Joint Mathematics Colloquium ICMAT-UAM-UC3M-UCM

Spectral sets, weak tiling and
Fuglede's conjecture

Máté Matolcsi
(Alfréd Rényi Institute of Mathematics)

12 November 2024
12:00
Aula Azul, ICMAT
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A bounded measurable set X in a d-dimensional Euclidean
space is called spectral if the function space L2(X) admits
an orthogonal basis of exponentials. The easiest example is
the unit cube, where elementary Fourier analysis tells you
that complex exponentials with integer frequencies form an
orthogonal basis.
Fuglede's conjecture stated that a set X is spectral if and
only if it tiles the space by translation. The conjecture was
recently proved for all convex bodies in all dimensions in a
joint work of Nir Lev and Mate Matolcsi. We will review the
proof, which includes the notion of weak tiling as a key
ingredient. Other results and open problems related to weak
tiling will also be mentioned