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
- Algebraic Topology and Knot Theory
- Applied Analysis, Control Theory and PDEs
- Combinatorics and Graph Theory
- Computational Mathematics
- Harmonic Analysis and Representation Theory
- Mathematics of Materials Science
- Probability
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:
- MATH 7240 (Topics in Algebraic Geometry)
- MATH 7260 (Homological algebra)
- MATH 7250 (RepresentationTheory)
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:
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:
Key Advanced Courses Offered at least on Alternate Years:
Key Advanced Courses Offered at least Every Third Year:
Applied Analysis, Control Theory and PDEs Key Common Core Courses:MATH 7311 Real Analysis; MATH 7210 Algebra-I; MATH 7510 Topology I Key Breadth Courses: Math 7320 Ordinary Differential Equations; MATH 7550 Differential Geometry; MATH 7360 Probability Theory; MATH 7350 Complex Variables; MATH 7330 Functional Analysis. Key Advanced Courses Offered Annually:
Key Advanced Courses Offered at least on Alternate Years:
- MATH 7386 Theory of Partial Differential Equations
- MATH 7366 Stochastic Analysis
- MATH 7380 Calculus of Variations
- MATH 7380 Optimal Control
- MATH 7380 Mathematical Topics in Systems Theory
Key Advanced Courses Offered at least Every Third Year:
- MATH 7380 Integral and Functional Equations
- MATH 7380 Asymptotic Analysis
- MATH 7380 Non-smooth Analysis
- MATH 7380 Evolution Equations
- MATH 7390 Harmonic Analysis I
Harmonic Analysis and Representation Theory Key Common Core Courses: MATH 7311 Analysis I; MATH 7510 Topology I. Key Breadth Courses: MATH 7550 Differential Geometry; MATH 7330 Functional Analysis; MATH 7350 Complex Analysis. Key Advanced Courses Offered at least on Alternate Years:
- MATH 7390 Harmonic Analysis I (Fourier Analysis)
- MATH 7390 Harmonic Analysis II (Classical Groups)
- MATH 7370 Lie Groups and Representation Theory
- MATH 7375 Wavelets
Key Advanced Courses Offered at least Every Third Year:
- MATH 7380 Seminar in Functional Analysis
- MATH 7390 Seminar in Harmonic Analysis
- MATH 7290 Representations of Finite and Compact Groups
- MATH 7690 Topological Groups
Probability Key Common Core Courses: MATH 7311 - Analysis I Key Breadth Courses: MATH 7360 Probability Theory; MATH 7330 Functional Analysis; MATH 7350 Complex Variables. Key Advanced Courses Offered Annually:
- MATH 7365 Applied Stochastic Analysis (offered in Fall)
- MATH 7366 Stochastic Analysis (offered in Spring)
Key Advanced Courses Offered at least on Alternate Years:
- MATH 7386 Theory of Partial Differential Equations
- MATH 7380 Optimal Control (offered in Fall)
Computational Mathematics Key Common Core Courses: MATH 7311 Analysis I; MATH 7510 Topology I. Key Breadth Courses: MATH 7330 Functional Analysis ; MATH 7350 Complex Variables; MATH 7710 Numerical Linear Algebra; MATH 7360 Probability Theory; MATH 7320 Ordinary Differential Equations. Key Advanced Courses Offered Annually:
Key Advanced Courses Offered at least on Alternate Years: