The diabolical bubbles of H. Minkowski
Speaker: Ramon van Handel, Princeton
Location: Warren Weaver Hall 1302
Date: Monday, December 11, 2023, 3:45 p.m.
In the late 1800s, in the course of his study of classical problems of
number theory, the young Hermann Minkowski discovered the importance of a
new kind of geometric object that we now call a convex set. He soon
developed a rich and beautiful theory for understanding such sets, laying
the foundations of convex geometry that are widely used to this day.
Among the most surprising observations of Minokwski's theory is that the
classical isoperimetric theorem-- which states that the ball has the
smallest surface area among all bodies of a given volume (a fact known
instinctively to any child who has played with soap bubbles)-- is just one
special case of a much more general phenomenon. When one fixes geometric
parameters other than surface area, Minkowski discovered that the
resulting "bubbles" can be strikingly bizarre -- they may have spikes
sticking out of them in any direction. A complete understanding of such
objects has remained a long-standing open problem, with major progress
being achieved only recently in joint work with Yair Shenfeld.
In this talk I will aim to discuss some of the history of these questions,
and to discuss how they arise in different guises across a surprisingly
wide range of mathematical disciplines including geometry, analysis,
algebra, combinatorics, and computational complexity.