Mathematical Conversations

The Unfinished Story of the Mahler Conjecture.

The polar body is a fundamental concept in functional and convex analysis, representing a special convex set associated with any convex subset of Euclidean space. One can think of the polar operation as, roughly speaking, the "inverse" of convex sets. This perspective naturally leads to the consideration of the volume product of a convex set and its polar, a quantity that is dimensionless and remains invariant under linear transformations.

One of the central problems in convex geometry, known as the Mahler conjecture, asks whether the cube minimizes this volume product among all centrally symmetric convex bodies. This seemingly simple question has remained open since its introduction in 1939, sparking extensive research and developments over the years. In this talk, I will briefly outline some of the history of the problem and share my (non-expert) understanding of several key concepts involved. Additionally, I will describe a surprising connection between Mahler's conjecture and symplectic measurements of convex domains, which prompted my initial interest in this question. 

Date & Time

February 05, 2025 | 6:00pm – 8:00pm

Location

Simons Hall Dilworth Room

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