Director, Graduate Studies
Faculty with Research Interests
For more information regarding faculty visit the Department of Applied Mathematics website.
The Department of Applied Mathematics puts mathematics to work solving problems in science, engineering, and society. Applied mathematicians investigate a wide variety of topics, such as how to construct methods for multi-criteria decision making (requiring discrete mathematics and statistics), analyzing the stochastic nature of financial markets (requiring probability/statistics, analysis, and optimization), understanding how liquids flow around solids, and how ions move in biological environments (requiring computational methods and analysis).
The department provides students with office space equipped with computers and full access to the university’s computer and library resources. The department also has a 128-core computer cluster for research purposes.
Research and Program Areas
The research and teaching foci of the Department of Applied Mathematics are primarily in five areas of modern applied mathematics: applied analysis, computational mathematics, discrete applied mathematics, statistics, and stochastics. These areas are briefly described in the following subsections, which also include the faculty with primary and secondary research interests in that area.
The applied analysis group studies mathematical problems arising from physical, chemical, geophysical, biophysical, and materials sciences. These problems are often described by time-dependent partial, ordinary, or integral differential equations, together with sophisticated boundary conditions, interface conditions, and external forcing. Nonlinear dynamical systems offer a geometrical and topological framework for detecting, understanding, and quantifying complex phenomena of these time-dependent differential equations. Partial differential equation theory allows us to correctly formulate well-posed problems and to examine behaviors of solutions, and thus also allows us to set the stage for efficient numerical simulations. Nonlocal equations arise from macroscopic modeling of stochastic dynamical systems with Lévy noise and from modeling long range interactions, and consequently give an understanding of anomalous diffusions.
Faculty with primary interests: Duan, Liu, Lubin, Ong, Tier
Faculty with secondary interests: Bielecki, Cassel, Cialenco, Hickernell, S. Li, X. Li, Nair, Schieber, Weening
Computer simulation is recognized as the third pillar of science, complementing theory and experiment. The computational mathematics research group designs and analyzes numerical algorithms and answers fundamental questions about the underlying physics. We construct and analyze algorithms for approximating functions and integration in high dimensions, and solving systems of polynomial equations. The emphasis is on meshfree methods, maximizing algorithm efficiency, avoiding catastrophic round-off error, overcoming the curse of dimensionality, and advancing adaptive computations to meet error tolerances. We develop accurate mathematical models and efficient numerical methods to investigate dynamics of interfaces. Our goal is to understand the underlying mechanisms that govern the process of pattern formation, i.e., growth and form. Examples include multi-phase flows in complex fluids, and vesicle deformation in bio-related applications such as drug delivery. We establish analytical and computational techniques for extracting effective dynamics from multiscale phenomena that are abundant in geophysical and biophysical systems.
Faculty with primary interests: Hickernell, S. Li, X. Li, Liu
Faculty with secondary interests: Cassel, Duan, Ong, Petrovic, Schieber
Discrete Applied Mathematics
The discrete mathematics group studies theoretical, algorithmic, and computational problems in the fields of graph theory, discrete optimization, combinatorics, and algebraic geometry, with applications in biology, computer science, physics, management sciences, and engineering. Network science, with fundamental concepts from graph theory and computational techniques from discrete optimization, is widely applied to problems arising in transportation/communication, distribution, and security of resources and information. Combinatorial search incorporates graph theory, set systems, and algorithms to tackle information-theoretic questions in topics such as message transmission, data compression, and identification of defective samples in a population. Computational algebra joins tools from algebraic geometry with randomized algorithms from discrete geometry to develop methods to solve systems of polynomial equations arising in statistical inference and mathematical modeling.
