Spherical Analytic Geometry

What are we after?

Imagine there’s a setting where mathematics proceeds in 3 steps:

1. Compute some homotopy invariants (e.g. cohomology)
2. Index these homotopy invariants using an analytic object (e.g. manifold)
3. Use analysis on the analytic object to compute numbers from the homotopy invariants (e.g. integrate/sum dimensions)

By homotopy invariants, we broadly refer to anything obtained by some construction in spectral algebraic geometry.

This suggests that the basic object of interest should be the mathematical object that implements number 2: some structure that indexes homotopy invariants using an analytic object. Examples of such things can come from

• $p$-adic rigid analytic geometry (prisms, perfectoid spaces, …)
• Geometric & Differential topology (Knot invariants, index theorems, …)
• Symplectic geometry (Floer homotopy)
• Quantum field theory (Arithmetic field theories, analytic Langlands, …)
• Classical and quantum information theory, thermodynamics

Hence, we’re probably after an object that’s more flexible than, say, “condensed $E_{\infty }$-ring” - Although this situation should appear in a foundational way, perhaps there are more refined ways of organizing this data, which then serves as a guide on how to construct this condensed-homotopy invariant. Indeed: what we’re after is a categorification of a “condensed $E_{\infty }$ ring”

Example Questions

Arithmetic Varieties and Algebraic Number Theory

To visualize the geometry of algebraic varieties over $\mathbb{Z}$, Mumford draws a picture of $Spec(\mathbb{Z}[t])$.

It’s the geometric space on which a function is given by a polynomial with $\mathbb{Z}$-coefficients. We see the vertical axis is labelled by elements of $\mathbb{Z}\simeq {\pi }_0(\mathbb{S})$ .

We can imagine generalizing this in two directions

• Move up along Postnikov tower of $\mathbb{S}$: have versions of this picture labelled by ${\tau }_{\le n}{\Omega }^{\infty }\mathbb{S}$. In other words, along the underlying/homotopy-$(n,0)$-categories underlying the $(\infty ,0)$-category of virtual finite sets. At each finite $n$ this is a precise and finite question.
• Move below $\mathbb{Z}$, to draw corresponding pictures for $MU,E_n^{\text{LT}}(l),KU,TMF,\mathbb{S},\cdots$.