Invariants and flow geometry

Sphere geometry and invariants

A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G′, connecting two simplices if they intersect. We prove that the Poincaré-Hopf value i(x) = 1−χ(S(x)), where χ(S(x)) is the Euler characteristics of the unit sphere S(x) of a vertex x in G1, agrees with the Gre...

Furrow geometry nnder surge and continuous flow

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.

Oswald Veblen Differential Invariants and Geometry

The problem of the comparative study of geometries was clearly outlined in a very general form by RIEMANN in his Habihtationschrift in 1854. After exphcitly recognizing the possibihty of discrete spaces, RIEMANN Hmited his discourse to continuous manifolds in the sense of Analysis Situs and defined what he meant by such manifolds. This amounted to assuming that the points of any neighborhood ca...

Gromov-Witten Invariants via Algebraic Geometry

Calculations of the number of curves on a Calabi-Yau manifold via an instanton expansion do not always agree with what one would expect naively. It is explained how to account for continuous families of instantons via deformation theory and excess intersection theory. The essential role played by degenerate instantons is also explained.