Notes on Worldsheet-Like Variables for Cluster Configuration Spaces

Notes on Complex Variables

Notes on Metric Spaces

These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. Still, you should check the corresponding sections in the book for a possibly different point of view on a few things. The main idea to ...

On Loop Spaces of Configuration Spaces

This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as “twistings” between the factors. The main homological results are given in terms of extensions of the “infinitesimal braid relations” or “universal YangBaxter Lie relations”.

Lecture Notes on Induction Variables

Once we have identified a loop (e.g., natural loop), one of the central questions about it is, which variables are induction variables of the loop. In for-loops, there are syntactical indicators if the step is of the form i++. But that alone does not make i a proper induction variable, because there could be further assignments to i. Furthermore, other variables could effectively be induction v...

Notes on Continuous Random Variables

Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes them from discrete random variables, which can take on only a sequence of values, usually integers. Typically random variables that represent, for example, time or distance will be continuous rather than discrete. Just as we descri...