## Research Areas

The study of computational mathematics has grown rapidly over the past 15 years and has allowed mathematicians to answer questions and develop insights not possible only 20-30 years ago. Modern computational methods require an in-depth knowledge of a variety of mathematical subjects which include linear algebra, analysis, ordinary and partial differential equations, asymptotic analysis, elements of harmonic analysis, and nonlinear equations.

Since computers are invaluable tools for an applied mathematician, students are expected to attain a highly professional level of computer literacy and gain a substantial knowledge of operating systems and hardware.

Computational mathematics courses include the study of computational linear algebra, optimization, numerical solution of ordinary and partial differential equations, solution of nonlinear equations as well as advanced seminars in wavelet and multi-resolution analysis.

Below is a list of faculty who work closely with this type of research:

Recent advances in our ability to quantitatively study biological phenomena have provided a tremendous number of exciting opportunities for applied mathematicians. The careful modeling, analysis, and simulation of these systems using the standard tools of applied mathematics has led to novel and non-intuitive insights into biology.

Furthermore, deeper understanding of the inherently complex and multiscale nature of biological systems, in many cases, requires the development of new mathematical tools, techniques, and methodologies (a challenge to which applied mathematics is particularly well suited). Research areas in APPM encompass immunology, infectious diseases, cardiology, neuroscience, and population genetics.

See the Mathematical Biology Group page

Below is a list of faculty who work closely with this type of research:

Mathematical geosciences encompass quantitative modeling, analysis, and simulation of all aspects of the Earth system. Our faculty's research intersects a broad range of geosciences: from the geodynamo to ocean circulation, from computational methods for seismic imaging to the impacts of weather on epidemiology, from tsunamis to stochastic weather generators. The complex and multiscale nature of geophysical systems, in many cases, requires the development of new mathematical models and simulation strategies, a challenge to which applied mathematics is particularly well suited.

Appropriate coursework includes analysis and computation, probability and statistics, as well as background courses in one of the sciences or engineering fields in which one intends to do research.

Below is a list of faculty who work closely with this type of research:

In recent years there has been an explosion of interest in the study of nonlinear waves and dynamical systems with analytical results often motivated by the use of computers. The faculty in the Program is actively and intensively involved in this growing field; research areas include integrable and near-integrable systems, conservative and dissipative chaos, as well as numerical computation.

Topics of interest include solitons, dispersive shock waves, integrable systems, cellular automata, pattern formation, qualitative structure and bifurcation theory, onset of chaos and turbulence, analytic dynamics, and transport phenomena. Program courses in this field include dynamical systems, nonlinear wave motion and many advanced seminars.

Suitable background courses are: analysis, computation, partial differential equations, and methods in applied mathematics. Valuable supplemental courses include mechanics and fluid dynamics.

Below is a list of faculty who work closely with this type of research:

- Mark Ablowitz
- David Bortz
- James Curry
- Bob Easton
- Ian Grooms
- Mark Hoefer
- Keith Julien
- Zachary Kilpatrick
- Congming Li
- James Meiss
- Juan Restrepo
- Harvey Segur
- Nancy Rodriguez

Physical Applied Mathematics is a term which generally refers to the study of mathematical problems with direct physical application. This area of research is intrinsically interdisciplinary. In addition to mathematical analysis, it requires a deep understanding of the underlying applications area, and usually requires knowledge and experience in numerical computation.

The Program's affiliated faculty have a wide variety of expertise in various areas of application, e.g. atmospheric and fluid dynamics, theoretical physics, plasma physics, genetic structure, etc. The course requirements of the Program are designed to provide students with a foundation for their study (analysis and computation).

The Program also requires supplemental courses in one of the science or engineering fields which are needed to begin doing thesis research in physical applied mathematics.

See the Dispersive Hydrodynamics Lab page, APPM's own fluid dynamics laboratory.

Below is a list of faculty who work closely with this type of research:

Almost all natural phenomena in the technological, biological, physical and social sciences have random components. Applied probability is the application of probabilistic methods to understand the random elements in real-life problems. Statistics is the science of using data, which typically arises from the randomness inherent in nature, to gain new knowledge.

Research areas of the applied math and affiliated faculty exhibit this interplay between mathematics and real-life problems. Areas of current interest include optimization of stochastic networks; the study of stochastic processes and stochastic differential equations in hydrology and telecommunications; probabilistic models, and statistical tests based on these models, in genetics and RNA sequencing; extreme value theory in estimation of maximal wind speeds.

Appropriate coursework includes analysis, probability and statistics, as well as background courses in one of the sciences or engineering fields in which one intends to do research.

Below is a list of faculty who work closely with this type of research: