The Department of Mathematics, Applied Mathematics, and Statistics conduct research in the following areas:
Algebra is the mathematics of operations like addition and multiplication of numbers, or composition and convolution of functions. Traditional objects of study in algebra include groups, rings, fields, and vector spaces, but any well-defined system of operations and equations can be subjected to algebraic manipulations and arguments. For example, one can study logic (via the connectives “and,” “or,” “implies,” and negation “not”) or differential equations (by algebraically defining a derivative and asking it to satisfy product and chain rules) in a purely algebraic fashion. Of particular interest in the department is category theory, in which the rules for composition of functions (the existence of identities and associativity) are abstracted to give a flexible system for studying problems in many areas of mathematics including traditional algebra, algebraic topology, or algebraic geometry.
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Bayesian statistics is a field of statistics which relies on the Bayesian interpretation of probability where probability expresses a degree of belief in an unknown quantity. The degree of belief is often based on prior knowledge, such as the result of previous experiments or personal belief, and can be updated after observing new data. The updated belief is mathematically expressed through the posterior distribution and Bayesian inferences are typically based on this. While the elegance of the Bayesian paradigm is undeniable, it has been regarded as a minor area in statistics until the late 20th century, mainly due to the difficulty of the posterior computation. With the development of high computational technology, however, it became one of the most rapidly growing fields in modern statistics.
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Continuum mechanics describe how a composite material or structure, rather than discrete particles, responds to applied forces. Mathematical models formulated using continuum mechanics are typically applied to systems with length scales ranging from a micron (cellular structures) to large (e.g. geophysical, planetary) scales. The equations of continuum mechanics involve conservation laws (of mass, energy, momentum et al) and maps from undeformed to deformed configurations. Continuum mechanics can be used to describe the deformation of elastic bodies, fluid motion, and more complex systems with micro-structure as well as fluid-structure interaction problems. Faculty interests include continuum modeling applied to physical, chemical, biological systems and numerical methods for simulating geophysical turbulence, transport and chemistry, cellular motility, and fluid-structure interaction models.
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Dynamical system theory is the study of processes that evolve over time. This evolution can be deterministic or stochastic discrete or continuous. Deterministic evolution means that all future states of the process are uniquely determined by its current state, stochastic evolution allows random transitions among system’s states. This concept encompasses a broad range of systems, models and applications in physics, chemistry, biology, engineering, life and social Sciences, and more.
In general, a dynamical system involves a state space and a rule, the generator of the system, that expresses how a choice of a current state determines its future states, system’s trajectory or called orbit Typical questions deal with structure and patterns of solutions, They include, sensitivity to initial state and/or other inputs (model parameters, external/driving sources et al); local and global dynamic patterns; parameter space analysis and qualitative changes that go under the name “bifurcation theory” , problems of control and optimization (can a dynamical system be brought to a desired state by external driving, what are efficient or optimal ways to accomplish it). Different concepts, methods and tools are employed in the study of dynamical systems from foundational mathematical tool, to numeric simulations, experiment, data analysis.
The faculty’s research interests cover a wide range of topics, including dynamical systems based modeling of biological systems ranging from networks of nerve cells to schooling fish, infectious diseases (individual and community level), chemistry and physiology.
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In inverse problems, the goal is to estimate unknown parameters from noisy observations of quantities that a related to the unknowns through a mathematical model. A characteristic feature of the inverse problems is their ill-posedness: the problem may have no solution, the solution may not be unique, and small errors in the data may propagate to huge errors in the estimated quantity. Inverse problems research addresses these problems in a systematic manner, either by appropriate regularization of the problem, or by using statistical methods recasting the problem as a question of inference. Inverse problems arise often in imaging or in context where the unknown of interest can be represent as an image, and many techniques developed for imaging science are closely related to those in inverse problems. However, the challenges in imaging problems have their specific nature, making imaging science an active research area of its own.
From pictures taken by smart phones to x-rays doctors use for diagnosis, images become an important part of our life. Imaging science addresses various problems related to images: image formation, image processing, image analysis. Image formation focuses on obtaining high quality images from indirect information such as hardware measurements. It is usually solved from an inverse problem perspective. Image processing refers to removing flaws such as blur, noise, scratch from images for better visualization or analysis. Automatic extraction of features of interest from images is one example of image analysis. Imaging research uses tools from a variety of mathematical and statistical subdisciplines such as variational analysis, partial differential equations, optimization, Bayesian statistics and more.
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Modeling and understanding living systems in quantitative terms is the new frontier in applied mathematics, characterized by the extreme complexity of the object of interest. In life sciences the analytic approach of dividing a system into well-understood simple units is often only a first step, as life itself is an emergent phenomenon based on a complex interplay of different simultaneous and interacting processes. For this reason, mathematical research in life sciences requires new ideas, and meaningful research is often the fruit of close collaboration with experts in biology and medical sciences.
The faculty’s research interests cover a wide range of topics, including complex adaptive biological systems, perinatal development of respiratory rhythms, signal transduction pathways (for gradient sensing chemotaxis and pattern formation), the development of the nervous system: modeling cellular level metabolism; connection of metabolic processes with medical imaging; infectious disease modeling (immunology, ecology of transmission, disease control, ployparasitism, drug resistance and genotyping techniques for monitoring drug-resistant mutants).
Methodologically, the mathematical research in biosciences is very wide, requiring skills from various areas such as differential equations, probability and statistics, dynamical systems and scientific computing.
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Numerical analysis is the study of the properties of algorithms for the computational solution of mathematical problems, including convergence rates and numerical stability when implemented in finite precision arithmetic. Scientific computing is concerned with the details of efficient and stable implementation of numerical schemes. Numerical analysis and scientific computing are very important in the design and implementation of algorithms for the simulation physical systems, in particular when analytic solutions are not available.
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Spatial Statistics is a branch of Statistics where the spatial location of the phenomena under study is of intrinsic interest or where spatial location is an integral part of a statistical model for the phenomena. Spatial data can be geo-statistical, aerial, or point patterns. Geo-statistical data arise when observations are made at fixed or pre-determined locations, and location coordinates are recorded as well as the phenomena of interest. For data where observations that are close tend to be more alike than observations that are further away, it is often useful to model dependence between observations as a function of the distance between them. Areal data arise when data are available on an aggregated level (sometimes for privacy reasons), such as zip-code of county level data. For such data it is often reasonable to assume that neighboring areas are more alike than areas that do not share a boundary. Point pattern data arise when the observed locations of a phenomenon is of main interest.
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Uncertainty quantification (UQ) aims at providing systematic methods to characterize and quantify the uncertainties arising when using complex models to analyze data. Uncertainties may arise from noise in the data, unknown or uncertain parameters, incompleteness of the model, discrepancies between the model and the reality, or from the turbulent or chaotic behavior of the system. Since UQ relies on analytic methods, probability theory, statistics, and numerical analysis, it bridges a wide range of different fields of statistics, mathematics and its applications. The UQ expertise in the department comprises Bayesian statistics, probability, chaotic and turbulent systems, and numerical analysis.