## Graduate Courses

**ACMS 50550. Functional Analysis**

(3-0-3)

This one semester course will cover selected topics in Functional Analysis. The theory will be built on Banach and Hilbert spaces and will be applied to selected examples from application including Laplace equations, heat equations, and wave equations. Tools and methods such as fixed point theorems, Dirichlet principle, Semi-group, etc. will be covered in the course.

**ACMS 50730. Mathematical and Computational Modeling in Biology and Physics**

(3-0-3)

Introductory course on applied mathematics and computational modeling with emphasis on modeling of biological problems in terms of differential equations and stochastic dynamical systems. Students will be working in groups on several projects and will present them in class at the end of the course.

**ACMS 60212. Advanced Scientific Computing**

(3-0-3)

This course covers fundamental material necessary for using high performance computing in science and engineering. There is a special emphasis on algorithm development, computer implementation, and the application of these methods to specific problems in science and engineering.

**ACMS 60395. Numerical Linear Algebra**

(3-0-3)

This course will cover numerical linear algebra algorithms which are useful for solving problems in science and engineering. Algorithm design, analysis and computer implementation will be discussed.

**ACMS 60610. Discrete Mathematics**

(3-0-3)

The course will provide an introduction into different subjects of discrete mathematics. Topics include (1) Graph Theory: Trees and graphs, Eulerian and Hamiltonian graphs; tournaments; graph coloring and Ramsey's theorem. Applications to electrical networks. (2) Enumerative Combinatorics: Inclusion-exclusion principle, Generating functions, Catalan numbers, tableaux, linear recurrences and rational generating functions, and Polya theory. (3) Partially Ordered Sets: Distributive lattices, Dilworth's theorem, Zeta polynomials, Eulerian posets. (4) Projective and combinatorial geometries, designs and matroids.

**ACMS 60620. Optimization**

(3-0-3)

Cross-listed with MATH 60620. Convex sets. Caratheodory and Radon's theorems. Helly's Theorem. Facial structure of convex sets. Extreme points. Krein-Milman Theorem. Separation Theorem. Optimality conditions for convex programming problems. Introduction to subdifferential calculus. Chebyshev approximations.

**ACMS 60630. Nonlinear Dynamical Systems**

(3-0-3)

Theory of nonlinear dynamical systems has applications to a wide variety of fields, from physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects--that it brings researchers from many disciplines together with a common language. A dynamical system consists of an abstract phase space or state space, whose coordinates describe the dynamical state at any instant; and a dynamical rule which specifies the immediate future trend of all state variables, given only the present values of those same state variables. Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution. A dynamical system can have discrete or continuous time. The discrete case is defined by a map and the continuous case is defined by a "flow. Nonlinear dynamical systems have been shown to exhibit surprising and complex effects that would never be anticipated by a scientist trained only in linear techniques. Prominent examples of these include bifurcation, chaos, and solitons. This course will be self-contained.

**ACMS 60640. Introduction to Mathematical Biology**

(3-0-3)

Theory of nonlinear dynamical systems has applications to a wide variety of fields, from mathematics, physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects--that it brings researchers from many disciplines together with a common language. Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution. A dynamical system can have discrete or continuous time. The discrete case is defined by a map and the continuous case is defined by a "flow." Nonlinear dynamical systems have been shown to exhibit surprising and complex effects. Prominent examples of these include bifurcation and chaos. Applications to population dynamics, cancer growth and spread of infection will be discussed amongst others. This course will be self-contained.

**ACMS 60650. Applied Partial Differential Equations**

(3-0-3)

Cross-listed with MATH 60650. Laplace equations: Green's identity, fundamental solutions, maximum principles, Green's functions, Perron's methods. Parabolic equations: Heat equations fundamental solutions, maximum principles, finite difference and convergence, Stefan Problems. First order equations: Characteristic methods, Cauchy problems, vanishing of viscosity-viscosity solutions. Real analytic solutions: Cauchy ¬Kowalevski theorem, Holmgren theorem.

**ACMS 60690. Numerical Analysis I**

(3-0-3)

A solid theoretical introduction to numerical analysis. Polynomial interpolation. Least squares and the basic theory of orthogonal functions. Numerical integration in one variable. Numerical linear algebra. Methods to solve systems of nonlinear equations. Numerical solution of ordinary differential equations. Solution of some simple partial differential equations by difference methods.

**ACMS 60790. Numerical Analysis II**

(3-0-3)

A solid introduction to numerical partial differential equations with an emphasis on finite difference methods for time dependent equations and systems of equations. Interpolation. Stability and convergence of solutions in systems of PDE arising in science and engineering. High-order accurate difference methods and Fourier methods. Well posed problems and general solutions for a variety of types of systems of equations with constant coefficients. Stability and convergence. Hyperbolic systems of equations.

**ACMS 60801. Statistical Inference**

(3-0-3)

A first graduate course in the theory of statistics. Basic estimation including unbiased, maximum likelihood and moment estimation; testing hypotheses for standard distributions and contingency tables; confidence intervals and regions; introduction to nonparametric tests and linear regression.

**ACMS 60842. Time Series Analysis**

(3-0-3)

This is an introductory and applied course in time series analysis. Popular time series models and computational techniques for model estimation, diagnostic and forecasting will be discussed. Although the book focuses on financial data sets, other data sets, such as climate data, earthquake data and biological data, will also be included and discussed within the same theoretical framework.

