Real-world mathematics problems usually are not amenable to complete abstraction. Therefore, they often do not fall within the admissible set of problems for "pure" mathematics research. Instead, applied mathematicians must work through these issues to respect the connections that any mathematical analysis will have to the application.

Because mathematics serves as the framework for so many different areas in science and engineering, applied mathematicians are continually increasing their breadth of knowledge in other fields.

In fact, applied mathematics research is usually assessed through two criteria: (1) the sophistication of the mathematics used and (2) the novelty of the application. At UC Merced, the faculty and students engage in research in the following areas:

### Mathematical areas

- asymptotic analysis and perturbation methods
- dynamical systems
- dynamics and design problems for circuits and electromagnetic media
- exponential integrators
- free boundary problems
- inverse problems
- nonlinear analysis
- nonlinear waves
- numerical linear algebra
- numerical optimization
- stochastic and statistical modeling
- waves in random media

**Application areas**

- acoustics
- bioelectrics
- Bose-Einstein condensates
- computational biology
- economics
- fluid dynamics
- image processing and compressed sensing
- mathematical physics
- plasma physics
- solar science
- ultrafast laser and nonlinear optics

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