Solve Real World Problems.

From the precise mechanics of the inner workings of living cells, to predicting how fluids and structures interact, Rensselaer mathematicians are continuing an establish tradition of harnessing mathematics to support enginneering and scientific pursuits for the good of our 21st-century world.


Biological sciences have undergone a great expansion, beginning in about the middle of the last century, and introduced a wealth of new areas studying systems ranging in size from the molecular to that of ecosystems. The tools of investigation in many areas of modern biology have grown to be increasingly quantitative and reliant on other sciences, particularly mathematics. The biosciences have thus become a rich source of mathematical problems, inspiring advances in modeling, analysis, and computational methods. Biomathematicians create these advances, as well as aid biological and medical scientists with the quantitative and predictive aspects of their discovery process.

The Rensselaer faculty work in a wide variety of areas in biomathematics, including neuroscience, DNA and RNA modeling, cellular systems and transport, sensory systems, disease modeling and diagnoses, organ imaging, and tissue mechanics. They have been using and advancing all three types of the above mathematical problem areas associated with biological or medical problems. Many of the research projects in this area involve close collaboration with researchers in the relevant biological or medical area.

Operations Research & Data Science

A central focus in operations research and data science is the study of real-world problems with the aim of making better decisions.  Analytical techniques, such as mathematical programming, machine learning, data mining, probability and statistics, and mathematical modeling and simulation, are used to formulate and solve mathematical models and optimization problems.  A common feature of the research is the collection and use of data, ranging from big data (massive collections of data) and network domain data (e.g. street maps) to abstract knowledge and assumptions about how processes work.  Applications of operations research and data science abound in many areas of engineering, business, science and medicine, leading to collaborative interdisciplinary research that is both interesting and challenging.

The research of Rensselaer faculty is driven by many compelling real-world problems.  Current work includes developing effective emergency responses to natural disasters, tracking infectious diseases (such as tuberculosis) so that they may be controlled effectively, and applying advanced data analysis for more reliable and efficient health-care solutions.  Our faculty are also using data analysis and modeling to increase the effectiveness of tracking motions in videos, to improve energy sustainability by increasing wind turbine output, and to analyze medical surfaces.  Methods of nonlinear bi-level programming, optimization, probabilistic and computational differential geometry have been developed for general data analysis that are applicable to wide range of problems in data science.

Geophysical & Environmental Modeling

Understanding how human activities impact the environment and ecosystems involves a web of interconnecting biological, chemical, and physical components. In addition to the disciplinary expertise required for each of these elements, mathematics plays a strong role in effectively analyzing and computing how the key drivers and parameters influence the various metrics of health of the ecosystem and environment. In particular, mathematical techniques and considerations are applied to derive quantitative representations of the complex models, to develop effective computational schemes which can accurately handle the multiple scales and equation types in the models, and to integrate historical and observational data into models which can help predict the effects of changes in policy and/or climate.

Particular aspects of geophysical and environmental modeling with which our faculty have been engaged include the analysis and simulation of turbulent wave interactions in the ocean, effective representations of chemical transport by flow structures in the ocean, the study of interacting vortex dynamics in planetary atmospheres, and spatio-temporal models for plant-herbivore interactions. The recently launched Jefferson Project at Lake George promises to deliver an unprecedented amount of data and information regarding the physical, chemical, and ecological state of the lake, and new collaboration opportunities for our faculty.

Inverse Problems & Imaging

Faculty Researchers:

For inverse problems, the sought after solution is indirectly related to the measurable data, and it is either impossible to obtain the data more directly or it is not desirable to do so. An important aspect of the problem is that the mathematical model of the process that produces the data is fundamentally used in the algorithm that produces the image. Challenges include: modeling of the physical problem, creating new mathematics for analysis of the model, identifying appropriate (often large) and/or rich data sets, and working with scientific computations and visualization aids. Much of the research in this area is connected with Rensselaer's Inverse Problems Center (IPRPI). Participants are from the Schools of Science and Engineering and share common mathematical interests and tools.  Common societal impacts are to human health and safety.

Researchers in the math department and IPRPI address some problems at the basic scientific level: for example, finding properties of the earth’s substructure from seismic measurements or determining material properties of mechanical or biological systems. Other problems focus on direct applications: finding tumors in biological tissue, distinguishing abnormal from normal tissue, identifying fault locations in earthquake active regions, establishing the integrity of dikes, locating objects concealed by vegetation cover and locating mines in the sea environment.  This work can involve a significant amount of mathematical modeling of the application problem; we note in particular that there is significant biomechanical modeling of tissue prior to addressing the inverse and imaging problem.

