Curricula by Research Specialty

Vaughn Jones Each student's curriculum must be approved in writing every semester by the Department. This page lists courses required for a doctoral student depending on the student's research specialty. (Students seeking an MS only should click on the desired MS degree at the page Graduate Degrees.) First-year doctoral students normally begin with the three Common Core courses:

One hour of Communicating Mathematics is required each semester of the first year as well. In order to Ph.D. qualify, each student must take at least four Breadth courses, of which at least two must come from List A. Each of the four breadth courses must be passed with at least a grade of B, and at least one must be passed with a grade of A.

List A: Algebra II (7211), Complex Variables (7350), Functional Analysis (7330), Topology II (7512), and Differential Geometry (7550).
List B: Ordinary Differential Equations (7320), Probability (7360), Graph Theory (7400), and Numerical Linear Algebra (7710),

Students planning to write a dissertation in pure mathematics are advised to take at least two courses from each of the areas of algebra, analysis, and topology.

In the listings below, please note that course numbers ending in the digits 80 or 90 are specialized topics courses. Thus a given number can represent many different special topics and it is the specific topics listed below which are being recommended according to research specialty. Please see the page Research Areas for descriptions of the various research areas. VIR courses are listed separately under Courses by Year.

Jump to Specialty:

Algebraic Number Theory, Algebraic Geometry and Representation Theory

Key Common Core Courses: MATH 7210 (Algebra I). Key Breadth Courses: MATH 7211 (Algebra II)

Key Advanced Courses Offered at least on Alternate Years:

Key Advanced Courses Offered at least Every Third Year:

Algebraic Topology and Knot Theory

Key Common Core Courses: MATH 7510 Topology I. Key Breadth Courses: MATH 7512 Topology II; Math 7550 Differential Geometry. Key Advanced Courses Offered Annually:

Key Advanced Courses Offered at least on Alternate Years:

MATH 7590 Cohomology Theory
MATH 7590 Knot Theory
Math 7590 Advanced topics in Knot Theory (quantum invariants, torsion, braids) Manifolds (3, 4, or higher dimensional)
MATH 7590 Differential Geometry II (Riemannian geometry, principal bundles,...)

Mathematics of Materials Science

(See also the course list for Applied Analysis, Control Theory, and Partial Differential Equations. Many of those courses are applicable also to materials science.) Key Common Core Courses: MATH 7311 Analysis I; MATH 7510 Topology I. Key Breadth Courses: MATH 7320 Differential Equations; MATH 7330 Functional Analysis; MATH 7350 Complex Analysis; MATH 7360 Probability Theory. Key Advanced Courses Offered Annually:

MATH 7325 Numerical Analysis
MATH 7384 Topics in the Mathematics of Materials Science (Topics will rotate by semester and will include composite materials, fracture mechanics, integral equations, and electro-magnetics.)

Key Advanced Courses Offered at least on Alternate Years:

Key Advanced Courses Offered at least Every Third Year:

MATH 7380 Non-smooth Analysis

Combinatorics and Graph Theory

Key Common Core Courses: MATH 7210 Algebra Key Breadth Courses: MATH 7400 Graph Theory and MATH 7211 Field Theory Key Advanced Courses Offered Annually: