By Prof. Nitin Nitsure from Bhaskaracharya Pratishthana, Pune

Abstract

Topology is popularly known as `rubber sheet geometry',
being the study of spaces and continuous maps that may not preserve
distances. Poincar\'e realized that many important topological
properties of spaces and maps are actually preserved by continuous
deformations, that is, `homotopies', and thus was born the subject of
Homotopy Theory around 1900 CE. Poincar\'e invented fundamental
groups, Emmy Noether invented homology groups, and Hurewicz invented
homotopy groups. Eilenberg, Steenrod, MacLane, Whitehead etc. made
further great advances that by 1960 made Homotopy Theory the backbone
of Algebraic Topology.  The years 1900 -- 1960 may be called the
classical period of Homotopy Theory.
    In the late 1960s, Quillen and others made the surprising
discovery that some of the basic structures of Homotopy Theory can be
abstracted into a categorical formulation, which applies to unexpected
other areas of mathematics, such as Commutative Algebra. This created
what we can call Modern Homotopy Theory. Developments arising from
this is an active area of mathematical research, with applications
ranging from Number Theory to Mathematical Physics. The modern
reformulation post 2000 of the basics of Algebraic Geometry by Lurie
and others makes heavy use of modern homotopy theory and allied
simplicial algebra and higher category theory.
   This lecture will give a rapid sketch of the development of some of
the above ideas, illustrated with simple examples.
It will be largely understandable by anybody who has a background in
topology and algebra at the advanced undergraduate level.
   This will be the inaugural talk of a planned series of seminars by
multiple speakers on modern homotopy theory.