Research

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.

An invitation to research

I’m going to be speaking about “Spherical Varieties and Tropical Geometry” in the Invitation to Mathematics seminar in two successive weeks:

  • Wednesday, October 29, 2014 – 4:10pm to 5:10pm in Cockins Hall 218
  • Wednesday, November 5, 2014 – 4:10pm to 5:10pm in Cockins Hall 218

Here are lecture slides for my first talk. and my second talk.

Here’s my abstract:

This is intended as a prospectus of research, and I’m looking for students who want to work on this project. A complex algebraic variety is called a spherical variety if it’s acted upon by a reductive group and there is a dense orbit under the action of a Borel subgroup. To begin, I will explain some generalities about spherical varieties and the convex bodies associated to them, mostly via examples. Then I will focus on the research of Jason Miller, some of which appears in his 2014 Ohio State dissertation. Then, making a fresh start, I will say something about tropical varieties, which are piecewise linear or skeletal versions of algebraic varieties; again there will be examples. Tropical varieties naturally live in the world of toric varieties, which are extremely special examples of spherical varieties. The goal of the project is to find notions akin to those of tropical geometry in the wider world of spherical varieties.

For a related reading list, see my earlier post.

Reading list for tropical geometry and spherical varieties

(image by Cowdery and Challas, featured in June 2009 Mathematics Magazine)
tropbezout

This is a reading and viewing list for a small group of people who are working to learn these topics. Anyone is welcome to use it, of course.