What is a p-adic quantum dilogarithm?

For zoom details please contact Pietere Moree (moree@mpim-bonn.mpg.de).

We will define p-adic quantum dilogarithms associated to a local field, and we will use them to construct invariants of ideally-triangulated 3-manifolds. The invariants depend on a local field and combinatorial adta associated to an ideal triangulation, are presented as multidimensional (Igusa-type) integrals reminiscent to motivic integration, and takes values in the monoid of Laurent polynomials in two variables with natural number coefficients. We will discuss connections of these invariants with classical geometry (the PSL_2 character variety of a knot), and give examples to illustrate the shape of the invariants for the three simplest knot complements of the 4_1, 5_2 and (-2,3,7) pretzel knot. Joint work in progress with Rinat Kashaev.