The Written Preliminary Exam is taken by all incoming AMCS graduate students at the University of Pennsylvania, just before the start of the fall semester (generally in late August). It plays three roles:

- It serves as a placement exam, to help determine whether students should begin with 500-level courses or with 600-level courses (or with a mixture).
- It is a requirement for each of the graduate degrees in applied mathematics, to ensure that those who receive graduate degrees have a solid mathematical foundation.
- It provides an incentive for incoming grad students to review basic material, which will then help them in their beginning graduate classes.

**PhD students** who do not pass the exam on their first attempt will have one more chance to pass it at the end of the spring semester (generally in late April or early May). PhD students who do not pass the exam by the end of their first year will be asked to leave the program.

**Masters students** who do not pass the exam on their first attempt will have two more opportunities to pass this exam. Students who fail the exam are **required** to retake the exam the next time it is offered. If a student does not do so, they will be given a score of zero, and lose that opportunity to retake the exam. While we strongly encourage master students to prepare before their arrival, intending to pass the exam on their first try, this first attempt serves primarily as a placement exam.

All students **who do not pass** the Written Prelim on the first try are strongly encouraged to take the Proseminar (MATH 504, 505), which helps to prepare you for this exam. You will also be directed to other 400/500-level math courses during your first year, to strengthen your problem-solving ability, and background in mathematics.

The written preliminary exam focuses on the material from an undergraduate mathematics program that is most important to those entering an applied mathematics graduate program. The exam is given in two 2.5-hour sessions, either on a single day or consecutive days. Each part consists of 6-8 problems.

The exam contains problems in linear algebra, advanced calculus, basic complex analysis, and probability. Some problems are computational, some ask for proofs, and some ask for examples or counterexamples. Each part of the exam contains a mixture of types of problems and a mixture of subjects.

**The key to success on the preliminary exams is practice! AMCS Preliminary Exam Examples**

Books that cover the material at an appropriate level are as follows. We list more than one book in some areas only to maximize the probability that the list contains a book you are familiar with.

**Linear algebra:** Strang, Gilbert “*Linear Algebra and its Applications”*

**Real analysis:** Rudin, Walter “*Principles of Mathematical Analysis*” or Strichartz, Robert “*The Way of Analysis*“

**Probability theory:** Ross, Sheldon “*A First Course in Probability*” or Hoel, Port and Stone “*Introduction to Probability Theory*“

**Complex Analysis:** Conway, John B. “*Functions of One Complex Variable”* or Bak and Newman “*Complex Analysis*” or Alfors, Lars “*Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable*“