Faculty with primary interests: Ellis, Kaul, Pelsmajer, Petrovic, Reingold, Stasi, Weening
The statistics research group works on applied problems in several areas of statistics from the theoretical, methodological, and computational points of view. Design and analysis of experiments with complex structures are used to help scientists gain higher-quality information from their lab work. Monte Carlo methods inform decisions depending on an unknown future by generating and analyzing a myriad of plausible scenarios. Algebraic statistics integrates algebra, geometry, and combinatorics into statistical modeling to provide better-fitting models for non-traditional data. Statistical network modeling and uncertainty quantification allow us to detect when certain data structures are more commonly observed than by chance. Bayesian statistics uses prior beliefs to inform statistical inference. We closely collaborate with scientists and engineers from many different disciplines such as mechanical, manufacturing, civil, and transportation engineering; social sciences; biology; neuroscience; business; and management.
Faculty with primary interests: Cialenco, Hickernell, Kang, Petrovic, Stasi
Faculty with secondary interests: Adler, Bielecki, Ellis, Gong
The research outcome of the stochastics group provides modeling tools for analysis, control, and numerical study of various stochastic systems that evolve in time and space, and are subject to randomness. Our study of structured dependence between stochastic processes helps to construct models of multivariate random dynamical systems with prescribed global structural features and prescribed marginal structural features. Random sequence comparison helps scientists to identify regions of similarity in the sequences of DNA, RNA, and proteins, or between strings in a natural language. Stochastic partial differential equations and stochastic dynamical systems serve as modeling tools for complex phenomena such as turbulent flows, climate change, and behavior of financial markets. Our research in the area of mathematical finance provides quantitative models of financial securities that allow pricing, hedging, and mitigating the risk of complex financial products.
Faculty with primary interests: Adler, Bielecki, Cialenco, Duan, Gong, Hickernell, Tier
Faculty with secondary interests: X. Li
Minimum Cumulative Undergraduate GPA
- Master's/Master of Science: 3.0/4.0
- Ph.D.: 3.5/4.0
Minimum GRE Scores
- Master's/Master of Science: 304 (quantitative + verbal), 2.5 (analytical writing)
- Ph.D.: 304 (quantitative + verbal), 3.0 (analytical writing)
Minimum TOEFL Scores
80/550 (internet-based/paper-based test scores)The applicants are also required to submit:
at least two letters of recommendation
a professional statement written in essay format (up to two pages long) that includes discussion of the reasons for pursuing graduate study, the applicant's academic background, relevant professional experience or related accomplishments to date, and applicant’s career goals
a curriculum vitae
Admission to the Master of Science and the Ph.D. program normally requires a bachelor’s degree in mathematics or applied mathematics. Candidates whose degree is in another field (for example, computer science, physics, or engineering) and whose background in mathematics is strong are also eligible for admission and are encouraged to apply. Candidates in the Ph.D. program must also have demonstrated the potential for conducting original research in applied mathematics. Students must remove deficiencies in essential undergraduate courses that are prerequisites for the degree program, in addition to fulfilling all other degree requirements.
For admission requirements of the individual degree, see the corresponding program’s graduate bulletin page.
Meeting the minimum or typical GPA test score requirements does not guarantee admission. GPA and test scores are just two of several important factors considered for admission to the program.
The director of graduate studies serves as temporary academic adviser for newly admitted graduate students in the Master of Science and the Ph.D. programs until an appropriate faculty member is selected as the adviser. Students are responsible for following all departmental procedures, as well as the general requirements of the Graduate College.
- Master of Applied Mathematics
- Master of Science in Applied Mathematics
- Doctor of Philosophy in Applied Mathematics
Joint Degree Programs
Measure Theory and Lebesgue Integration; Metric Spaces and Contraction Mapping Theorem, Normed Spaces; Banach Spaces; Hilbert Spaces.
Bounded Linear Operators on a Hilbert Space; Spectrum of Bounded Linear Operators; Fourier Series; Linear Differential Operators and Green's Functions; Distributions and the Fourier Transform; Differential Calculus and Variational Methods.
