# Hebrew University Topology and Geometry Seminar

February 24, 2016

1100-1235

Ross building, Seminar Room 70A

## Mikhail Katz

*Bar Ilan University
*

##
Determinantal variety and bi-Lipschitz equivalence

**Abstract:**
The unit circle viewed as a Riemannian manifold has
diameter (not 2 but rather) $\pi$, illustrating the difference between
intrinsic and ambient distance. Gromov proceeded to erase the
difference by pointing out that when a Riemannian manifold is embedded
in $L^\infty$, the intrinsic and the ambient distances coincide in a
way that is as counterintuitive as it is fruitful. Witness the
results of his 1983 Filling paper. Gromov exploited this embedding to
prove a universal upper bound for the systole of an essential (e.g.,
aspherical) manifold, and created an entirely new area of research
around the invariants called filling radius and filling volume, which
is active until today with recent contributions by Larry Guth and
others.

In the context of the manifold of nonsingular matrices, Asaf Shachar
asked when the intrinsic and the ambient (Euclidean) metrics are
bilipschitz equivalent. The answer turns out to hinge on the
structure of the stratification of the determinantal variety.