## Program Milestones

• We suggest students complete the fundamental courses in their first year of study to provide a firm foundation for our advanced courses and for their foray into mathematical research.

Each of our 500-level courses are offered in Fall or Spring, so students should plan accordingly to ensure sequenced courses are taken in the correct order. Below we offer 3 sample programs for first year students.

Sample 1. For MATH students with interest in Algebra, Geometry or Topology

 Fall Spring 510a – Algebra 1 510b – Algebra 2 525a – Real Analysis 1 520 – Complex Analysis 540 – Topology 535a – Differential Geometry 595 – Practicum in Teaching 500 – Graduate Colloquium

Sample 2: For MATH students with interest in Analysis, Differential Equations or Probability

 Fall Spring 507a – Probability 1 541a – Statistics 1 525a – Real Analysis 1 520 – Complex Analysis 555a – PDE 565a – ODE 595 – Practicum in Teaching 500 – Graduate Colloquium

Sample 3: For AMAT students

 Fall Spring 555a – PDE 502a – Numerical Analysis 1 505a – Applied Probability 1 541a – Statistics 1 525a – Real Analysis 1 525b – Real Analysis 2 595 – Practicum in Teaching 500 – Graduate Colloquium
• Our written screening examinations test mastery of fundamental domains of mathematics. Passing the associated course is necessary but not sufficient preparation for the exams.

Topics covered and copies of past exams are on our website, at https://dornsife.usc.edu/mathematics/graduate-exams/. We strongly encourage student to focus on writing solutions to past exams clearly and completely.

##### MATH students
• must pass THREE of the following exams:
• Algebra, based on Math 510a and Math 510b
• Real and Complex Analysis, based on Math 525a and Math 520
• ODE and PDE, based on Math 555a and Math 565a
• Geometry and Topology, based on Math 535 and Math 540
• Probability and Statistics, based on Math 507a and Math 541a
• should pass ONE of Algebra or Real and Complex Analysis by August of their 2nd year;
• should pass ALL THREE written exams after 5 semesters (that is, January of their 3rd year).
##### AMAT students
• must pass FOUR exams among the following options:
• Real Analysis, based on Math 525a
• Statistics, based on Math 541a
• Applied Probability, based on Math 505a
• Numerical Analysis or PDE, based on Math 502a or Math 555a
• should pass TWO of the required exams by August of their 2nd year;
• should pass ALL FOUR written exams after 4 semesters (that is, August of their 3rd year)

Students may be given additional time to complete exams if they are making steady progress and if the student’s (temporary) advisor recommends it.

• You should try to choose an advisor in your 2nd year, and certainly by early into your 3rd year.

There is no one way to find an advisor. We suggest you:

• go to classes taught by various faculty;
• attend the area seminars;
• discuss advisors with older students;
• speak with faculty in office hours or at tea;
• request a reading course (Math 590) on a special topic.

The student-advisor relationship is much like an apprenticeship, so good fit between student and advisor is important. We recommend you keep an open mind about your research area in order to find the best advisor for you.

• For MATH, the student prepares a 10-page written research proposal with references which must be approved by the committee at least one week prior to the exam. The oral is a 45 minute presentation in which the student defends/discusses the proposal. No original research is required. MATH students should take their oral exam within 2 semesters after finishing their written exams. It must be taken by the end of the 7th semester at the latest.

For AMAT, the oral is more demanding: the student must have enough research for a partial thesis which is submitted to the committee at least two weeks prior to the exam. The oral is a 45 minute presentation in which the student defends the research and discusses the proposal for continued research. AMAT students should take their oral within 3 semesters after finishing their written exams. It must be taken by the end of their 4th year at the latest.

The oral exam committee consists of five faculty members:

• Committee chair must be the advisor
• 3 additional faculty in Mathematics
• 1 faculty member at USC outside of Mathematics

Students must submit the Appointment or Change of Qualifying Exam Committee form to the Mathematics Graduate Advisor at least 2 weeks prior to taking the exam. For more details about this process, please consult the Ph.D. Handbook.

• The student must give a public lecture (thesis defense) on their research, to be approved by their dissertation committee.

The Dissertation Committee consists of three faculty members:

• Committee chair must be the advisor
• 1 additional faculty in Mathematics
• 1 faculty member at USC outside of Mathematics

This defense should take place at least two months before the end of the semester in which they intend to receive their degree, so that they can complete all of the graduate school Thesis and Submission Deadlines on time.