STP 421 – Probability Theory
Spring 2016
Instructor: Dr. Jay Taylor, office: PSA 447; phone:
9652641; email: jetaylo6@asu.edu
Time: Tuesdays and
Thursdays 3:004:15
Location: LSA 101
Office
Hours: Wednesdays 11:001:00 in PSA 447.
Text: Six
Sources of Collapse by
Charles Hadlock (MAA, 2012).
Course Description: This course will use probability theory and mathematical modeling to investigate how complex systems collapse. Sudden and often catastrophic losses of order occur in many settings, including structural collapses (bridges and buildings); ice melting and water boiling; cell, organ and organismal death; decline and extinction of once common species (the passenger pigeon and the American chestnut); ecosystem collapse and desertification; business failures (Enron and Lehman Brothers); market crashes (tulip mania, the housing market bubble); and the collapse of societies, governments and empires (the Anasazi, Yugoslavia). Our approach will be to build mathematical models of collapsing systems and to then look for features that may be shared in common across different settings. Following Hadlock (2012), we will examine six mathematical themes, including rare events and extreme value theory; crowd behavior and group dynamics; competition and evolution; bifurcations and loss of stability; oscillations and feedback; and network structure and percolation. Particular emphasis will be placed on probability theory, stochastic models and simulation.
Prerequisites: Formally, these include three semesters of calculus, up through multivariate calculus. In practice, you should be familiar with differentiation (product, quotient and chain rules), integration (definite and indefinite integrals, substitution, integrationbyparts), Taylor series expansions and Jacobians. If your calculus is rusty, then be sure to review it at the beginning of the semester, as we will make extensive use of these techniques in the second half of the course.
Attendance and quizzes: Class attendance is not mandatory, but there will be regular (i.e., at least weekly) quizzes which will usually be given at the beginning of each class session. I will drop your three lowest quiz scores when determining your final grade, so you can miss up to three of these without penalty. Missed quizzes cannot be made up, but you will not be penalized for excused absences (e.g., for serious illness, universitysanctioned activities, etc.) provided that you inform me in a timely manner.
Exercises: These will be posted on the course web page at the following link, along with their solutions. You are welcome to work in groups, but you should write up your solutions individually and you should always give credit if your solution came from another source, such as a textbook or an online resource, or from one of your classmates. Please note that late assignments will only be accepted at the instructor's discretion and no assignments will be accepted once the solutions have been posted.
Group projects: Each student is expected to participate in a group research project exploring a topic or question related to the mathematics of collapse in depth. A list of possible research projects will be provided at this link, but groups are welcome to design their own projects with the consent of the instructor. Each group should contain three to five students and you should inform me of the composition of each group by 28 January 2016. Students unattached after this date will be randomly allocated to groups. A project proposal (1 page) is due on 11 Feb 2016. This should clearly describe the question that you are going to investigate and how you will do so and it should also list five references. A progress report (1 page) is due on 24 March 2016. This should briefly describe what you have done so far and what remains to be done. The final writeup (1015 pages) is due on 26 April 2016. This should be structured like a published research paper and include an introduction, a methods section, a results and discussion section, and a bibliography with at least 20 references, which each group member has read. All writing should be your own; in particular, avoid quoted text. Each group member should submit three copies of the final report, which will be distributed for peer review, and the group should also provide me with a single copy which I will read. Grades will be based on the progress report (20 points) and the peer and instructor reviews of the final report (80 points).
Grading: Course grades will be based on quizzes (50%), exercises (30%) and the group project (20%). A score of 90% and above will guarantee an A, 75 – 89% will guarantee a B, 60 – 74% will guarantee a C, and 40 – 59% will guarantee a D.
ASU Policy on Academic Integrity: `Academic honesty is expected of all students in all examinations, papers, laboratory work, academic transactions and records. The possible sanctions include, but are not limited to, appropriate grade penalties, course failure (indicated on the transcript as a grade of E), course failure due to academic dishonesty (indicated on the transcript as a grade of XE), loss of registration privileges, disqualification and dismissal. For more information, see http://provost.asu.edu/academicintegrity.'
Course notes: These are posted here. In addition, I have prepared a short document summarizing the most important concepts in probability theory. This is the material that you would be expected to have mastered should you take a more advanced probability or statistics course in the future.
Date 
Topic 
Reading 
12 Jan 
Overview: order, randomness and the mathematics of collapse 
Hadlock 1.11.4 
14 Jan 
Probability spaces and the laws of probability 
Taylor 1.11.4 
19 Jan 
Conditional probabilities and independence 
Taylor 2.12.2 
21 Jan 
The law of total probability and Bayes' formula 
Taylor 2.32.4 
26 Jan 
Discrete random variables 
Taylor 3.1 
28 Jan 
Continuous random variables 
Taylor 3.2 
2 Feb 
Expectations and moments 
Taylor 3.3 
4 Feb 
How to simulate random events 

9 Feb 
Normality and the central limit theorem 
Hadlock 2.12.4 
11 Feb 
Entropy and information 
Taylor 5.15.2; Frank (2009) 
16 Feb 
Extreme value distributions 
Hadlock 2.5; notes 
18 Feb 
Extreme value distribution theorem 

23 Feb 
Heavytailed distributions 
Hadlock 2.62.7; notes 
25 Feb 
Stable distributions and Levy flight 

1 March 
Stable distributions and Levy flight 

3 March 
The Levy flight foraging hypothesis 

8 March 
Spring Break 

10 March 
Spring Break 

15 March 
The Levy flight foraging hypothesis 

17 March 
Phase transitions: the Ising model and ferromagnetism 

22 March 
Phase transitions: the Ising model and ferromagnetism 

24 March 
Phase transitions: the Ising model and ferromagnetism 

29 March 
Phase transitions: the Ising model and ferromagnetism 

31 March 
Group projects 

5 April 
The Artificial Anasazi project 
Hadlock 3.43.6; Axtell et al. (2002) 
7 April 
Evolution and natural selection 
Hadlock 4.44.5; notes 
12 April 
Quasispecies and error thresholds 

14 April 
Bifurcations, epidemics and herd immunity 
Hadlock 6.3 
19 April 
Branching processes and biological invasions 
Taylor 7.1 
21 April 
Catastrophic regime shifts and tipping points 
Hadlock 6.4; Scheffer & Carpenter (2003) 
26 April 
Predatorprey dynamics 
Hadlock 5.4 
28 April 
Chaos and unpredictability 
Hadlock 6.5; Schaffer & Kot (1986) 