Review Articles Assignment in Ordinary Differential Equations

Several years ago, I started working with a new graduate student, who was richly decorated with all sorts of academic awards, local and national, for his academic achievements in mathematics. He was an excellent researcher, brilliant mathematician, and gifted student. I was therefore stunned to discover that his writing was nothing short of dreadful! I realised that science students, and mathematics students in particular, obtain next to no training in scientific writing - or any writing, for that matter. These bright students however, are expected to go on to jobs where they will need to write progress reports or technical reports, or to complete a graduate degree which requires the completion of a lengthy thesis. Clearly, scientific writing should be part of their training, and our current undergraduate education is sorely lacking in this area. 

So I decided to introduce short review articles (2 pages) into the curriculum of my Ordinary Differential Equations class. My goal in doing so is two fold.

First, I want the students to learn how to write scientifically. This includes basic writing skills such as having a logical flow of ideas through the article, beginning paragraphs with sentences that introduce the topic of the paragraph, and ending paragraphs with sentences that lead meaningfully to the next paragraph. For scientific writing, it is also important that the students learn to back up all of their statements with articles, to refer to journal articles rather than websites, how to reference a journal article or textbook, and to never make vague and flowery statements that sound grand but impart no useful information.

Second, I want the students to interact with current research literature, and see how widely applicable is the subject matter of the course. By the time the students have written their second research article, they have become familiar with search engines such as Web of Science, they have learned that they can decipher at least the main message in some mathematics and/or science papers, and they have seen a wide variety of applications of the course material. In particular, they are able to select applications that interest them, broadening the selection that I am able to offer in class.

The students’ reactions to this article assignment has been very interesting. Many of the students who are generally in the top half of a mathematics class groan and complain that the reason they are in science or mathematics is because they don’t know how to write. Their discomfort is further testament to the importance of the exercise. Then there is always another group of students, often belonging to the lower half of the class in terms of math marks, who are delighted that they are finally able to show off a skill that they do have in a mathematics class! There are, of course, some students who are both excellent mathematicians and excellent writers, and they too are pleased to be able to gain some recognition for their writing skills.

Ideally, I the good writers end up working together with the students who do better in the language of mathematics, and collaborate across the different typ

Path Integral Methods for Stochastic Differential Equations

A pedagogical paper (Path Integral Methods for Stochastic Differential) on how to use path integral and diagrammatic methods to solve stochastic differential equations perturbatively. The paper was originally written as a companion to a lecture by Carson Chow on the same topic at the 2009 mathematical neuroscience workshop in Edinburgh. The slides are available on the website.

The material could be suitable for a section in a “methods in applied math” or an “applied dynamical systems” course for first year graduate students or advanced undergraduates. It forgoes any semblance of rigor in favor of demonstrating practical ways to compute quantities such as moments and cumulants for nonlinear SDEs. We have also used these same methods to compute fluctuations and correlations around mean field theory in deterministic high dimensional dynamical systems although we do not explicitly cover this topic in this paper. The methods we introduce are well established in statistical mechanics and quantum field theory but are not well known in dynamical systems. We hope that this paper takes the mystery out of path integrals and Feynman diagrams and show that they are merely a convenient means to organize a perturbation expansion that can be applied to a variety of problems.

Nonlinear dynamics and Chaos: Lab Demonstrations

This 1994 video shows six laboratory demonstrations of chaos and nonlinear phenomena, intended for use in a first course on nonlinear dynamics. Steven Strogatz explains the principles being illustrated and why they are important.

The demonstrations are:
(1) A tabletop waterwheel that is an exact mechanical analog of the Lorenz equations, one of the most famous chaotic systems;
(2) A double pendulum, a paradigm of chaos in conservative systems;
(3) Airplane wing vibrations and aeroelastic instabilities, as exemplars of Hopf bifurcations;
(4) Self-sustained oscillations in a chemical reaction;
(5) Using synchronized chaos to send secret messages; and
(6) Composing musical variations with a chaotic mapping.

Strogatz is joined by his colleagues Howard Stone, John Dugundji, Irving Epstein, Kevin Cuomo, and Diana Dabby.

Introductory Computational Neuroscience

These materials are a work in progress that can be used as the basis of an introductory computational neuroscience course that is intended to be a roughly 60/40 mixture of hands-on lab work and lecturing. The original one semester, 3 credit hour course (http://www.ni.gsu.edu/~rclewley/Teaching/CompNeuro/NEUR4030.html) was intended for mid-level undergraduates with either less rigorous math backgrounds or unfamiliarity with computer programming, or both. It is based mostly on Hugh Wilson's book, "Spikes, Decisions, and Actions", with some additional material from Izhikevich's book, "Dynamical systems in neuroscience". The course provides a visually- and conceptually-oriented introduction to the basics of qualitative analysis of differential equations and modeling neural systems. It focuses on electrophysiology of single neurons and activity in small networks, while trying to inspire interest in more sophisticated applications such as neural coding, cortical pattern formation and synchrony as components for memory and cognition, motor control, etc.
In particular, the scripts are modernized versions of the original Matlab codes from Wilson's book. They have been converted into Python, and in most cases take advantage of the ODE solving, phase plane, and bifurcation analysis capabilities of PyDSTool (http://pydstool.sf.net). The materials represent only a sample of the original codes, and in some cases are re-interpretations rather than direct translations. The codes are also not rigorously organized, but they will be a good starting point for new course development. The repository includes some homework and project ideas.
As a work in progress, the intention is for the materials to be customized or improved through community participation using the git version control system (http://youtu.be/SCZF6I-Rc4I). Feel free to fork and contribute to the project through github, and submit pull requests to update the Master version so that others may benefit from valuable contributions.
An index of the materials is provided in files_info.txt. The repository can be found at https://github.com/robclewley/compneuro.

Graduate (MAGIC) Course on Ergodic Theory

Prize winner, Teaching DS Competition, 2013

These notes form a 10-lecture course on ergodic theory and its applications to hyperbolic dynamical systems. The level of material is suitable for beginning graduate students in mathematics who want to either gain an overview of various aspects of ergodic theory, or want to gain a more detailed understanding of the ergodic theory of hyperbolic dynamical systems via thermodynamic formalism. 

Topics covered include: examples of dynamical systems, uniform distribution mod 1, invariant and ergodic measures, ergodic theorems and recur- rence, measure-theoretic and topological entropy, thermodynamic formalism, the ergodic theory of hyperbolic dynamical systems.

The course consists of 10 lectures. Each lecture has a set of slides (slides link) and an accompanying set of more detailed lecture notes (notes link). The slides were developed to be used in a MAGIC course (see below), but can also be used for self-study for those who only want/need an overview of the subject. The lecture notes provide details (proofs, additional commentary, etc) for those who are interested in the technical details, and are intended to be used as self-study material. The lecture notes also contain exercises.

MAGIC (=Mathematics Access Grid: Instruction and Collaboration) is a network of 19 mathematics departments at universities in the UK and provides graduate level courses to first year PhD students in pure and applied mathematics. The courses are intended to widen students’ knowledge of mathematics, rather than provide training in their research area. The lectures are given via Access Grid video-conferencing technology. I have given a course on ergodic theory using these notes as part of MAGIC every year since its inception in 2008.