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The Atiyah-Singer theorem is a major achievement of twentieth century mathematics. It has inspired a lot of work and people started to develop generalizations of this theorem. I would like to know the current state of the theory and current central problems in index theory. Let me mention some possible generalisations:

  1. More than one operator This approach was initiated by Atiyah himself in the index theorem for families of operators. Possible generalizations are index theorems for foliations (for easy foliations which are fibrations it implies the index theorem for families).

  2. Higher index theory This line of research concerns manifolds which are $G$-spaces and equivariant operators or more generally, noncompact (complete, Riemannian) manifolds (as in the work of John Roe). The prominent examples are universal covers of non-simply connected manifolds (where the fundamental groups acts).

  3. Non-smooth manifolds Examples include combinatorial manifolds, Lipschitz manifolds and quasiconformal manifolds. This direction was developed mostly by Nicolae Teleman.

  4. Non-commutative geometry This whole discipline is heavily inspired by index theory: in fact the points mentioned above use the machinery of noncommutative geometry. There is an abstract local index theorem for spectral triples of Connes-Moscovici.

What are most important current directions and trends in index theory? What are the most important problems in index theory?

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    $\begingroup$ There's also index theory on certain noncompact manifolds. See in particular the Atiyah-Patodi-Singer theorem for manifolds with cylindrical ends, or Mrowka-Ruberman-Saveliev's recent end-periodic index theorem. This isn't quite the same as your second bullet point, I don't think. $\endgroup$
    – mme
    Commented Mar 28, 2016 at 23:36

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Let me add some more current trends to the list.

  1. Local and smooth index theory. Starting from a geometric index problem, one regards the cohomological index as a form on the underlying manifold, rather than just a cohomology class. One regards the index itself as an element of a smooth $K$-group, which includes differential form data obtained from the local index. As an example, the smooth family index of a family of Dirac operators typically involves an $\eta$-form.

  2. Indices of non-elliptic operators. An important example is Bismut's hypoelliptic Dirac operator.

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