Course Recommendations

Graduate Program Course Recommendations

  • Math 555, 601 and 631 are strongly recommended for most students.
  • The Fall of your first year, you should plan to take 771 (required) and three additional courses.
  • The Spring of your first year you should plan to take four courses.

Typical graduate course sequences for various focus areas are listed below.  Discuss your course selections with your mentor/advisor before registering each semester; the guidelines below are coarse, your mentor/advisor will help you identify the high priority courses.  In many cases it is a good idea to take the course if you plan to cover the topic on your Oral Qualifying Exam.

It may be that one (or more) of the courses that your mentor recommends has a title/syllabus similar to a course that you took elsewhere.  And as a result you may wish to skip the course.  In that case, arrange a short (say 30-60 minute) oral exam with the professor teaching the class.  The purpose of the exam is for the professor to assess whether or not you know the material in the breadth and depth that they would expect of a student working in the subject (or a related area).  If you pass, the professor writes a short note/email to the DGSA and a waiver will be added to your file; otherwise you take the course.  [Certainly we do not want to waste students' time with courses that they do not need.  However there have been cases in which it became apparent (during the course of the student's thesis work) that these taken-elsewhere-courses did not cover the material in the rigor/depth/breadth necessary to leave the student well prepared for thesis work.  This puts them at a competitive disadvantage, may delay graduation, and is burdensome for the advisor.]

Applied Mathematics & Numerical Analysis

  • First Year: 553 Asymptotics and Perturbation Methods, 555 ODE, 561 Numerical Linear Algebra, Optimization and Monte Carlo Simulation, 557 Intro to PDE, 563 Applied Computational Analysis, 575 Mathematical Fluid Dynamics, 577 Mathematical Modeling, 631 Real Analysis, 641 Probability Theory
  • Second Year: 541 Applied Stochastic Processes, 553 Asymptotics and Perturbation Methods, 575 Mathematical Fluid Dynamics, 641 Probability Theory, 651 Hyperbolic PDE, 661 Numerical PDE I, 663 Numerical PDE II, 653 Elliptic PDE, 690-40 Topics in Probability

Algebra & Number Theory

  • First Year: 601 Groups, Rings and Fields, 602 Commutative Algebra, 605 Algebraic Number Theory, 611 Algebraic Topology I, 631 Measure and Integration, 633 Complex Analysis, 636 Analytic Number Theory
  • Second Year: 603 Representation Theory, 612 Algebraic Topology II, 620 Smooth Manifolds, 621 Differential Geometry, 625 Riemann Surfaces, 627 Algebraic Geometry

Analysis

  • First Year: 545 Stochastic Calculus, 555 ODE, 557 Intro to PDE, 631 Real Analysis, 633 Complex Analysis, 641 Probability Theory
  • Second Year:  545 Stochastic Calculus, 553 Asymptotics and Perturbation Methods, 635 Functional Analysis, 641 Probability Theory, 651 Hyperbolic PDE, 653 Elliptic PDE

Geometry & Topology

  • First Year: 555 ODE, 631 Real Analysis, 601 Groups, Rings and Fields, 602 Commutative Algebra, 611 Algebraic Topology I, 620 Smooth Manifolds621 Differential Geometry633 Complex Analysis
  • Second Year: 603 Representation Theory, 612 Algebraic Topology II, 653 Elliptic PDE, 627 Algebraic Geometry, PHYSICS 781 Quantum Field Theory, 527 General Relativity, 690-20 Topics in Differential Geometry, 605 Number Theory

Probability

The essential Probability courses are 541 Applied Stochastic Processes, 545 Stochastic Calculus, 631 Real Analysis, 641 Probability Theory.  Other important Probability courses include 553 Asymptotics and Perturbation Methods, 555 ODE, 557 Intro to PDE,  561 Numerical Linear Algebra, Optimization and Monte Carlo Simulation, 633 Complex Analysis, 635 Functional Analysis.  Students should select other courses in Analysis, Applied Mathematics and Numerical Analysis based on their interests. For example students working in Stochastic Analysis should take courses in PDE. In general, strong computational skills are valuable.