COLLOQUIUM David Krejčiřík "Is the optimal rectangle a square? "

par Prof. David Krejčiřík (Czech Technical University in Prague)

jeudi 4 avr. 2024, 16:00 → 17:00 Europe/Paris

Salle René Baire (IMB)

We give a light talk on optimality of shapes in geometry and physics. 
First, we recollect classical geometric results that the disk 
has the largest area (respectively, the smallest perimeter) 
among all domains of a given perimeter (respectively, area). 
Second, we recall that the circular drum has the lowest 
fundamental tone among all drums of a given area or perimeter 
and reinterpret the result in a quantum-mechanical language of nanostructures. 
In parallel, we discuss the analogous optimality of square 
among all rectangles in geometry and physics. 
As the main body of the talk, 
we present a joint work with Freitas in which we show that 
the disk actually stops to be the optimiser for elastically supported membranes, 
disproving in this way a long-standing conjecture of Bareket's. 
We also present our recent attempts to prove the same spectral-geometric properties 
in relativistic quantum mechanics. 
It is frustrating that such an illusively simple 
and expected result remains unproved 
and apparently out of the reach of current mathematical tools.