# Mathematics Colloquium

#### 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.

**Synopsis:**

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.