**Speaker**: Nianzi Li (李念梓), Shanghai Jiao Tong University

**Date**: Wed, Jun 27, 2018

**Time**: 15:30 - 17:00

**Venue**: 3323A

**Title**: Cycle representation theory

**Abstract**:

This report is a review of the genesis and developments of cycle representation theory over last few decades. The purpose of cycle representation theory is to simplify the description of Markov processes. While traditional tools to describe and investigate Markov processes are stochastic matrices and weighted directed graphs, this theory provides some new insights, which uses weighted directed circuits (cycles) to study Markov processes and focuses more on the geometric and topological aspects. First we start with the definition of circuit processes, that is, we show how to generate Markov processes by a collection of directed circuits, which can be viewed as a finite sequence of points on a directed circle, and the corresponding weights. On the other hand, given a Markov process, we wish to find some weighted circuits, such that the process generated by these circuits is exactly the given one. We consider the discrete finite case, the denumerable case, continuous parameter case and higher-order case respectively and develop the corresponding representation algorithms.

**Slides**:
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