Department of Politics, Princeton University

**Instructors:**

- Brendan Cooley
- 4th year Ph.D. student (International Relations, Formal & Quantitative Methods)
- [email protected]
- Office: 300 Fisher Hall
- Office Hours: TBA

- Dan Gibbs
- 4th year Ph.D. student (American Politics, Formal & Quantitative Methods)
- [email protected]
- Office: 025 Corwin Hall
- Office Hours: TBA

**Faculty Advisor:** Kristopher Ramsay

**Piazza Discussion Page:** https://piazza.com/princeton/summer2018/pol500/home

**Description:** This course is designed to prepare incoming Politics Ph.D. students for POL 502 (Mathematics for Political Scientists) and other coursework in Formal & Quantitative Methods. It covers the fundamentals of calculus, probability theory, linear algebra, and real analysis.

The course consists of two parts. The *summer component* be completed by students remotely and will run from June 15 to August 16. Students will read course material, watch video lectures, and complete weekly problem sets on topics in calculus, probability theory, and linear algebra. The *core component* will take place in Princeton from August 27 to September 7. Class will meet twice daily and students will complete daily problem sets.

**Prerequisites:** The course assumes no background knowledge beyond high school algebra. It is self-contained and designed to serve as an introduction or refresher for the material.

**Course Goals:** The goal of the course is to introduce students to mathematical tools that they will use to develop and test social scientific theory. Our focus is on building the skills necessary to conduct comparative static analysis and empirically test the comparative statics generated by a theory. This also serves as an introduction to thinking mathematically. We will cover a lot of ground, but the course is meant to expose students to topics that will be revisit in greater depth in POL 502, POL 571, and other courses in Formal and Quantitative Methods.

**Course Structure:** We will post a problem set every Friday during the summer component. The schedule below will contain links to the assignments and relevant video lectures. We encourage students to post questions about the material in the video lectures and the assignments to the Piazza discussion page. We will do our best to answer them quickly. We also encourage students to discuss problems and course material on the discussion page. Please sign up for the Piazza page if you have not already done so.

The core course will meet weekdays between August 27 and September 10 (Labor Day exempted) with a morning session and an afternoon session each day. The morning session will meet from *10:00am to 11:50am*, and the afternoon session will meet from *1:30pm to 3:20pm*. We will post a daily problem set at the conclusion of each afternoon session in the schedule section below.

**Problem Sets:** Students are required to complete the weekly problem sets during the summer period and the daily problem sets during the camp period. Problem sets during the summer component are due by *midnight each Thursday* (see table of deadlines below). Completed problem sets can be uploaded using the linked Dropbox file requests in the “Due” column of the summer schedule. We understand that summer often features travel and other activities that get in the way of doing math. Please get in touch if you don’t think you’ll be able to submit assignments by their due date, and we’ll work on an alternative assignment schedule for you.

Problem sets during the camp component are due by *10:00am the morning after they are distributed*. Students are encouraged to work together on the problem sets, but are expected to write up their solutions individually. The camp will conclude with a final exam given in class at *10:00am on September 10*.

LaTeX is a typesetting system that will likely become a big part of your life as a graduate student. This summer is a good time to get familiar with it. We ask that you begin submitting typed problem sets by *week 3 of the summer component.* Below, we provide links to tutorials to help you get started. Feel free to post questions about how to use LaTeX on Piazza. Learning is a slog at the beginning, but it will make your life much easier down the road.

**Exams:**

- August 27: Pre-Exam covering summer topics (calculus, probability, and linear algebra)
- September 10: Final Exam

**Grading:**

- 9 Summer Problem Sets (15%)
- 9 Core Problem Sets (15%)
- Pre-Exam (30%)
- Final Exam (40%)

**MS**(Required) Moore, Will H. and David A. Siegel.*A Mathematics Course for Political and Social Research***S**(Optional) Strang, Gilbert.*Introduction to Linear Algebra*.**MM**(Optional) McCarty, Nolan and Adam Meirowitz.*Political Game Theory: An Introduction*

**Summer Component**

The summer component will follow closely the material in Moore and Siegel’s *A Mathematics Course for Political and Social Research*. Siegel has produced video lectures to accompany the textbook. We list the relevant book sections and video lectures for each section of the summer component in the table below. We recommend students watch the video lectures before reading the relevant sections of the textbook. However, different parts of the course will likely require different levels of attention depending on each student’s background.

