There are two related programs in Applied Mathematics leading to the PhD degree - Mathematical Science and Applications-Oriented Mathematics. These two programs provide frameworks for the study of mathematics and its interactions with science and engineering. In addition, the members of the applied mathematics group have interests in certain collateral areas. Current topics in the Mathematical Science program include:

**Mathematical Science:**

*
Scattering theory, wave propagation, statistical mechanics,
electrodynamics, acoustics, plasmas, magnetohydrodynamics, elasticity,
critical phenomena, fluid mechanics, geophysical fluid dynamics,
and mathematical biology.
*

**Prelim #6. Mathematical Science Cluster:**

Students may prepare for the written prelim in Applied Mathematics by taking the two of the following three courses:

- Math 574 Applied Optimal Control
- Math 580(590) Introduction to the Mathematics of Fluid Dynamics
- Math 586(590) Computational Finance

**Collateral Areas:**
Scientific supercomputing, control theory, parallel computational
control, parallel scheduling, stochastic modeling, queuing theory
and computer performance evaluation, numerical analysis, and symbolic
computation.

The department will also work with other departments on combined study programs and joint degrees to meet individual needs and special interests of students. The department's broad spectrum of activities includes group theory, classical and functional analysis, differential geometry and topology, statistics and probability, and computational science. Joint programs of any of these and Applied Mathematics may be arranged.

**Applied Mathematics Advisor:**
Students' programs of doctoral study in Applied Mathematics should
be made in close consultation with an Applied Mathematics advisor
and will depend upon their interests and research area. Programs
should be arranged so that 500-level courses leading to the two
written prelims and to the fulfillment of the minor requirement
are taken early.

**Doctoral Minor Requirement:**
The doctoral minor requirement should be designed in consultation
with an Applied Mathematics advisor and in accordance with the
department regulations. The minor typically consists of a sequence
of two 500 level courses either in the department or in an outside
department. If the minor courses are in the department, the two
courses may be chosen from the list issued by the Graduate Studies
Committee. Typically these courses are required for one of the
preliminary examinations in clusters outside of applied mathematics,
such as Combinatorics, Algorithms and Complexity, Computational
Science, Analysis, etc. Any other sequence of the department's
courses or courses in an outside department must be approved in
advance by the Director of Graduate Studies. A minor in an outside
department is recommended for students interested in a specific
application area such as plasma physics, fluid dynamics, elasticity,
scattering, or neuroscience.

This is one of two cluster examinations required for the Applied Mathematics Option. The examinee is required to answer at least 3 out of 6 questions form material covered in other Mathematical Sciences courses offer in recent terms. Usually 9 or more questions will be offered on the exam. A perfect score consists of answering 5 questions correctly. The questions deal with the mathematical formulation and solution of problems stated in physical contexts. The basic topics for the examination are discrete and continuum mechanics, electromagnetics, scattering theory, wave propagation, diffusion theory, applied optimal control theory and computation, computational and mathematical finance, mathematical biology, and problems from other physical sciences and engineering. Some exams in the file indicate the intent and level.

**Topics in Mathematical Models in the Sciences:**

*Optimal Control*- Optimal control theory, calculus of variations, maximum principle, dynamic programming, feedback control, linear systems with quadratic criteria, singular control, stochastic differential equations, Gaussian and Poisson noise, stochastic control.*Fluid mechanics*- Navier-Stokes equations for viscous flow, Euler equations for inviscid flow; Prandtl boundary layer.*Computational Finance*- Pricing of derivative instruments such as options, interest rates and other contracts; computation of fair market prices.

**Former Topics in Mathematical Science:**

*Classical mechanics*- generalized coordinates, central force motion, electrostatics, potential theory, energy conservation, nonlinear vibrations.*Elasticity*- static and dynamic problems including linear elastic waves; beam theory; biharmonic equation; Beltrami-Michell equation; Poisson's equation.*Biology processes*- population dynamics, (logistic and age structure equation), interacting populations (predator-prey and competition).

**Course References:**

- David J. Acheson,
*Elementary Fluid Dynamics*, Oxford Applied Mathematics and Computing Science Series, Oxford University Press, 1990 (Math 590 Friedlander). - R. F. Stengel,
*Optimal Control and Estimation*, Dover Paperback, 1994 (Math 574). - D. E. Kirk,
*Optimal Control Theory: An Introduction*, Prentice-Hall, 1970 (Math 574). - C. C. Lin and L. Segel,
*Mathematics Applied to Deterministic Problems*(Math 580). - L. Segel,
*Mathematics Applied to Continuum Mechanics*(Math 580). - P. Wilmott, Howison, Dewynne,
*The Mathematics of Financial Derivatives: A Student Introduction*, Cambridge, 1995 (Math 590/58X?).

**General References:**

- Courant & Hilbert,
*Methods of Mathematics Physics, Vol. 2*

**Web Source: http://www.math.uic.edu/~hanson/prelmspsyl.html**

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