Liviu Nicolaescu, University of Notre Dame
Title: A probabilistic proof of the Gauss-Bonnet-Chern theorem
Abstract: Suppose that E → M is an oriented real vector
bundle of even rank 2r over a smooth compact oriented
manifold M of
dimension m ≥ 2r. To a metric h and compatible connection
∇ the Chern-Weil construction associates
the Euler form
e(E,h,∇)∈ Ω2r(M).
I will show that there is a Gaussian probability measure on the space
of smooth section of E such that the expected value
of the zero
locus of such a random section is equal, in the sense of currents, to
the Euler form e(E,h,∇).
In the process I will also
describe how to recover probabilistically the geometry of the triplet (E,h,∇).