A common problem in low-dimensional topology is to ask how the topology and geometry of a manifold changes if you glue a solid torus into one of its torus boundary components (also known as Dehn filling) or more generally, if you glue a handlebody into a higher genus boundary component. One topological version of this problem is to ask how the isotopy classes of Heegaard surfaces change. Every Heegaard surface for the unfilled manifold becomes a Heegaard surface for the filled manifold, but there may be other properly embedded non-Heegaard surfaces that also become Heegaard surfaces if you cap them off after the filling. In particular these new Heegaard surfaces may have lower genus, so the Heegaard genus of the manifold could drop after filling. The quintessential example of this is a knot complement in the 3-sphere: There are knot complements with arbitrarily high Heegaard genus, but if you Dehn fill to produce the 3-sphere, then the genus drops to zero.
Of course, for such a manifold there is exactly one filling that produces the 3-sphere and one can ask how much the genus can drop for the other fillings. There are examples where Heegaard genus drops by one for a line of slopes, and the resulting Heegaard surfaces are often called horizontal. However, Moriah-Rubinstein  (and later Rieck-Sedgwick ) showed that there are only finitely many slopes for which the genus can drop by more than one (and only finitely many lines of slopes where it drops by one.) As far as I know there are no examples where there are two slopes for which the genus drops by more than one. So one can ask:
Question: Is there a 3-manifold with Heegaard genus , a torus boundary component and two slopes on such that Dehn filling along each slope produces a 3-manifold with Heegaard genus less than or equal to ?
Note that this is closely related to the Berge conjecture, which asks which knots in the 3-sphere have a Dehn surgery that produces a 3-manifold with Heegaard genus one. If you could prove the answer to this question is “no” then you would reduce the Berge conjecture to a question of which tunnel-number-one knots (i.e. those with Heegaard genus two) have lens space surgeries. However, I’m more inclined to bet on “yes” for this one. Perhaps it’s because I’ve been spending too much time trying to construct manifolds with multiple non-isotopic Heegaard splittings. But whichever answer turns out to be the case will be very interesting.
For gluing in higher genus handlebodies, things get much more complicated because there is no longer a single slope that determines the gluing. Tao Li  has generalized Moriah-Rubinstein’s result to higher genus, but rather than having a finite number of fillings in which the genus may drop, Li requires that the (infinitely many) meridian curves for the glued in handlebody have distance in the curve complex above some bound from a finite collection loops in the boundary. (The difficulty has to do with the fact that disjoint essential loops in the torus are parallel, but disjoint essential loops in high genus surfaces may not be.) It would be interesting to find examples where the Heegaard genus drops after gluing in handlebodies in two inequivalent ways, but I don’t know the best way to generalize the question above.
 Moriah, Yoav; Rubinstein, Hyam, Heegaard structures of negatively curved 3-manifolds. Comm. Anal. Geom. 5 (1997), no. 3, 375–412.
 Rieck, Yo’av; Sedgwick, Eric, Persistence of Heegaard structures under Dehn filling. Topology Appl. 109 (2001), no. 1, 41–53.