  In this paper, we describe a new type of surgery for non-compact Riemann
surfaces that naturally appear when colliding two holes or two sides of the
same hole in an orientable Riemann surface with boundary (and possibly orbifold
points). As a result of this surgery, bordered cusps appear on the boundary
components of the Riemann surface. In Poincar\'e uniformization, these bordered
cusps correspond to ideal triangles in the fundamental domain. We introduce the
notion of bordered cusped Teichm\"uller space and endow it with a Poisson
structure, quantization of which is achieved with a canonical quantum ordering.
We give a complete combinatorial description of the bordered cusped
Teichm\"uller space by introducing the notion of maximal cusped lamination, a
lamination consisting of geodesics arcs between bordered cusps and closed
geodesics homotopic to the boundaries such that it triangulates the Riemann
surface. We show that each bordered cusp carries a natural decoration, i.e. a
choice of a horocycle, so that the lengths of the arcs in the maximal cusped
lamination are defined as $\lambda$-lengths in Thurston--Penner terminology. We
compute the Goldman bracket explicitly in terms of these $\lambda$-lengths and
show that the Mapping Class Group acts as a generalized cluster algebra
mutation. From the physical point of view, our construction provides an
explicit coordinatization of moduli spaces of open/closed string worldsheets
and their quantization.
