Real enumerative geometry in R^n with smooth tangency constrains

In the talk we shall describe a point of view on real enumerative
geometry, which links real algebraic geometry and smooth topology.
In particular, we ask to count real algebraic varieties in R^n that pass
through a generic configuration of real points P and are tangent to a
given collection of generically immersed submanifolds X of R^n. The
geometric number of counted varieties usually depends on the
configuration (P,X), so one can try to count algebraic varieties with
certain signs, so that the total algebraic number (after the "crossing wall"
corrections) is locally constant in the complement of some discriminant
in the space of configurations (P,X). We shall explain how one can find
such sign rules in the case of the following concrete problem: how many
real plane rational curves of degree d pass through a generic configuration
P of 3d−2 real points and are tangent to a given immersed curve X in general
position? In this case, the crossing wall jumps term corresponds to reducible,
cuspidal and special nodal curves and is expressed as a degree one
finite type invariant of X. In addition, we shall discuss a higher dimensional
generalization of this problem.