This shall be mainly a reading course intended for 1st year Ph.D. students. The choice of topics pursued shall depend very much on the interests of the audience.
Pre-requisites : Point-set topology and Fundamental group (Roughly, upto Ch. 9 of Munkres - Topology : A First Course), Basic matrix theory and algebra.
Syllabus (Tentative) :
Part I:
1) Simplicial Complexes : Examples, Simplicial Approximation, Edge group, More properties (Eg., Joins, Links, et al).
2) Simplicial Homology : Homology groups, Betti numbers, Computation, Euler-Poincare formula, Exact Sequences.
References : [M] [K], [E-H], [A]
Part II:
3) Simplicial Morse Theory : k-connected complexes, Morse inequalities, Gradient vector fields, Applications. References : [J], [F]
4) Combinatorial Laplacians : Properties, Spectrum, Hodge decomposition. References : [H-J], [JLYY]
Part III:
(Time-permitting and audience-willing, some of these topics will be pursued).
5) A-Homotopy theory.
6) Knot theory.
7) Graphs on surfaces.
8) Persistent Homology.
References :
[A] M.A. Armstrong, Basic Topology.
[E-H] H. Edelsbrunner and J. L. Harer, Computational Topology : An Introduction.
[F] R. Forman, An user’s guide to discrete Morse theory.
[H-J] D. Horak and J. Jost, Spectra of combinatorial Laplace operators on simplicial complexes.
[J] J. Jonsson, Simplicial complexes of graphs.
[JLYY] X. Jiang, L. Lim, Y. Yao, Y. Ye, Statistical ranking and combinatorial Hodge theory
[K] D. Kozlov, Combinatorial Algebraic Topology.
[M] J. R. Munkres Elements of Algebraic Topology.
Apart from these references, we shall source the required material from other books, articles or lecture notes also.
Additional Material :