Research by members of the Department of Applied Mathematics involves the development and use of mathematics—mathematical modeling, analysis, and computation—in order to solve real-world problems. Each faculty member's research is identified with at least one of six primary research areas listed below.

## 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:

Primary Faculty:

Secondary Faculty:

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. Our faculty use techniques in partial and ordinary differential equations, applied analysis, stochastic processes, probability theory, and numerical analysis.

Furthermore, a 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: cell migration, ecology, infectious diseases, neuroscience, and genetics.

Mathematical biology courses include the study of modeling in the biological sciences, computational neuroscience, mathematical computational biology, as well as an advanced seminar series.

See the Mathematical Biology Group page for additional information.

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

Primary Faculty:

Secondary Faculty:

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:

Primary Faculty:

Secondary Faculty: (None currently)

The Applied Mathematics Department's research in nonlinear PDEs and dynamics is intrinsically interdisciplinary, i.e., involves the study of mathematical problems with direct physical application. Physically inspired research problems invariably lead to complex nonlinear phenomena so that, in addition to mathematical analysis, their solution requires a deep understanding of the underlying application area, and often requires knowledge and experience in numerical computation. The faculty in Applied Mathematics specializing in this area of research generally work on problems in nonlinear waves, dynamical systems, partial differential equations and applications. Topics of interest include wave motion, solitons and traveling waves, dispersive shock waves, integrable systems, pattern formation, qualitative structure and bifurcation theory, dynamics on networks, and transport phenomena. Application areas include numerous areas of physics (fluid dynamics, condensed matter, optics, plasma), biology (neural systems, ecology), and sociology (crime, social networks).

Courses in this field include dynamical systems, nonlinear waves, and many advanced seminar courses. Research talks in this field are regularly offered in the Dynamics Seminar and the Nonlinear Waves Seminar.

Suitable background coursework includes analysis, numerical analysis, partial differential equations, mathematical modeling, and methods of applied mathematics.

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

Primary Faculty

Secondary Faculty

Statistics and Data Science are the studies of empirical inquiry. Statistical Science focuses on the development of data analytic methods that are ubiquitous and applicable in all sciences, as well as the theoretical underpinnings supporting such approaches. Data Science focuses on the application of such methods to data problems, including development of implementations and associated studies of computational aspects. Of chief importance is the role of interdisciplinary research to solve scientifically impactful problems, which often motivate the need for new statistical methodology. Faculty members working in Statistics and Data Science cover a breadth of expertise and domain knowledge including Bayesian computation, epidemiology, statistical climatology, statistics for energy science, signal processing and image analysis, networks, machine learning for physical systems, uncertainty quantification as well as the study of collaborative research.

Appropriate coursework includes statistics, probability, computation and mathematical analysis 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:

Primary Faculty:

Secondary:

The Stochastic Processes and Applications (SPA) research group aims to develop and apply probabilistic tools to model, predict, and analyze randomness in real-life phenomena. As such SPA encompasses various areas of both theoretical and applied probability, including Bayesian networks, computational biology, computational probability, discrete probability, mathematical finance, Markov processes, Markov chain Monte Carlo (MCMC) algorithms, optimal stopping, stochastic control, stochastic differential equations, and random graphs.

Students interested in this area who have little to no exposure to probability are encouraged to first take Applied Probability (APPM 3570/STAT 3100).

Appropriate coursework for research in this area includes Data Assimilation in High Dimensional Dynamical Systems (APPM 4/5510), Mathematical Statistics (APPM 5520), Stochastic Analysis for Finance (APPM 4/5530), Time Series (APPM 4/5540), Markov Processes, Queues, and Monte Carlo Simulation (APPM 4/5560), Random Graphs (APPM 4/5565), and Introduction to Stochastic Processes (APPM 6550). These courses may be complemented with special topics courses such as "Measure-theoretic Probability," "Stochastic Simulation," and "Stochastic Differential Equations."

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

Primary Faculty:

Secondary Faculty: