## Curriculum for Ph.D. of Mathematics with a Concentration in Applied and Computational Mathematics

This page explains the course options for the ACM track and recommened path for the Quals and Comps. Course pairings that form coherent sequences are so designated for convenience of reference later on. There are no sequence requirements *per se*. A general mathematical descriptor is used, followed in parentheses by the Bulletin course title and number.

**S1. **Real and Complex Analysis (Analysis I & II, MATH 703-704)

**S2. **Foundation of Computational Mathematics (Foundations of Computational Mathematics I, MATH 708, and Foundations of Computational Mathematics II, MATH 709)

**S3. **Mathematical Modeling (Applied Mathematics I & II, Math 720-721)

**S4. **Theory of Advanced Differential Equations (Differential Equations I & II, MATH 723-724)

**S5. **Numerical Differential Equations (Numerical Differential Equations I & II, MATH 726-727)

**S6. **Functional Analysis (Functional Analysis I, MATH 756, followed by Applied Functional Analysis, MATH 755)

**S7. ** Probability Theory (MATH 710-711, cross-listed as STAT 810-811)

**S8. ** Approximation Theory (MATH 725) and Nonlinear Approximation Theory (MATH 729)

Basic Year 1 Sequences for the Admission to Candidacy Examination (also known as the Qualifying Examination or Qual):

**S1. ** Real and Complex Analysis (Analysis I & II, MATH 703-704). All Mathematics Ph.D. students are examined on this sequence.

**S2. ** Foundations of Computational Mathematics I & II (MATH 708-709).

Students who have been examined on MATH 701-702 as a qualifying sequence instead of **S2**, and have been admitted to candidacy, may use **S2** as part of their breadth requirement (area **B4**), or they can choose a year-long numerical sequence from breadth areas **B5** (see below).

Years 1 and 2 Sequences for the Comprehensive Examination:

The basic areas that can be chosen for the Comprehensive Exam are divided into two groups. The first comprehensive sequence must be chosen from group **G1**. The second comprehensive sequence can be chosen from either **G1** or **G2**. *These sequences distinguish the ACM students*. The third comprehensive sequence must be chosen from group **G2**, but not from group **G1**, or it can be any other approved sequence in Mathematics, which does not overlap with the first two sequences chosen.

As for any student in the Mathematics Ph.D. program, the three course sequences upon which the Comprehensive Exam is based should be determined by the student in consultation with the prospective major professor (dissertation supervisor). At least one Comprehensive Exam sequence should be in the student’s intended area of research specialization. The examiners, one of whom must be chosen from outside the department, will also serve on the student’s doctoral committee. Selection of the sequences must have the approval of the Graduate Director.

**G1. ** Core areas: Applied Mathematics and Differential Equations:

**a) S3 **

**b) S4 **

**c) S5 **

**G2. **Emphasis Electives: areas most closely relevant to Applied and Computational Mathematics, arranged into coherent sequences:

a**) S8 **

b**) **Fourier Analysis / Wavelets (MATH 750-751)

c**) **Discrete Mathematics (MATH 774-775)

d**) **Graph Theory (MATH 776-777)

e**) **Optimization (MATH 722) and Discrete Optimization (MATH 770)

f**) S6 **

g**) S7 **

** **h**) **A combination of two closely related courses at least one of which is listed as Selected Topics in Applied Mathematics (Math 728x).

Breadth Requirement

Every student must take at least 12 credit hours of 700-level courses (four one-semester courses), with a final grade of B or better** outside of the areas **selected for the Comprehensive Examination. The four additional courses should be chosen in consultation with the major professor. It is recommended that the student is exposed to at least seven of the following eleven areas. Since the Comprehensive Examination normally covers three of these areas already, the four required courses can easily be chosen to achieve this breadth. To indicate the potential overlap with Comprehensive Examination areas, the year-long sequences as labeled above are indicated, but for the purposes of this requirement, only one course from any particular sequence in an outside area is needed.

**B1.** Differential Equations (**S4** = **G1(b)** or similar topics drawn from ODE’s, PDE’s, integral equations)

**B2.** Numerical Methods for Differential Equations (**S5** = **G1(c)** or similar topics drawn from finite difference methods, finite element and volume methods, variational methods, multigrid methods)

**B3.** Mathematical Modeling (**S3 = G1(a)** or similar topics drawn from dimensional analysis, scaling, perturbation methods, calculus of variations, financial mathematics, mathematical models of continua, gas dynamics, complex fluids, multiscale analysis)

**B4.** Numerical Analysis (**S2,** if not already used for the Admission to Candidacy Examination, **G2(h),** or similar topics drawn from approximation of functions, functionals, and operators, methods for linear and nonlinear systems of equations, preconditioning techniques, eigenvalue problems, domain decomposition methods)

**B5.** Scientific Computing (**G2(e)** or **G2(h),** or similar topics drawn from parallel computing, molecular dynamics simulation, data structure, numerical optimization, GPU computing, visualization, Monte Carlo simulation, etc.)

**B6.** Approximation Theory, Fourier Analysis, Wavelets (**S8 = G2(a)**, or **G2(b),** or similar topics)

**B7.** Discrete Mathematics, Graph Theory (**G2(c)** or **G2(d)**) or similar topics)

**B8.** Stochastic Methods (**S7 = G2(g)**, or similar topics drawn from stochastic differential equations, stochastic methods, probability, learning theory)

**B9.** Real, Complex, and Functional Analysis, Analytic Number Theory (**S7 = G2(f)** or any of MATH 752, 754, 757, 782, 783)

**B10.**Algebra, Geometry, Topology, Logic, and Number Theory other than Analytic (any of MATH 701, 702, 73x, 74x, 76x, 780, 784, 785)

**B11.** Interdisciplinary Applications (chemical, physical, biological, geological, nanosciences, etc. at the graduate level)

**Remarks: **

**a) **Each course taken for the breadth requirement can be classified in only one breadth area.

**b) **The courses from the qualifying sequences used in the Admission to Candidacy Examination cannot be applied for the breadth requirement.

**c) **The breadth area of Selected Topics courses (728x) will be designated when the courses are announced.

**d) **Relevant courses from other departments can be approved for the areas **B1-B10**.