Basic model equations describing wave propagation, diffusion and potential functions; characteristics, Fourier transform, Green function, and eigenfunction expansions; elementary theory of partial differential equations; Sobolev spaces; linear elliptic equations; energy methods; semigroup methods; applications to partial differential equations from engineering and science.
Basic theory of systems of ordinary differential equations; equilibrium solutions, linearization and stability; phase portraits analysis; stable unstable and center manifolds; periodic orbits, homoclinic and heteroclinic orbits; bifurcations and chaos; nonautonomous dynamics; and numerical simulation of nonlinear dynamics.
Analytic functions, contour integration, singularities, series, conformal mapping, analytic continuation, multivalued functions.
The course provides a systematic approach to modeling applications from areas such as physics and chemistry, engineering, biology, and business (operations research). The mathematical models lead to discrete or continuous processes that may be deterministic or stochastic. Dimensional analysis and scaling are introduced to prepare a model for study. Analytic and computational tools from a broad range of applied mathematics will be used to obtain information about the models. The mathematical results will be compared to physical data to assess the usefulness of the models. Credit may not be granted for both MATH 486 and MATH 522.
The goal of the course is for students to learn how to use applied mathematics methods and skills to analyze real-world problems and to communicate their results in a non-academic setting. Students will work in groups of 2 or 3 to study and analyze problems and then provide useful information to a potential client. The time distribution is flexible and includes discussions of problems, presentation of needed background material and the required reports, and presentations by the teams. Several small projects will be examined and reported on.
Concepts and methods of gathering, describing and analyzing data including statistical reasoning, basic probability, sampling, hypothesis testing, confidence intervals, correlation, regression, forecasting, and nonparametric statistics. No knowledge of calculus is assumed. This course is useful for graduate students in education or the social sciences. This course does not count for graduation in any mathematics program. Credit given only for one of the following: MATH 425, MATH 476, or MATH 525.
The purpose of this course is to introduce students to the theory and application of supervised and reinforcement learning to big data problems in finance. This course emphasizes the various mathematical frameworks for applying machine learning in quantitative finance, such as quantitative risk modeling with kernel learning and optimal investment with reinforcement learning. Neural networks are used to implement many of these mathematical frameworks in finance using real market data.
Basics of computation with systems of polynomial equations, ideals in polynomial rings; solving systems of equations by Groebner bases; introduction to elimination theory; algebraic varieties in affine n-space; Zariski topology; dimension, degree, their computation and theoretical consequences.
Matrix algebra, vector spaces, norms, inner products and orthogonality, determinants, linear transformations, eigenvalues and eigenvectors, Cayley-Hamilton theorem, matrix factorizations (LU, QR, SVD).
Introduction to both theoretical and algorithmic aspects of linear optimization: geometry of linear programs, simplex method, anticycling, duality theory and dual simplex method, sensitivity analysis, large scale optimization via Dantzig-Wolfe decomposition and Benders decomposition, interior point methods, network flow problems, integer programming. Credit may not be given for both MATH 435 and MATH 535.
Random events and variables, probability distributions, sequences of random variables, limit theorems, conditional expectations, and martingales.
This is an introductory course in stochastic processes. Its purpose is to introduce students into a range of stochastic processes, which are used as modeling tools in diverse field of applications, especially in the business applications. The course introduces the most fundamental ideas in the area of modeling and analysis of real World phenomena in terms of stochastic processes. The course covers different classes of Markov processes: discrete and continuous-time Markov chains, Brownian motion, and diffusion processes. It also presents some aspects of stochastic calculus with emphasis on the application to financial modeling and financial engineering.
This course will introduce the student to modern finite dimensional stochastic analysis and its applications. The topics will include: a) an overview of modern theory of stochastic processes, with focus on semimartingales and their characteristics, b) stochastic calculus for semimartingales, including Ito formula and stochastic integration with respect to semimartingales, c) stochastic differential equations (SDE's) driven by semimartingales, with focus on stochastic SDE's driven by Levy processes, d) absolutely continuous changes of measures for semimartingales, e) some selected applications.
