In these lectures I will outline a proof of the Atiyah-Singer Index theorem. I will begin with the definition of the symbol $\sigma_l$ of a linear differential operator $L$ and the notion of ellipticity of $L$. I go on to define the analytic index $i_a(L)$ of a linear elliptic differential operator $L$ on a compact manifold $M$. Then I proceed to the definition of the $K$-theoretic element $\kappa(L)\in K(M)$ associated to the elliptic operator $L$ via its symbol and the cohomology class $\tau_L (M) \in H^*(M)$. The topological index $i_t(L)$ is defined as $<\tau_L(M), [M]>$ where $[M]$ is the fundamental class of the (connected) compact (oriented ) manifold. The index theorem assets that $i_a(L)=i_t(L)$. I then give a proof following essentially the ideas in the announcement of the theorem by Atiyah and Singer. The proof however differs significantly from those in the Palais seminar which carries out in detail the outline given in the announcement. Unlike the Palais seminar I will not be using the sophisticated machinery of Pseudo differential operators but stay in the confines of Differential operators. A second point of difference is in the proof of the 'bordism invariance' where I will be using the ideas developed by McKean and Singer rather than results about coercive boundary value problems. The proof as outlined by Atiyah and Singer is along the following lines. They show that $i_a(L)$ depends only on $\tau (L)$ and conclude from that it suffices to prove the theorem for the special class 'twisted index' operators on even dimensional manifolds. Then using the bordism in variance they further reduce the problem to special manifolds where the equality of the analytic and topological indices are well known theorems. This is also essentially what I will be doing. The Atiyah-Bott-Patodi proof of the index theorem follows a different path after reducing the problem to the case of special operators, following a method proposed by McKean and Singer. I will say something about this proof towards the end of these lectures.
I will begin with the definition $i_t(L)$ of the analytic index of a Linear Elliptic Differential operator $L$ on a smooth compact manifold. I will then describe the symbol of $L$ and the $K$-theoretic element $\kappa (L)$ associated to it and then the cohomology class $\tau(L)$ on $M$ defined using it
Schedule:
Dates: 24, 29 and 31 July 2024
Time: 10:30 AM to 12:00 PM (IST)
Zoom Link: https://icts-res-in.zoom.us/j/91833525623?pwd=kgRqG7GGzqA810Lvib3gfVUqxXbt9d.1
Meeting ID: 918 3352 5623
Passcode: 242526
For any queries contact us at academicoffice@icts.res.in