Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re (f) is Morse, we prove the existence of a stratified gradient-like vector field for Re (f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimension m (p)+ ns as was conjectured by Goresky and MacPherson.

We define a new combinatorial class of triangulations of closed 3manifolds, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ² in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric <i>g</i> _<i>ij</i> keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably large", then the solution metric <i>g</i> _<i>ij</i> converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature <i>R</i>(<i>t</i>, <i>x</i>) goes to positive infinity as (<i>t</i>, <i>x</i>) tends to the blowup points, and a flow with surgery has

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.

We show that solutions of Thurston's equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston's equation and a commutative ring associated to triangulated 3-manifolds. It is shown that solutions to homogeneous equations over the real numbers are critical points of an entropy function.