Applied Mathematics and Computational Sciences/Math
The field of applied mathematics and computational sciences deals with the use of mathematical concepts and computational techniques in various fields of science and engineering. It is utilized in almost every discipline of science, engineering, industry, and technology, and has become an indispensable component. The computational science approach aims at understanding and solving problems mainly through the analysis of mathematical models combined with numerical simulations on computer.
Applied mathematics and computational sciences is a highly interdisciplinary field. The numerical simulation techniques are mostly developed and analyzed in the field of numerical analysis in mathematics. The modeling of and applications to specific scientific problems require disciplinary knowledge and expertise. The need of massive amounts of calculations and data processing calls for development in computer and information sciences. This is a time of opportunity as traditional boundaries between disciplines are breaking down, especially around data and computations.
The expertise of mathematics and computational sciences is in dire need, as access to unprecedented amounts of information and computing resources is creating new opportunities, working together in multidisciplinary teams, to actively engage with and to change the world around them. Mathematics and computing sciences are extensively applied in economics, biology, medical science as well as quantitative social science including global health, environmental science and humanities (for example, digital media). New application areas are constantly being discovered and established techniques are being applied in new ways and in emerging fields. Consequently, a wide variety of career opportunities are open to people with mathematical and computational talent and training.
Major Requirements
(Not every course listed is offered every semester, and the course list will be updated periodically. Please refer to the online Course Catalog for Courses offered in 2019-2020.)
Divisional Foundation Courses
Option 1: only applicable to Class of 2022 who have taken INTGSCI 101 & 102
Option 2: only applicable to Class of 2022 who have taken INTGSCI 101
Option 3: Applicable to Class of 2023 and any student who has not taken INTGSCI 101
Interdisciplinary Courses
Disciplinary Courses
Electives
Courses listed in the table below are recommended electives for the major and the course list will be updated periodically. Students can also select other courses in different divisions as electives.
MATH 101 Calculus (4 credits)
This course covers the elements of basic calculus using introductory Newtonian physics both as a source of example problems and as the paradigmatic application of calculus to the description of natural phenomena. Newton’s fundamental laws of motion are framed in mathematical terms involving derivatives, so calculus techniques are essential to the analysis and prediction of natural phenomena. The application of calculus to Newtonian physics also serves as a platform for analogous reasoning about models of social, political, and economic systems.
Prerequisite(s): Familiarity with standard elements of algebra, geometry, and elementary functions (trigonometric, exponential, and logarithmic) at the high school level, or consent of the instructor.
MATH 201 Multivariable Calculus (4 credits)
Main topics of this course include vectors and vector functions, the geometry of higher dimensional Euclidean spaces, partial derivatives, multiple integrals, line integrals, vector fields, Green’s Theorem, Stokes’ Theorem and the Divergence Theorem.
Prerequisite(s): MATH 101
INTGSCI 101: Integrated Science 1 (4 credits)
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.
Prerequisite(s): MATH 101
INTGSCI 102: Integrated Science 2 (4 credits)
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.
Prerequisite(s): INTGSCI 101
BIOL 110 Integrated Science – Biology (4 credits)
Integrated Science-Biology employs five themes that describe properties of life and will be reiterated over again in Integrated Science-Biology: Organization (Structure and Function), Cycling of Energy and Matter, Information (Genetic Variation), Homeostasis (Interactions), and Evolution. These themes will be unified under the organizational principles of the Scientific Methods, formulating hypothesis and testing hypothesis with experiments. Students in Integrated Science-Biology will develop the understanding of key concepts in the context of cross-talks with chemistry and physics. While no previous knowledge is required, some background is advantageous.
Prerequisite(s): None
CHEM 110 Integrated Science – Chemistry (4 credits)
With an integrated approach, this course examines basic concepts and fundamental principles in chemistry based on the laws of physics. The course starts with an introduction to the static structures of atoms, molecules and matter including life itself, followed by an exploration of the dynamical and collective processes during chemical reactions. It explains how atoms, the basic building blocks of matter, interact with each other and construct the world around us, how subatomic electrons modulate the chemical properties of elements, and how the rearrangement of atoms during chemical reactions gives rise to astonishing phenomena in nature. Centered on topics in chemistry, this course not only prepares students for upper-level disciplinary courses, but also helps students develop an interdisciplinary molecular perspective, which allows them to tackle problems in various fields such as condensed matter physics, molecular biology, medicine, materials science and environmental science. While no previous knowledge is required, some background is advantageous. Not open to students who have credits for both INTGSCI 101 and 102 or CHEM 120
Prerequisite(s): None
CHEM 120 Core Concepts in Chemistry: An Environmental Perspective (4 credits)
Current challenges and opportunities in environmental science require a foundational knowledge of core concepts in chemistry. In this course, students will learn core chemical concepts including properties of gases and solutions, thermodynamics, kinetics, equilibrium, electrochemistry and nuclear chemistry as they apply to the understanding of ozone depletion, photochemical smog, climate change, acid deposition, dissolved oxygen, pH, alkalinity and alternative energy sources.
Prerequisite(s): INTGSCI 101, MATH 101
PHYS 121 Integrated Science – Physics (4 credits)
This course is about how to view the world from the perspective of classical mechanics, based on an understanding of the core concepts and theoretical laws. As a science foundation course, it helps students appreciate the elegant simplicity of the universal laws governing the complex systems surrounding us, and it teaches an important approach to identifying, formulating, and solving problems encountered in the physical world. The course begins with the core concepts of classical mechanics ̶ time, space, mass, force, work, energy, momentum ̶ and the physical laws that link them with each other. Students first learn Newton’s laws and the universal law of gravitation as they apply to point mass systems. Subsequently, basic concepts of oscillation and waves, rigid body motion, fluid mechanics, thermodynamics and statistical mechanics are introduced, illustrated with real-life examples (e.g., physics of cooking, biosphere as a thermal engine) to help students integrate different science foundation courses by themselves. While no previous knowledge of physics is required, some background is advantageous. Not open to students who have credits for both INTGSCI 101 and 102.
Prerequisite(s): MATH 101. Not open to students who have credits for both INTGSCI 101 and 102.
MATH 302 Numerical Analysis (4 credits)
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.
Prerequisite(s): MATH 201, MATH 202
MATH 303 ODE and Dynamical Systems (4 credits)
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.
Prerequisite(s): MATH 201, MATH 202
MATH 403 Partial Differential Equations (4 credits)
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.
Prerequisite(s): MATH 303
MATH 404 Stochastic Modeling & Computing (4 credits)
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.
Prerequisite(s): MATH 205
MATH 405 Methods for Data Analysis (4 credits)
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.
Prerequisite(s): MATH 202
MATH 406 Mathematical Modeling (4 credits)
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.
Prerequisite(s): MATH 403, or consent of instructor
MATH 202 Linear Algebra (4 credits)
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.
Prerequisite(s): MATH 201
MATH 205 Probability and Statistics (4 credits)
This course serves as an introduction to probability theory and statistics. It covers basic concepts of the probabilistic description of independent events, some types of probability distributions that frequently arise, some statistical measures used to characterize probability distributions, the central limit theorem, common types of processes and the distributions they generate, the statistics typically employed for testing the explanatory power of a model or hypothesis.
Prerequisite(s): MATH 101
MATH 307 Complex Variable (4 credits)
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.
Prerequisite(s): MATH 201, MATH 202
MATH 308 Real Analysis (4 credits)
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.
Prerequisite(s): MATH 101 or 101H
MATH 401 Abstract Algebra (4 credits)
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.
Prerequisite(s): MATH 202
MATH 301 Advanced Introduction to Probability (4 credits)
Advanced introduction to basic, non-measure theoretic probability. Topics include random variables with discrete and continuous distributions. Independence, joint distributions, conditional distributions, generating functions, Bayes' formula, and Markov chains. Rigorous arguments are presented for the law of large numbers, central limit theorem, and Poisson limit theorems.
Prerequisite(s): MATH 201
MATH 306 Number Theory (4 credits)
Divisibility properties of integers, prime numbers, congruences, quadratic reciprocity, number-theoretic functions, simple continued fractions, rational approximations; contributions of Fermat, Euler, and Gauss.
Prerequisite(s): MATH 201
MATH 408 Differential Geometry (4 credits)
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.
Prerequisite(s): MATH 201, MATH 202
MATH 409 Topology (4 credits)
Elementary introduction to topology. Topics include surfaces, covering spaces, Euler characteristic, fundamental group, homology theory, exact sequences.
Prerequisite(s): MATH 202
MATH 450 Measure and Integration (4 credits)
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.
Prerequisite(s): MATH 308
STATS 301 Statistics (4 credits)
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.
Prerequisite(s): MATH 201 and MATH 205