## Graduate Program Course Recommendations

**MATH****555**,**MATH 601**and**MATH 631**are strongly recommended for most students.- The Fall of your first year, you should plan to take
**MATH 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:
**MATH 553**Asymptotics and Perturbation Methods,**MATH 555**ODE,**MATH 561**Numerical Linear Algebra, Optimization and Monte Carlo Simulation,**MATH 557**Intro to PDE,**MATH 563**Applied Computational Analysis,**MATH 575**Mathematical Fluid Dynamics,**MATH 577**Mathematical Modeling,**MATH 631**Real Analysis,**MATH 641**Probability Theory - Second Year:
**MATH 541**Applied Stochastic Processes,**MATH 553**Asymptotics and Perturbation Methods,**MATH 575**Mathematical Fluid Dynamics,**MATH 641**Probability Theory,**MATH 651**Hyperbolic PDE,**MATH 661**Numerical PDE I,**MATH 663**Numerical PDE II,**MATH 653**Elliptic PDE,**MATH 690-40**Topics in Probability

### Algebra & Number Theory

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

### Analysis

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

### Geometry & Topology

- First Year:
**MATH 555**ODE, 631 Real Analysis,**MATH 601**Groups, Rings and Fields,**MATH****602**Commutative Algebra,**MATH****611**Algebraic Topology I,**MATH 620**Smooth Manifolds,**MATH****621**Differential Geometry,**MATH****633**Complex Analysis - Second Year:
**MATH 603**Representation Theory,**MATH 612**Algebraic Topology II,**MATH 653**Elliptic PDE,**MATH 627**Algebraic Geometry,**PHYSICS 781**Quantum Field Theory,**MATH 527**General Relativity,**MATH 690-20**Topics in Differential Geometry,**MATH 605**Number Theory

### Probability

The essential Probability courses are **MATH 541** Applied Stochastic Processes, **MATH 545** Stochastic Calculus, **MATH 631** Real Analysis, **MATH 641** Probability Theory. Other important Probability courses include **MATH 553** Asymptotics and Perturbation Methods, **MATH 555** ODE, **MATH 557** Intro to PDE, **MATH 561 **Numerical Linear Algebra, Optimization and Monte Carlo Simulation, **MATH 633** Complex Analysis, **MATH 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.