Let (X,0) be a complex surface germ with an isolated singularity at the origin of C^n and f:(X,0)->(C,0) be a non constant holomorphic function. We will study the inner metric of the Milnor fibers f^(-1)(t) by defining a family of metric invariants called inner-rates. After  that i will state and prove the inner-rates formula which provide us a concrete way of computing the inner rates and, on the other hand, relate them with the polar curves of f. We will end the talk with an application of the inner rates to compute the integral of the gaussian curvature on the Milnor fibers.

Given a knot in the 3-sphere, one can associate some knot homologies with it, among which knot Floer homology and (reduced) Khovanov homology. Motivated by Fox-Milnor's obstruction for slice knots, we prove that all knots in a certain family have these homologies with total rank being a square integer. Moreover, we conjecture that the rank of knot Floer homology is congruent to 1 modulo 8 for all slice knots, and we prove this fact for a subfamily of slice knots, namely fusion number 1 ribbon knots. This is a joint work with Hockenhull and Willis, and partially also with Dunfield and Gong.

It is known that convex bodies in the model 3-spaces of constant curvature are rigid with respect to the induced intrinsic metric on the boundary. This story has two classical chapters: the rigidity of convex polyhedra and the rigidity of smooth convex bodies, though there is also a common generalization obtained by Pogorelov.
Similarly to this, Jean-Marc Schlenker proved that hyperbolic metrics with smooth strictly convex boundary on a compact hyperbolizable 3-manifold M are rigid with respect to the induced metric on the boundary (and also with respect to the dual metric). It is reasonable to expect that similar results should hold also for polyhedral boundaries, and eventually for general convex boundaries. Curiously enough, no polyhedral counterparts were proven up to now, though some partial progress was made by François Fillastre. One of the main difficulties is that convex hyperbolic cone-metrics on the boundary of M (which is a standard intrinsic description of what we expect to be the induced metric on a polyhedral boundary) might admit not so polyhedral realizations, which are hard to handle or to exclude. A prototypical example is the boundary of a convex core bent along an irrational lamination.
I will present a recent work proving the rigidity (and the dual rigidity) of hyperbolic metrics on M with convex polyhedral boundary under mild additional assumptions. As another outcome, it follows that convex cocompact hyperbolic metrics on the interior on M with the convex cores that are "almost polyhedral" are globally rigid with respect to the induced metric on the boundary of the convex core, and are infinitesimally rigid with respect to the bending lamination. This is a step towards conjectures of William Thurston.