Orateur
Description
Noncommutative geometry and the notion of topos are two mathematical concepts
that provide complementary perspectives on the structure of a space. In this talk, I will
begin by explaining, as simply as possible, these two concepts and what makes them
unique. The originality of noncommutative geometry can be directly perceived through
the existence of an intrinsic time evolution of a noncommutative space. The originality
of toposes can similarly be perceived through the intuitionistic logic associated with a
topos. It is the metric structure, embodied by a representation—as operators in Hilbert
space—of coordinates and the length element, that allows noncommutative geometry
to engage with reality, namely the structure of space-time at the infinitesimally small
scale as revealed by contemporary physics through the Standard Model. As for
toposes, it is the additional structure of a sheaf of algebras that enables geometry to
manifest beyond topology.
In the second part of the talk, I will explain how the spectrum of the ring of integers
can be understood through these two geometric lenses. The connection between
these two approaches rests on an extension of class field theory that sheds light on
the analogy established by Mumford and Mazur between knots and prime numbers.
The spectral perception of the ring of integers naturally emerges from the study of
the zeros of the Riemann zeta function, thereby revealing deep structures at the
interface of arithmetic, topology, and geometry
