**Title:** Ultra-Products and Chromatic Homotopy Theory

**Speaker:** Tomer Schlank

**Speaker Info:** MIT

**Brief Description:**

**Special Note**:

**Abstract:**

Let $C_{p,n}$ be the $K(n)$-local category at height $n$ and prime $p$. These categories are of great interest to the stable homotopy theorist since they serve as a the “associated graded” pieces of the chromatic filtration on the category of spectra. It is a well known observation that for a given height $n$ certain “special” phenomena happen only for small enough primes. Further, in some sense, the categories $C_{p,n}$ become more regular and algebraic as $p$ goes to infinity for a fixed $n$. The goal of this talk is to make this intuition precise.Given an infinite sequence of mathematical structures, logicians have a method to construct a limiting one by using “ultra-products”. We shall define a notion of “ultra-product of categories” and then describe a collection of categories $D_{n,p}$ which will serve as algebro-geometric analogs of the $K(n)$-local category at the prime $p$.

Then for a fixed height $n$ we prove:

$$\prod_p^{\mathrm{Ultra}} C_{n,p} \cong \prod_p^{\mathrm{Ultra}} D_{n,p}.$$

If time permits we shall describe our ongoing attempts to use these methods to get a version of the $K(n)$-local category corresponding to formal Drinfeld modules (instead of formal groups).

This is a joint project with N. Stapleton and T. Barthel.

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