Because Moore and Siegel’s treatment of linear algebra is somewhat terse, we recommend that students who have not previously taken linear algebra purchase Strang’s *Introduction to Linear Algebra*. Video lectures accompanying Strang’s course are available through MIT’s Open CourseWare, as a supplement to the material in the textbook. We have listed the relevant book chapters in the table below.

Post | Due | Topic | Video | Reading | Solutions |
---|---|---|---|---|---|

Jun 15 | Jun 21 | Building Blocks, Functions, Limits, Continuity | MS 1-2 | MS 1-4 | Solutions |

Jun 22 | Jun 28 | Differentiation | MS 3-4 | MS 5-6 | Solutions |

Jun 29 | Jul 5 | Integration | MS 5 | MS 7 | Solutions |

Jul 6 | Jul 12 | Optimization | MS 6 | MS 8 | Solutions |

Jul 13 | Jul 19 | Probability (Introduction) | MS 7 | MS 9 | Solutions |

Jul 20 | Jul 26 | Probability (Distribution Functions) | MS 8-9 | MS 10-11 | Solutions |

Jul 27 | Aug 2 | Scalars, Vectors, and Matrices | MS 10.1-10.6, 10.8 | MS 12.1-12.3.4, 12.4 (S 1) | Solutions |

Aug 3 | Aug 9 | Solving Systems of Equations | MS 11.1-11.5 | MS 13.1-13.2.2 (S 2.1-2.4) | Solutions |

Aug 10 | Aug 16 | Matrix Inversion and Determinants | MS 10.7, 11.6-11.8 | MS 12.3.5-12.3.7, 13.2.3-13.2.4 (S 2.5-2.7, 5) | Solutions |

**Core Component**

The core component introduces student to real analysis, the study of the real number system and its properties. This is the realm in which political scientists spend almost all of their time, so its useful to understand the fundamentals of how it works and why. It also serves as an introduction to the *language* of mathematics. Once these ideas have been introduced, we take the calculus concepts covered in the summer component to a multidimensional environment, introduced in the summer component.

The course will provide students with basic skills necessary to express theoretical ideas in the language of mathematics, formally derive comparative static propositions about those ideas, and marry those propositions to data.

The appendix of McCarty and Meirowitz’s book contains a nice treatment of many of the topics we’ll cover, and is a useful companion text to this part of the course.

Date | Morning Session | Afternoon Session | Problem Set | Submit |
---|---|---|---|---|

Aug 27 | Pre-Exam (Closed Book) | Exam Debrief, Summer Review | Problem Set | Submit |

Aug 28 | Proofs and Logic | Ordered Sets | Problem Set | Submit |

Aug 29 | Metric Spaces (I) | Metric Spaces (II) | Problem Set | Submit |

Aug 30 | Linear Spaces | Normed Linear Spaces | Problem Set | Submit |

Aug 31 | Functions: Continuous | Functions: Monotone, Linear, Convex/Concave | Problem Set | Submit |

Sep 4 | Inner Product Spaces, Orthogonality, Projection | Functions: Smooth | Problem Set | Submit |

Sep 5 | Optimization: Unconstrained | Optimization: Equality Constrained | Problem Set | Submit |

Sep 6 | Optimization: Existence and Uniqueness | Comparative Statics | Problem Set | Submit |

Sep 7 | Exam Review (I) | Exam Review (II) | Problem Set | Submit |

Sep 10 | Final Exam (Closed Book) |

As mentioned above, we ask that students begin submitting typed problem sets in week 3 of the summer component. That requires learning LaTeX, a typesetting system that facilitates drafting mathematical documents. First thing’s first, you’ll need to download a LaTeX distribution and a text editor suitable for drafting LaTeX documents.

**Distributions**

**Text Editors** (Choose One)

Loading Jennifer Pan’s problem set template into your text editor of choice will give you a sense of how .tex documents are constructed. Both TeXmaker and TeXstudio have buttons that will compile your .tex document into a clean .pdf.

The tricky and painful part will be getting comfortable with LaTeX’s syntax. Googling questions about LaTeX syntax will often turn up results on StackExchange, many of which can be very helpful. We suggest spending some time in broader tutorials before diving into specifics.

We have benefited greatly from the materials developed by our predecessors:

- Amanda Kennard (2017)
- Korhan Kocak (2017)
- Saurabh Pant (2015-2016)
- Ted Enamorado (2015-2016)
- Gabriel Lopez-Moctezuma (2014)
- Yuki Shiraito (2013-2014)
- Jidong Chen (2012-2013)
- Peter Buisseret (-2012)