Igor Shparlinski: Bilinear forms with Kloosterman sums

We start with a general introduction to the area which is nowadays
known as Klostermania, where the main goal is showing nontrivial
cancellations between Kloosterman sums in some families.

We outline some new bounds on bilinear sums with Kloosterman sums and
also with some similar sums. In particular, these bounds improve some
recent results of V. Blomer, E. Fouvry, E. Kowalski, Ph. Michel and
G. Milicevic (2014-2016). As a result we improve the error term in the
asymptotic formula for mixed moments of $L$-series associated with
Hecke eigenforms.

We also discuss further extensions of this method (jointly with
Tianping Zhang and Kui Liu), which improve some recent results of
R. Nunes (2016).

Finally, we outline some possible arithmetic applications of these bounds.