STP 421 – Probability Theory

Spring 2016

Instructor: Dr. Jay Taylor, office: PSA 447; phone: 965-2641; e-mail: jetaylo6@asu.edu
Time: Tuesdays and Thursdays 3:00-4:15
Location: LSA 101
Office Hours: Wednesdays 11:00-1:00 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, integration-by-parts), 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, university-sanctioned 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 write-up (10-15 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 1.1-1.4

14 Jan

Probability spaces and the laws of probability

Taylor 1.1-1.4

19 Jan

Conditional probabilities and independence

Taylor 2.1-2.2

21 Jan

The law of total probability and Bayes' formula

Taylor 2.3-2.4

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

MATLAB notes; random_walk_2d.m

9 Feb

Normality and the central limit theorem

Hadlock 2.1-2.4

11 Feb

Entropy and information

Taylor 5.1-5.2; Frank (2009)

16 Feb

Extreme value distributions

Hadlock 2.5; notes

18 Feb

Extreme value distribution theorem

notes

23 Feb

Heavy-tailed distributions

Hadlock 2.6-2.7; notes

25 Feb

Stable distributions and Levy flight

notes

1 March

Stable distributions and Levy flight

notes

3 March

The Levy flight foraging hypothesis

Humphries et al. (2010)

8 March

Spring Break


10 March

Spring Break


15 March

The Levy flight foraging hypothesis

notes

17 March

Phase transitions: the Ising model and ferromagnetism

notes

22 March

Phase transitions: the Ising model and ferromagnetism

notes

24 March

Phase transitions: the Ising model and ferromagnetism

notes

29 March

Phase transitions: the Ising model and ferromagnetism

notes

31 March

Group projects


5 April

The Artificial Anasazi project

Hadlock 3.4-3.6; Axtell et al. (2002)

7 April

Evolution and natural selection

Hadlock 4.4-4.5; notes

12 April

Quasispecies and error thresholds

Bull et al. (2005)

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

Predator-prey dynamics

Hadlock 5.4

28 April

Chaos and unpredictability

Hadlock 6.5; Schaffer & Kot (1986)