The study of applied mathematics and computational sciences deals with the use of mathematical concepts and computational techniques in various fields of science and engineering. It has become an indispensable component of almost every discipline of science, engineering, industry, and technology.

The fundamental concepts and tools of calculus, probability, and linear algebra are essential to modern sciences, from the theories of physics and chemistry that have long been tightly coupled to mathematical ideas, to the collection and analysis of data on complex biological systems. Given the emerging technologies for collecting and sharing large data sets, some familiarity with computational and statistical methods is now also essential for modeling biological and physical systems and interpreting experimental results. MF1 is an introduction to differential and integral calculus that focuses on the concepts necessary for understanding the meaning of differential equations and their solutions. It includes an introduction to a software package for numerical solution of ordinary differential equations.

This course focuses on the concept of energy and its relevance for explaining the behavior of natural systems. The conservation of energy and the transformations of energy from one form to another are crucial to the function of all systems, including familiar mechanical devices, molecular structures and reactions, and living organisms and ecosystems. By integrating perspectives from physics, chemistry, and biology, this course helps students see both the elegant simplicity of universal laws governing all physical systems and the intricate mechanisms at play in the biosphere. Topics include kinetic energy, potential energy, quantization of energy, energy conservation, cosmological and ecological processes.

The fundamental concepts and tools of calculus, probability, and linear algebra are essential to modern sciences, from the theories of physics and chemistry that have long been tightly coupled to mathematical ideas, to the collection and analysis of data on complex biological systems. Given the emerging technologies for collecting and sharing large data sets, some familiarity with computational and statistical methods is now also essential for modeling biological and physical systems and interpreting experimental results. MF2 is an introduction to probability and statistics with an emphasis on concepts relevant for the analysis of complex data sets. It includes an introduction to the fundamental concepts of matrices, eigenvectors, and eigenvalues.

This course focuses on the collective behavior of systems composed of many interacting components. The phenomena of interest range from the simple relaxation of a gas into an equilibrium state of well-defined pressure and temperature to the emergence of ever increasing complexity in living organisms and the biosphere. The course provides an overview of some fundamental differences between traditional disciplines as well as indications of how they complement each other some important contexts. Topics include thermodynamic (statistical mechanical) equilibrium, fundamental concepts of temperature, entropy, free energy, and chemical equilibrium, driven systems, fundamentals of biological and ecological systems.

Integrated Science 3 emphasizes the physics and chemistry concepts of oscillating systems, waves, and fields, and includes applications to human perception. In addition to their fundamental importance to physics and chemistry proper, these ideas are essential for developing an awareness of the principles employed by engineers in the construction of the electrical and optical devices that are ubiquitous in modern civilization. Topics include harmonic oscillators, sound waves, light, and reaction-diffusion patterns.

Integrated Science 4 has more of a chemistry/biology emphasis, with physics brought to bear as needed. It treats topics relevant to understanding organisms, biochemical engineering, and the environment. Topics include evolution, modern biology, ecosystems, hydrology, and climate.

Scientific Writing and Presentations cover some of the areas of scientific communication that a scientist needs to know and to master in order to successfully promote his or her research and career. Students will learn to recognize and construct logical arguments and become familiar with the structure of common publication formats. It will help students to advance their skills in communicating findings in textual, visual and verbal formats for a variety of audiences.

Theory of ordinary differential equations with some of the modern theory of dynamical systems. Topics include differential equations and linear systems of DEs, the general theory of nonlinear systems, the qualitative behavior of two-dimensional and higher-dimensional systems, and applications in various areas.

Introductory course on numerical analysis. Topics include: Development of numerical techniques for accurate, efficient solution of problems in science, engineering, and mathematics through the use of computers. Linear systems, nonlinear equations, optimization, numerical integration, differential equations, simulation of dynamical systems, error analysis.

* Choose two courses from the following four courses:

Topics include heat, wave, and potential equations: scientific context, derivation, techniques of solution, and qualitative properties. Topics to include Fourier series and transforms, eigenvalue problems, maximum principles, Green’s functions, and characteristics.

Second course on probability focusing on stochastic process and stochastic simulations.

Topics include discrete-time and continuous-time Markov chains, Poisson processes and renewal theory, branching processes, generating random numbers and variates, Monte Carlo simulation, statistical analysis of simulation results, variance reduction techniques, etc.

Introduction to techniques used in the construction, analysis, and evaluation of mathematical models. Individual modeling projects in biology, chemistry, economics, engineering, medicine, or physics. Mathematical techniques such as nondimensionalization, perturbation analysis, and special solutions will be introduced to simplify the models and yield insight into the underlying problems.

Geometry of high dimensional data sets. Linear dimension reduction, principal component analysis, kernel methods. Nonlinear dimension reduction, manifold models. Graphs. Random walks on graphs, diffusions, page rank. Clustering, classification and regression in high- dimensions. Sparsity. Computational aspects, randomized algorithms.

Partial differentiation, multiple integrals, and topics in differential and integral vector calculus, including Green’s theorem, the divergence theorem, and Stokes’s theorem.

Systems of linear equations and elementary row operations, Euclidean n-space and subspaces, linear transformations and matrix representations, Gram-Schmidt orthogonalization process, determinants, eigenvectors and eigenvalues; applications.

Introduction to analysis of functions of complex variables. Topics include complex numbers, analytic functions, complex integration, Taylor and Laurent series, theory of residues, argument and maximum principles, conformal mapping.

An introduction to the principles and concepts of abstract algebra. Abstract algebra studies the structure of sets with operations on them. The course studies three basic kinds of “sets with operations on them”, called Groups, Rings, and Fields, with applications to number theory, the theory of equations, and geometry.

Topological structure of the real number system; rigorous development of one-variable calculus including continuous, differentiable, and Riemann integrable functions and the Fundamental Theorem of Calculus; uniform convergence of a sequence of functions.

Courses listed below are recommended electives for the major. Students can also select other courses in different divisions as electives.

Introduction to analysis of functions of real variables. Topics include Lebesgue measure and integration; L^p spaces; absolute continuity; abstract measure theory; Radon-Nikodym Theorem; connection with probability; Fourier series and integrals.

A first course to differential geometry focusing on the study of curves and surfaces in 2- and 3-dimensional Euclidean space using the techniques of differential and integral calculus and linear algebra. Topics include curvature and torsion of curves, Frenet-Serret frames, global properties of closed curves, intrinsic and extrinsic properties of surface, Gaussian curvature and mean curvatures, geodesics, minimal surfaces, and the Gauss-Bonnet theorem.

Elementary introduction to topology. Topics include surfaces, covering spaces, Euler characteristic, fundamental group, homology theory, exact sequences.

An introduction to the concepts, theory, and application of statistical inference, including the structure of statistical problems, probability modeling, data analysis and statistical computing, and linear regression. Inference from the viewpoint of Bayesian statistics, with some discussion of sampling theory methods and comparative inference. Applications to problems in various fields.

First course in probability. Topics include: Probability models, random variables with discrete and continuous distributions. Independence, joint distributions, conditional distributions. Expectations, functions of random variables, central limit theorem.

Graduates will be prepared to begin careers in research and development, teaching, finance, technology and management and other areas, or to pursue graduate study in the United States, China or internationally.