# Lecture notes “Aspects of the Monster Tower Construction”

Here are lecture notes and slides from the 4-lecture series delivered by me and Susan Colley at the Workshop on Nash Blow-up and Semple Tower, II in June 2019 at KU Leuven. The full title is “Aspects of the Monster Tower Construction: Geometric, Combinatorial, Mechanical, Enumerative.” The notes will serve as the kernel of a book that we are now (slowly) writing along with Corey Shanbrom.

Here’s a link to the full 51-page lecture notes:

Leuven lecture notes

Here are the lecture slides

Leuven Lecture 1

Leuven Lecture 2

Leuven Lecture 3

Leuven Lecture 4

Everything in the notes is built on the “Basic construction.” At the lectures we handed out this one-page reminder of the construction:

Basic construction

Here is our Introduction:

In these lectures we survey the construction and use of the monster tower (also known as the Semple tower) in three distinct areas of mathematics.

Lecture 1: The Monster Tower (Kennedy) — This lecture will explain how three seemingly different situations lead to the same construction:

1. Compactifying curvilinear data (algebraic geometry)
2. Studying Goursat distributions (differential geometry)
3. Analyzing a truck with trailers (mechanics and control theory)

Lecture 2: Combinatorial Aspects (Colley) — We explain a natural system of coordinate charts on the monster space. We show how to lift (prolong) a curve in the base into the tower. We explain a natural coarse stratification of the monster, catalogued by a simple system of code words.

Lecture 3: Mechanical Aspects (Kennedy) — A version of the monster tower construction creates the natural configuration space for a truck with trailers. We explain the model and survey some important features, including Lie brackets of its basic vector fields, its singular configurations, and its dynamics.

Lecture 4: Enumerative Aspects (Colley) — We begin with an introduction (via examples) to the subjects of enumerative geometry and intersection theory. Specializing to the enumeration of contacts between plane curves, we present a strategy for counting such contacts and illustrate it with a quadruple contact formula we once proved. The ideas behind this formula lead to a discussion of the orbits of the monster space, and to the idea of appropriately lifting a family of curves.