Here is an easily described, but very difficult, problem that I
(and a number of other people) really would like to see solved during
our life times. The basic problem is to compute the dimension of a certain vector space.
Suppose we are given a triangulation (a tessellation by triangles where any two triangles share at most a common edge or a common vertex) of a polygonal domain in the plane. Let $S$ be the space of
once differentiable functions on that triangulation that on each
triangle can be represented as a bivariate polynomial of degree 3. $S$ is clearly a vector
space. It is known that the dimension of that space is greater than or
equal to $3V_B + 2V_I + 1 + \sigma$ where $V_B$ is the number of
boundary vertices, $V_I$ is the number of interior vertices, and
$\sigma$ is the number of singular vertices of the underlying
triangulation. A singular vertex is an interior vertex that has
exactly four edges attached where those edges form two parallel pairs.
(In other words, a singular vertex is the intersection of the
diagonals of a convex quadrilateral.) It is known that generically
the dimension of $S$ equals the given expression, and there is no case known
where the dimension is larger than that expression.
*Many people
conjecture that the lower bound equals the dimension for all
triangulations. Prove, or disprove, that conjecture.*

Some background: triangulations are the natural generalization of a
partition of an interval to two variables, and $S$ is a spline space
with potential for a wide range of practical problems, such as data
fitting or solving partial differential equations. The problem has
been known among approximation theorists since the early 1970s, and
despite efforts by a number of people the problem is still unsolved.
For more information on spaces like $S$ see the recent book by Lai and
Schumaker [*Spline Functions on Triangulations*, Cambridge University Press] in particular section 9.9. The basic issue
with the kind of spline space considered here is that the dimension
depends not just on the topology of the triangulation, but also on its
geometry. An arbitrarily small change of the location of the vertices
can change the dimension. If the polynomial degree is four (instead
of three) it is known that the dimension can change only when a vertex
switches between being singular and non-singular. On the other hand,
if the polynomial degree is two (instead of three) many configurations
other than singular vertices are known where the dimension changes
with the geometry. So the polynomial degree three straddles the
boundary, and we would like to know to which camp it belongs.

greaterthan or equal to..." --- is that correct? – Noam D. Elkies Aug 6 '11 at 3:11