre for Mathematics. It has enough material common to the syllabus followed by several Universities and IIT’s in the country and goes beyond. Nevertheless it has different flavour liked by variety of students. I have published a book in which one-third of the content is roughly the present course. This book is followed by several universities abroad also for their course.INTENDED AUDIENCE : Anybody who would like to get trained in Algebraic Topology such as Computer scientists, Electrical , Aerospace engineers and mathematicians, and physicists.PREREQUISITES : Point Set Topology is pre-requisite. Exposure to Basics of Linear algebra and Group theory is preferred. Preferably attended Algebraic Topology Part-I course offered by the same instructorINDUSTRIES SUPPORT : All IIT’s, IISERs , TIFR and Universities in India.

Syllabus

Week-1:What is Algebraic Topology: Notion of Homotopy.
Week-2:Categories and Functors Week-3:Basic Homological Algebra
Week-4:Singular Homology Week-5:Computations with simple objects Week-6:Review of Simplicial complexes and Simplicial homoloy
Week-7:Euler Characteristic, Lefschetz Number
Week-8:Relation between fundamental groups and 1st homology
Week-9:Applications homology: Brouwer’s invariance of domain
Week-10:Applications continued. Lefschetz Fixed point theorem
Week-11:CW complexes and CW homology
Week-12:Discussion for future topics.

Taught by

Prof. Anant R. Shastri

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