This course is about modeling, analysis, simulation and prediction of dynamical behavior of complex systems under random influences. The mathematical models for such systems are in the form of stochastic differential equations. It is especially appropriate for graduate students who would like to use stochastic methods in their research, or to learn these methods for long term career development. Topics include white noise and colored noise, stochastic differential equations, random dynamical systems, numerical simulation, and applications to scientific, engineering and other areas.
This course introduces various methods for understanding solutions and dynamical behaviors of stochastic partial differential equations arising from mathematical modeling in science, engineering, and other areas. It is designed for graduate students who would like to use stochastic methods in their research or to learn such methods for long term career development. Topics include the following: Random variables; Brownian motion and stochastic calculus in Hilbert spaces; Stochastic heat equation; Stochastic wave equation; Analytical and approximation techniques; Stochastic numerical simulations via Matlab; and applications to science, engineering, and other areas.
This course introduces the basic time series analysis and forecasting methods. Topics include stationary processes, ARMA models, spectral analysis, model and forecasting using ARMA models, nonstationary and seasonal time series models, multivariate time series, state-space models, and forecasting techniques.
This is an introductory course in mathematical finance. Technical difficulty of the subject is kept at a minimum by considering a discrete time framework. Nevertheless, the major ideas and concepts underlying modern mathematical finance and financial engineering are explained and illustrated.
Topological spaces, continuous mappings and homeomorphisms, metric spaces and metrizability, connectedness and compactness, homotopy theory.
A graduate-level introduction to modern graph theory through existential and algorithmic problems, and the corresponding structural and extremal results from matchings, connectivity, planarity, coloring, Turán-type problems, and Ramsey theory. Proof techniques based on induction, extremal choices, and probabilistic methods will be emphasized with a view towards building an expertise in working in discrete applied mathematics.
A graduate-level course that introduces students in applied mathematics, computer science, natural sciences, and engineering, to the application of modern tools and techniques from various fields of mathematics to existential and algorithmic problems arising in discrete applied math. Probabilistic methods, entropy, linear algebra methods, Combinatorial Nullstellensatz, and Markov chain Monte Carlo, are applied to fundamental problems like Ramsey-type problems, intersecting families of sets, extremal problems on graphs and hypergraphs, optimization on discrete structures, sampling and counting discrete objects, etc.
Development of the calculus of tensors with applications to differential geometry and the formulation of the fundamental equations in various fields.
Point-set theory, compactness, completeness, connectedness, total boundedness, density, category, uniform continuity and convergence, Stone-Weierstrass theorem, fixed point theorems.
Graduate level introduction to probabilistic methods, including linearity of expectation, the deletion method, the second moment method and the Lovasz Local Lemma. Many examples from classical results and recent research in combinatorics will be included throughout, including from Ramsey Theory, random graphs, coding theory and number theory.
Algebraic structures are present in a broad variety of statistical contexts, involving both parametric and non-parametric statistical models for continuous and discrete random variables. A broad range of algebraic tools is used to better understand model structure, improve statistical inference, and explore new classes of models. The course offers an overview of fundamental theoretical constructions relevant to some of the more popular recent applications in the field: exact conditional test for discrete data, likelihood geometry, parameter identifiability and model selection, network models with applications to social sciences and neuroscience, and phylogenetics and tree-based evolutionary models in biology.
Theory of sampling distributions; principles of data reduction; interval and point estimation, sufficient statistics, order statistics, hypothesis testing, correlation and linear regression; introduction to linear models. Credit given only for one of MATH 425, MATH 476, MATH 525, or MATH 563.
This course introduces the basic statistical regression model and design of experiments concepts. Topics include simple linear regression, multiple linear regression, least square estimates of parameters; hypothesis testing and confidence intervals in linear regression, testing of models, data analysis and appropriateness of models, generalized linear models, design and analysis of single-factor experiments.
In addition to the theoretical constructs in financial mathematics, there are also a range of computational/simulation techniques that allow for the numerical evaluation of a wide range of financial securities. This course will introduce the student to some such simulation techniques, known as Monte Carlo methods, with focus on applications in financial risk management. Monte Carlo and Quasi Monte Carlo techniques are computational sampling methods which track the behavior of the underlying securities in an option or portfolio and determine the derivative's value by taking the expected value of the discounted payoffs at maturity. Recent developments with parallel programming techniques and computer clusters have made these methods widespread in the finance industry.
Random vectors, sample geometry and random sampling, generalized variance, multivariate normal and Wishart distributions, estimation of mean vector, confidence region, Hotelling's T-square, covariance, principal components, factor analysis, discrimination, clustering.
Various type of designs for laboratory and computer experiments, including fractional factorial designs, optimal designs and space filling designs.
Categorical data analysis, contingency tables, log-linear models, nonparametric methods, sampling techniques.
The wealth of observational and experimental data available provides great opportunities for us to learn more about our world. This course teaches modern statistical methods for learning from data, such as regression, classification, kernel methods, and support vector machines.
This course surveys industrial and scientific applications of data analytics with case studies including exploration of ethical issues. Students will learn how to prepare data for analysis, perform exploratory data analysis, and develop meaningful data visualizations. They will work with a variety of real world data sets and learn how to prepare data sets for analysis by cleaning and reformatting. Students will also learn to apply a variety of different data exploration techniques including summary statistics and visualization methods.
In this project-oriented course, students will work in small groups to solve real-world data analysis problems and communicate their results. Innovation and clarity of presentation will be key elements of evaluation. Students will have an option to do this as an independent data analytics internship with an industry partner.
Many mathematical problems cannot be solved analytically or by hand in a reasonable amount of time; so, turn to mathematical software to solve these problems. Popular examples of general-purpose mathematical software include Mathematica, MATLAB, the NAG Library, and R. Researchers often find themselves writing mathematical software to demonstrate their new ideas or using mathematical software written by others to solve their applications. This course covers the ingredients that go into producing mathematical software that is efficient, robust, and trustworthy. Students will write their own packages or parts of packages to practice the principles of reliable mathematical software.
Rigorous introduction to the theory of Bayesian statistical inference and data analysis including prior and posterior distributions, Bayesian estimation and testing, Bayesian computation theories and methods, and implementation of Bayesian computation methods using popular statistical software.
Fundamentals of matrix theory; least squares problems; computer arithmetic, conditioning and stability; direct and iterative methods for linear systems; eigenvalue problems. Credit may not be granted for both Math 577 and Math 477. Prerequisite: An undergraduate numerical course, such as MATH 350 or instructor permission.
Polynomial interpolation; numerical solution of initial value problems for ordinary differential equations by single and multi-step methods, Runge-Kutta, Predictor-Corrector; numerical solution of boundary value problems for ordinary differential equations by shooting method, finite differences and spectral methods. Credit may not be granted for both MATH 578 and MATH 478. Prerequisite: An undergraduate numerical course, such as MATH350 or instructor's consent.
This course is concerned with a branch of complexity theory. It studies the intrinsic complexity of numerical problems, that is, the minimum effort required for the approximate solution of a given problem up to a given error. Based on a precise theoretical foundation, lower bounds are established, i.e. bounds that hold for all algorithms. We also study the optimality of known algorithms, and describe ways to develop new algorithms if the known ones are not optimal.
Various elements, error estimates, discontinuous Galerkin methods, methods for solving system of linear equations including multigrid. Applications.
This course is a continuation of Math 485/548. It introduces the student to modern continuous time mathematical finance. The major objective of the course is to present main mathematical methodologies and models underlying the area of financial engineering, and, in particular, those that provide a formal analytical basis for valuation and hedging of financial securities.
This course is concerned with the design and implementation of optimal trading strategies. In particular, it covers the mean-variance portfolio selection problem, utility maximization, statistical arbitrage and pairs trading, market making, and the optimal liquidation problem. The analysis includes such important features as: the construction and usage of predictive signals, finding a trade-off between risk and return, accounting for transaction costs, and market impact. The available mathematical tools and models are presented in each case, and they include: methods for solving constrained optimization problems, stochastic control and the Hamilton-Jacobi-Bellman equation, hidden Markov models, arrival processes, PCA, and time-series analysis. An important part of the course is the implementation of trading algorithms on a computer, using both simulated and real market data. (3-0-3)
The course covers basics of the modern interest rate modeling and fixed income asset pricing. The main goal is to develop a practical understanding of the core methods and approaches used in practice to model interest rates and to price and hedge interest rate contingent securities. The emphasis of the course is practical rather than purely theoretical. A fundamental objective of the course is to enable the students to gain a hands-on familiarity with and understanding of the modern approaches used in practice to model interest rate markets.
This is an advanced course in the theory and practice of credit risk and credit derivatives. Students will get acquainted with structural and reduced form approaches to mathematical modeling of credit risk. Various aspects of valuation and hedging of defaultable claims will be presented. In addition, valuation and hedging of vanilla credit derivatives, such as credit default swaps, as well as vanilla credit basket derivatives, such as collateralized credit obligations, will be discussed.
This is an advanced course on quantitative risk management. The major concepts and ideas from the modern risk management will be explained and illustrated. The course builds upon general theory of risk measures and performance measures and addresses the current regulatory requirements for market participants.
This course introduces numerical methods, especially the finite difference method for solving different types of partial differential equations. The main numerical issues such as convergence and stability will be discussed. It also includes introduction to the finite volume method, finite element method and spectral method. Prerequisite: An undergraduate numerical course such as MATH 350 and MATH 489 or consent of instructor.
Fundamentals of multivariate meshfree radial basis function and moving least squares methods; applications to multivariate interpolation and least squares approximation problems; applications to the numerical solution of partial differential equations; implementation in Matlab.
Prerequisite: Instructor permission required.
The course is for students in the Master of Applied Mathematics program who have an approved summer internship at an outside organization. This course can be used in place of Math 523 subject to the approval of the director of the program.
Current research topics presented in the department colloquia and seminars.
The course is part of the capstone experience for students in the Master of Applied Mathematics program. Students will work in groups of 2 or 3 to study and analyze a real-world problem.
This course provides the foundation of how to teach mathematics in the context of introductory undergraduate courses. The course is designed to encourage participation and cooperation among the graduate students, to help them prepare for a career in academia, and to help convey the many components of effective teaching.
Course content is variable and reflects current research in combinatorics.
Course content is variable and reflects current research in graph theory.
Course content is variable and reflects current research in computational mathematics.
Course content is variable and reflects current research in applied analysis.
Course content is variable and reflects current research in stochastic.
Successful project management links the basic metrics of schedule adherence, budget adherence, and project quality. But, it also includes the 'people components' of customer satisfaction and effective management of people whether it is leading a project team or successfully building relationships with co-workers. Through course lectures, assigned readings, and case studies, the basic components of leading, defining, planning, organizing, controlling, and closing a project will be discussed. Such topics include project definition, team building, budgeting, scheduling, risk management and control, evaluation, and project closeout.
This course presents strategies for scientists to use when engaging a variety of audiences with scientific information. Students will learn to communicate their knowledge through correspondence, formal reports, and presentations. Students will practice document preparation using report appropriate formatting, style, and graphics. Written assignments, discussion questions, and communication exercises will provide students with a better understanding of the relationship between scientists and their audiences whether in the workplace, laboratory, etc.
Lectures and discussion relating to ethics relating to the health professions. This course exposes students to current ethical and social issues surrounding health care, including health care provider and patient interactions and institutional considerations using case study examples.