The marriage of analytic power to geometric intuition drives many of today's mathematical
advances, yet books that build the connection from an elementary level remain scarce. This …

Geometric Measure Theory: A Beginner's Guide, Fifth Edition provides the framework
readers need to understand the structure of a crystal, a soap bubble cluster, or a universe …

Randomized smoothing is the current state-of-the-art defense with provable robustness
against $\ell_2 $ adversarial attacks. Many works have devised new randomized smoothing …

Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions that is
different from elliptic and hyperbolic geometry (and from the Minkowskian geometry of …

To the families of geometric measures of convex bodies (the area measures of Aleksandrov‐
Fenchel‐Jessen, the curvature measures of Federer, and the recently discovered dual …

A sharp quantitative version of the anisotropic isoperimetric inequality is established,
corresponding to a stability estimate for the Wulff shape of a given surface tension energy …

The present book contains a proof of essentially one theorem (Theorem 1.9) about the
asymptotic shape of a droplet. That alone accounts for such a long time it took to write the …

This classic text serves as a tool for self-study; it is also used as a basic text for
undergraduate courses in differential geometry. The author's ability to extract the essential …

Define a “Liouville domain” to be a compact exact symplectic manifold with contact-type
boundary. We use embedded contact homology to assign to each four-dimensional Liouville …

This chapter is devoted to volume large deviation principles, as opposed to surface large
deviation principles. While the former are reminiscent of the case of iid random variables …