Curviliner enumerative geometry

Enumerative Geometry

Given a complex projective variety V (as defined in [1]), we wish to count the curves in V that satisfy certain prescribed conditions. Let f C denote complex projective -dimensional space. In our first example, V = f C2, the complex projective plane; in the second and third, V is a general hypersurface in f C of degree 2 − 3. Call such V a cubic twofold when  = 3 and a quintic threefold wh...

Enumerative Real Algebraic Geometry

Two well-defined classes of structured polynomial systems have been studied from this point of view—sparse systems, where the structure is encoded by the monomials in the polynomials fi—and geometric systems, where the structure comes from geometry. This second class consists of polynomial formulations of enumerative geometric problems, and in this case Question 1.1 is the motivating question o...

Enumerative Real Algebrai Geometry

Enumerative geometry and knot invariants

We review the string/gauge theory duality relating Chern-Simons theory and topological strings on noncompact Calabi-Yau manifolds, as well as its mathematical implications for knot invariants and enumerative geometry.

Enumerative Geometry for Real Varieties

Of the geometric figures in a given family satisfying real conditions, some figures are real while the rest occur in complex conjugate pairs, and the distribution of the two types depends subtly upon the configuration of the conditions. Despite this difficulty, applications ([7],[28],[32]) may demand real solutions. Fulton [11] asked how many solutions of an enumerative problem can be real, and...