Curricula by Research Specialty
Each student's curriculum must be approved every semester by the Department. This page provides advice about courses needed by a doctoral student depending on the student's eventual dissertation 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 core-1 courses: Math 7200 (Algebra), 7311 (Analysis I), and 7510 (Topology I). One hour of Communicating Mathematics is required each semester of the first year as well. In the second term, each doctoral student selects three of the six core-2 courses (with a fourth required by the end of the second year for breadth): 7210 (Field Theory), 7312 (Analysis II), 7320 (Ordinary Differential Equations), 7400 (Graph Theory), 7512 (Topology II), and 7550 (Differential Geometry).
Jump to Specialty:
- Algebraic Number Theory and Algebraic Geometry
- Algebraic Topology and Knot Theory
- Applied Mathematics in Materials Science
- Combinatorics and Graph Theory
- Applied Analysis, Control Theory and PDEs
- Harmonic Analysis and Representation Theory
- Probability
Algebraic Number Theory and Algebraic Geometry
Key Core-1 Courses: 7200 (Geometric and Abstract Algebra)
Key Core-2 Courses: 7210 (Algebra I)
Key Advanced Courses Offered Annually:
- 7211 (Algebra II)
Key Advanced Courses Offered at least on Alternate Years:
- 7280 (Topics in Commutative Algebra)
- 7290 (Topics in Algebraic Number Theory)
Key Advanced Courses Offered at least Every Third Year:
- 7280 (Topics in Algebraic Geometry)
- 7290 Homological algebra
- 7290 Representations of Finite and Compact Groups
Algebraic Topology and Knot Theory
Key Core-1 Courses: Math 7510 Topology I.
Key Core-2 Courses: Math 7512 Topology II; Math 7550 Differential Geometry.
Key Advanced Courses Offered Annually:
- Math 7520 Algebraic Topology
- Math 7590 Geometric Topology
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,...)
Applied Mathematics in 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 Core-1 Courses: Math 7311 Analysis I; Math 7510 Topology I.
Key Core-2 Courses: Math 7312 Analysis II; Math 7320 Differential Equations.
Key Advanced Courses Offered Annually:
- Math 7330 Functional Analysis
- Math 7350 Complex Analysis
- Math 7360 Probability Theory
- Math 7380 Numerical Analysis
- Math 7384 Topics in Applied Analysis in 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:
- Math 7380 Calculus of Variations
- Math 7386 Partial Differential Equations
Key Advanced Courses Offered at least Every Third Year:
- Math 7380 Non-smooth Analysis
Combinatorics and Graph Theory
Key Core-1 Courses: Math 7200 Algebra
Key Core-2 Courses: Math 7400 Graph Theory and Math 7210 Field Theory
Key Advanced Courses Offered Annually:
- Math 7490 Discrete Optimization
Key Advanced Courses Offered at least on Alternate Years:
- Math 7490 Selected Topics on Graph Theory
Key Advanced Courses Offered at least Every Third Year:
- Math 7490 Matroid Theory
Applied Analysis, Control Theory and PDEs
Key Core-1 Courses:Math 7311 Real Analysis; Math 7200 Linear Algebra; Math 7510 Topology I
Key Core-2 Courses: Math 7320 Ordinary Differential Equations; Math 7312 Measure and Integration.
Key Advanced Courses Offered Annually:
- Math 7350 Complex Variables
- Math 7330 Functional Analysis
- Math 7380 Numerical Methods & Applications
Key Advanced Courses Offered at least on Alternate Years:
- Math 7380 Partial Differential Equations
- Math 7360 Probability Theory
- Math 7390 Stochastic Analysis
- Math 7380 Calculus of Variations
- Math 7380 Optimal Control
- Math 7550 Differential Geometry
Key Advanced Courses Offered at least Every Third Year:
- Math 7390 Applied Analysis in the Materials Sciences
- 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 Core-1 Courses: Math 7311 Analysis I; Math 7510 Topology I.
Key Core-2 Courses: Math 7312 Analysis II; Math 7550 Differential Geometry.
Key Advanced Courses Offered Annually:
- 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 7390 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
Key Core-1 Courses: Math 7311 - Analysis I
Key Core-2 Courses: Math 7312 - Measure and Integration
Key Advanced Courses Offered Annually:
- Math 7360 Probability Theory (offered in Fall)
- Math 7380 Stochastic Analysis and Applications to Mathematical Finance (offered in Fall)
- Math 7330 Functional Analysis (offered in Fall)
- Math 7390 Stochastic Analysis (offered in Spring)
Key Advanced Courses Offered at least on Alternate Years:
- Math 7380 Partial Differential Equations
- Math 7350 Complex Variables
- Math 7380 Optimal Control (offered in Fall)
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