Electromagnetics, Optics & Plasmas

Electromagnetism is a fundamental branch of physics and a key component of many natural and engineered systems. Since its inception, electromagnetism has been a rich source of fundamental mathematical problems, especially in the theory and numerical computation of solutions to partial differential equations. In the past several decades, much research effort has been focused on phenomena arising from the interaction of electromagnetic fields, such as light, and underlying optical media, such as plasmas, glass fibers, gasses or crystals composed of active atoms, or artificial composites containing materials with different response properties. Analytical, asymptotic, and computational methods developed by applied mathematicians have proven to be important for investigating and understanding these phenomena.

Researchers at Rensselaer are investigating fundamental problems in electromagnetic wave propagation using these three classes of techniques, while at the same time further developing the mathematical tools. They have contributed to fields ranging from exactly solvable nonlinear partial differential equations used in optical pulse propagation, modeling of electromagnetic responses of composite materials and ionized plasmas, to highly accurate computational algorithms. These algorithms are designed to preserve important features of the underlying physical systems which for example enable accurate simulations of the problems and even of their long-time asymptotic behavior. These computational tools can be deployed on some of the largest computers in the world in order to help describe basic questions relating to electromagnetic phenomena.

High Performance Computing & Numerical Analysis

Research in high performance computing and numerical analysis involves the development of algorithms designed to compute solutions of difficult, often nonlinear, mathematical problems spanning a wide range of applications. An essential element of the research concerns the analysis of the algorithms, which seeks to uncover important properties of the methods, such as stability and convergence, so that the algorithms can be employed with a clear understanding of their behavior and accuracy. To obtain well-resolved solutions of problems in complex multidimensional configurations, high-performance computers, such as those available at Rensselaer’s CCI, are often needed, and an exciting aspect of the research involves implementing algorithms effectively and efficiently for such platforms.

Faculty in the Department of Mathematical Sciences are active in this area of research. Recent work has led, for example, to the development of a wide class of stable and efficient partitioned algorithms for fluid-structure interaction problems, high-order accurate variational finite-difference methods, exactly divergence-free central discontinuous Galerkin finite-element methods, energy-conserving or asymptotically preserving high-order methods for kinetic equations, upwind methods for hyperbolic equations in second-order form, high-order conservative methods for Maxwell’s equations, and numerical methods in differential geometry.

Acoustics, Combustion & Fluid-Structure Interactions

Research in acoustics, combustion and fluid-structure interactions share a common theme of mathematical modeling and numerical simulation of problems in fluid and solid mechanics. Applications in these areas often involve wave propagation in complex constitutive materials, such as acoustic wave propagation in non-uniform media, detonation in heterogeneous explosives, or nonlinear deformation in elastic solids, among others. Mathematical models in these areas involve systems of partial differential equations, usually nonlinear and of hyperbolic type, together with matching conditions at interfaces in multi-material applications. For the latter applications, the dynamics of interfaces separating materials is an important feature of the problem, and developing stable and robust numerical tools for their accurate simulation is essential.

Researchers at Rensselaer are tackling interesting modeling issues related to long-range acoustic wave propagation in the ocean interacting with multicomponent sediments at the ocean floor, high-speed compressible flow in multiphase reactive materials, and compressible and incompressible fluids interacting with deformable structures, such as blood flow in veins and arties. A significant aspect of the active research concerns the development of numerical algorithms for the accurate solution of the model equations, and faculty in the math department are among the leaders in developing advanced simulation tools in these exciting areas of research.

High-Dimensional Stochastic Modeling, Analysis & Simulation

A common situation in modern scientific research is that the number of factors affecting quantities of interest (such as the shape of a biomolecule, changes in regional climate, or the biodiversity of the ecosystem in a certain lake) is so large that a model comprising all of them would be analytically and/or computationally intractable. A typical way of modeling such systems is to include a manageable number of explicit dynamical variables, and represent the others by some suitable statistical or stochastic terms. Much of the recent mathematical research in such stochastic systems involves the appropriate formulation of such models when the number of retained degrees of freedom is still large, and the development of analytical methods and computational approaches to characterize how the key quantities of interest are impacted by the combination of the nonlinear dynamics of the explicit variables and their stochastic driving by unresolved variables.

High-dimensional stochastic systems are being developed and explored by Rensselaer faculty in the context of microbiology, geophysics, optics, and epidemiology. Examples of recent and ongoing research includes the impact of network topology and statistics on the synchrony of a neuronal network, social influence dynamics on random networks, hydrodynamic fluctuations in suspensions of swimming microorganisms, propagation of light through a disordered active medium, the interaction of molecular motor proteins in intracellular transport, and turbulent dynamics of waves in the ocean.