## 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 Manifolds,**621**Differential Geometry,**633**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.