**12/16/2010**

**Henryk Iwaniec
Rutgers University**

**Weyl's sums for roots of quadratic congruences**

It is known that the roots of congruences for a fixed irreducible quadratic polynomial are equidistributed. This statement translates to getting cancellation in the corresponding sum of Weyl's sums. In a recent work by W. Duke, J. Friedlander and H. Iwaniec we succeeded to get cancellation (so also the equidistribution) in very short sums of Weyl's sums relatively to the discriminant. The spectral theory of metaplectic automorphic forms is the basic tool, of which some special aspects will be the subject of this talk. Numerous applications of the result will be also discussed.