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## Concentrations

Applied Inverse Problems

A combination of human genome information and ground-breaking progress of new emerging technology has contributed to a significant amount of innovation in research. This, in conjunction with recent developments in computational capabilities, has allowed the scientific community to address unanswered questions in the life sciences and social sciences. Inverse problems play a key role in these developments by providing a solid foundation for translating complex systems into functional mathematical models. An inverse problem asks for the identification of the variable coefficients of a partial differential equation or integral equation when a certain measurement of a solution of that equation is available. Inverse problems have found numerous applications, including tumor identification, skin cancer detection, hypothermia treatment, identification of impact loads, material properties of functionally graded materials, delamination, detection in composite laminates, identifying implied volatility to predict markets, car windscreen modeling, image processing, detection of cracks in beams and plates, tomography, and many others.

Biomedical Mathematics

Mathematical modeling has become an important tool in understanding human physiology. Our program will aim to train graduate students to acquire the analytical and quantitative skills needed to design, develop, implement, analyze, validate, use, and interpret mathematical models in biomedical areas while also providing an immersion experience in an application area. In this way, students will be positioned to contribute in important ways to biomedical research in a broad community while retaining a solid focus on using appropriate and well-justified mathematical approaches. Academic and industrial research addressing challenges such as understanding the behavior of complex organs, characterizing cell-signaling pathways, identifying biophysical causes of diseases, developing personalized medical treatments, and making better use of clinical data will benefit from an increased number of mathematically trained researchers.

Discrete Mathematics

The study of network-like structures is becoming central to an increasing array of scientific fields, and the graphs and combinatorial structures of discrete mathematics are essential tools for modeling these networks. The goal of the Discrete Mathematics concentration is to train Ph.D. students who will be leaders in the application of discrete mathematics to model a host of real-world problems. The emerging importance of complex networks marks an exciting and essential time to codify discrete mathematical modeling as a discipline. Researchers in fields that have not traditionally been aligned with discrete mathematics, including the biological sciences, medicine, and the social sciences, are finding greater utility in the tools of discrete mathematics. Ph.D. students in the Discrete Mathematics Concentration will have a strong foundation in discrete mathematics along with a mathematical modeling mindset that seeks to apply mathematics for the advancement of research in other fields.

Dynamical Systems and Fluid Dynamics

The fields of discrete and continuous dynamical systems have revolutionized the way in which mathematical models are developed and analyzed in a vast array of applications. In particular, fluids play a prominent role within many of the major economic sectors, such as energy, healthcare, industrials and materials. Understanding fluids, which encompass liquids, gases, and plasma, and their vast array of behaviors is critical to improving and developing a variety of applications. The ability to create and analyze mathematical models that accurately describe fluids therefore plays a crucial part in the overall development of science, technology, and industry. The Dynamical Systems and Fluid Dynamics concentration will provide an opportunity for students to apply fundamental knowledge to nonlinear phenomena and fluids-related problems, with research projects focused on the mathematical challenges inherent in understanding and predicting the behavior of biomedical fluids, aerosol dispersion, oil recovery, climate, natural disasters, and fuel efficiency, among others.

Geometry, Relativity, and Gravitation

Modeling and computation are critical to the study of relativistic phenomena. This includes the study of black-hole collisions, the dynamics of matter in the vicinity of black holes, and the detection and interpretation of gravitational waves. In the Geometry, Relativity, and Gravitation Concentration, students will learn the fundamentals of general relativity, the geometry of manifolds associated with general relativity, advanced computational techniques to solve the general relativistic field equations, and advanced statistical techniques required to analyze relativistic phenomena through gravitational wave data analysis. Students specializing in numerical simulations of black hole space-time and black-holes with matter will develop expertise in analyzing and solving complex nonlinear coupled systems of hyperbolic partial differential equations (including high-resolution shock-capturing techniques) and nonlinear elliptical differential equations. In addition to a sound understanding of General Relativity, students specializing in gravitational wave data analysis will master advanced signal processing and statistical inference techniques required to detect signals buried in noisy data and to make quantitative statements about the parameters of those signals.