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This article develops several main results for a general theory of homological algebra in
categories such as the category of sheaves of idempotent modules over a topos. In the …

We report on the following highlights from among the many discoveries made in
Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the …

We provide a potential conceptual reason for the positivity of the Weil functional using the
Hilbert space framework of the semi-local trace formula of Connes (Sel Math (NS) 5 (1): 29 …

We first explain the link between the Berry-Keating Hamiltonian and the spectral realization
of zeros of the Riemann zeta function of the first author, and why there is no conflict at the …

Segal's Γ-rings provide a natural framework for absolute algebraic geometry. We use G.
Almkvist's global Witt construction to explore the relation with J. Borger-geometry and …

Toward building tropical analogues of adic spaces, we study certain spaces of prime
congruences as a topological semiring replacement for the space of continuous valuations …

We define vector bundles for tropical schemes, and explore their properties. The paper
largely consists of three parts (1) we study free modules over zero-sum free semirings, which …

Noncommutative geometry has its roots both in quantum physics and in pure mathematics.
One of its main themes is to explore and investigate “new spaces” that lie beyond the scope …

Text We construct, for specific imaginary quadratic number fields with class number 1, an
arithmetic site of Connes-Consani type. The main difficulty here is that their constructions …