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A Non-Archimedean Approach to Stratifications

Immanuel Halupczok

Abstract

These are notes from a mini-course about the main results of arXiv:2206.03438: I explain how, using suitable valued fields, one obtains a natural notion of canonical stratifications (of e.g. algebraic subsets of $\mathbb{R}^n$). I also explain how the same techniques yield more invariants of singularities, and I present an application to Poincaré series. While some rudimentary knowledge of model theory is useful, the notes should also be accessible without such knowledge. In particular, they contain an introduction to the non-standard analysis needed for this approach.

A Non-Archimedean Approach to Stratifications

Abstract

These are notes from a mini-course about the main results of arXiv:2206.03438: I explain how, using suitable valued fields, one obtains a natural notion of canonical stratifications (of e.g. algebraic subsets of ). I also explain how the same techniques yield more invariants of singularities, and I present an application to Poincaré series. While some rudimentary knowledge of model theory is useful, the notes should also be accessible without such knowledge. In particular, they contain an introduction to the non-standard analysis needed for this approach.
Paper Structure (41 sections, 25 theorems, 25 equations, 30 figures)

This paper contains 41 sections, 25 theorems, 25 equations, 30 figures.

Table of Contents

  1. The goal
  2. A wish list for stratifications
  3. Some (unsuccessful) trials to define almost translation invariance
  4. The classical solution and our new approach
  5. Non-standard analysis
  6. Adding infinitesimal elements to $\mathbb{R}$
  7. A concrete example of $\mathbb{R}^*$: Hahn series
  8. The nsa-principle in action
  9. $\mathbb{R}^*$ as a valued field
  10. Defining a valuation on $\mathbb{R}^*$
  11. Valuations of tuples and valuative balls
  12. Hahn fields are spherically complete
  13. Being almost equal -- the leading term structures $\operatorname{RV}^{(n)}$
  14. Risometries
  15. Maps which almost preserve translation invariance
  16. ...and 26 more sections

Key Result

Proposition 2.2.2

$\mathbb{R}(\!(t^{\mathbb{Q}})\!)$ is real closed (meaning it is not algebraically closed, but it becomes algebraically closed after adjoining a square-root of $-1$).

Figures (30)

  • Figure 1: Example \ref{['exa.kein']}
  • Figure 2: Example \ref{['exa.pt']}
  • Figure 3: Example \ref{['exa.gerade']}
  • Figure 4: Example \ref{['exa.singsing']}
  • Figure 5: Example \ref{['exa.whit']}
  • ...and 25 more figures

Theorems & Definitions (71)

  • Definition 1.1.1
  • Definition 1.1.2
  • Example 1.1.3
  • Example 1.1.4
  • Example 1.1.5
  • Example 1.1.6
  • Example 1.1.7
  • Remark 1.1.9
  • Remark 1.1.11
  • Definition 1.2.1
  • ...and 61 more