I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This …

Background See WP-article on F_1 = F_{un} = Field with one element (and also this MO question). Paraphrasing someone: we do not know what is it, but it is not a field :). For this …

There are various frameworks around which enlarge the category of rings to include more exotic objects such as the 'field with one element,' $\mathbb{F}_1$. While these frameworks …

As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general: $\bullet …

Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute alge …

I'm reading Deitmar's paper on Schemes over $\mathbb{F}_1$. Proposition 2.4. states that for a scheme $X$ over $\mathbb{F}_1$ there is a bijection between $X(\mathbb{F}_1)$ and the …

I've heard of this many times, but I don't know anything about it. What I do know is that it is supposed to solve the problem of the fact that the final object in the category of …

$$\frac{1}{\mu(B)}\int_B \vert x \vert d\mu(x) = \frac{1}{p+1}$$ This formula holds for the unit ball in $\mathbb{Q_p}$. This formula also holds for $\mathbb{R}$ when $p=1$. Shou …

What should affine and projective schemes over $\mathbf{F}_1$ be?

If doing geometry over $\mathbb F_p$ means also using its algebraic closure, it must be interesting to talk about the algebraic closure of $\mathbb F_1$ - the field with one elemen …

Coproducts of modules over an algebraic monad $\Sigma$ are described in Section 4.16.14/15 in Durov's thesis. It is claimed there that for $\Sigma$-modules $M,N$, the set $M \copro …

Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are …

Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus {0}$ is very similar to the condition for a commutative ring to be a field. This analogy is also us …

What are the obstacles that prevent from defining Spec$\mathbb{Z}$ in absolute geometry? By absolute geometry I mean the geometry over the field with one element F1.

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?