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BIRS Workshop Lecture Videos

The tangent groupoid and hypoelliptic operators van Erp, Erik


In the 1980s, Alain Connes gave a conceptually appealing proof of the Atiyah-Singer index theorem by means of the so-called "tangent groupoid". Over time, the tangent groupoid was generalized to more complicated analytic settings. I will discuss the role played by groupoids in work on the index problem for hypoelliptic differential operators on contact manifolds. The groupoid perspective proved to be very fruitful in pointing the way to a solution of this problem. A non-trivial hurdle in this case is to construct the correct groupoid. The (by now) standard construction of "adiabatic groupoids" does not give the desired object. The integration theorem for Lie algebroids of Crainic-Fernandes provides the necessary technical tool for our construction.

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