**ACMS 60850. Applied Probability**

(3-0-3)

A thorough introduction to probability theory. Elements of measure and integration theory. Basic setup of probability theory (sample spaces, independence). Random variables, the law of large numbers. Discrete random variables (including random walks); continuous random variables, the basic distributions and sums of random variables. Generating functions, branching processes, basic theory of characteristic functions, central limit theorems. Markov chains. Various stochastic processes, including Brownian motion, queues and applications. Martingales. Other topics as time permits.

**ACMS 60852. Statistical Methods in the Biological and Health Sciences**

(3-0-3)

This course surveys the statistical methods used in biological and biomedical research. Topics include study designs commonly used in health research including case-control, cross-sectional, prospective and retrospective studies; statistical analysis of different types of data arising from biological and health research including categorial data analysis, count data analysis, survival analysis, linear mixed models, lab data, and diagnostic tests. Design and analysis of clinical trials, relative risk assessment, statistical power and sample size calculations will also be covered by the class. Additional topics of introduction to statistical genetics and bioinformatics might also be covered. Students are expected to have basic knowledge in statistics, such as random variables, distributions, estimation, and hypothesis testing.

**ACMS 60875. Statistical Methods in Data Mining**

**(**3-0-3)

Data mining is widely used to discover useful patterns and relationships in data. We will emphasize on large complex datasets such as those in very large databases or web-based mining. The topics will include data visualization, decision trees, association rules, clustering, case based methods, etc.

**ACMS 60880. Experimental Design**

(3-0-3)

A thorough survey of topics in the design of experiments.

**ACMS 60882. Applied Linear Models**

(3-0-3)

A comprehensive treatment of the theory of multiple regression analysis and an introduction to the generalized linear model. Least squares estimators. The Gauss-Markov theorem. Maximum-likelihood estimation of parameters, confidence, and prediction intervals. Residuals analysis: detecting heteroskedasticity and autocorrelation. The Box-Cox transformation. Principal components regression. Models for a binomial response. Models for a categorical response. Analysis of contingency tables using generalized linear models.

**ACMS 60883. Applied Generalized Linear Models**

(3-0-3)

Methods and applications of statistical techniques for analyzing categorical data in two and higher dimensions. Specific topics include tests of association for multiway contingency tables, logistic regression, loglinear models, and regression models for ordinal data. The theory of generalized linear models will be emphasized, as well as estimation methods such as the Newton-Raphson and Fisher scoring. Concepts will be illustrated using examples with real datasets from the social and biological sciences. Calculations will be performed in R.

**ACMS 60885. Applied Bayesian Statistics**

(3-0-3)

A comprehensive treatment of the statistical analysis from a Bayesian perspective. Modern computational tools such as MCMC are emphasized. The principles of Bayesian analysis are described with an emphasis on practical rather than theoretical issues, and illustrated with actual data. A variety models are considered, including linear/nonlinear regression, hierarchical models, generalized linear model, and mixed models. Issues on data collection, model formulation, computation, and model checking and sensitivity analysis are also covered.

**ACMS 60890. Statistical Methods for Financial Risk Management**

(3-0-3)

This course is an introduction to some of the models and statistical methodology used in the practice of managing market risk for portfolios of financial assets. Throughout the course, the emphasis will be on the so-called loss distribution approach, a mapping from the individual asset returns to portfolio losses. Methodology presented will include both univariate and multivariate statistical modeling, Monte Carlo simulation, and statistical inference. This course will make heavy use of the R statistical computing environment.

**ACMS 60901. Mathematical Finance I**

(3-0-3)

An introduction to the contract terms, payoffs, and classical pricing theories for a variety of financial instruments: stocks, bonds, foreign exchange, and their derivatives: forwards, futures, options, swaps. Classical theory of options pricing. Put-call parity. The multiperiod binomial model. Discrete time pricing of American options. Continuous time models for stock prices and interest rates. Explicit derivation of the Black-Scholes option pricing formula. Limitations of the Black-Scholes model.

**ACMS 60902. Mathematical Finance II**

(3-0-3)

An introduction to the more recent advances in the continuous modeling of asset prices and pricing contingent claims. Solution to Heston’s stochastic volatility model. Merton’s jump-diffusion model. Introduction to Lévy processes. Stochastic calculus for jump processes. Hedging in incomplete markets.

**ACMS 60922. Quantitative Methods for Investment**

**(**3-0-3)

An introduction to quantitative methods used by "buy side" organizations for their investment decisions. After introducing different asset classes, financial instruments as well as investment companies, advanced topics include financial econometrics, tests of return/volatility predictability, modern portfolio theory, portfolio optimization, modeling transaction data, and the factor pricing models. Throughout the course, statistical predictive methods and numerical algorithms will be emphasized.

**ACMS 60932. Statistical Inference forFinance**

(3-0-3)

Review of general concepts of statistical estimation theory: bias, mean-squared error, consistency, maximum likelihood estimation, standard error, and interval estimation. Estimating stock price volatility and parameters in binomial pricing models. Overview of financial time series analysis including ARMA, GARCH, and stochastic volatility modeling and estimation. The loss distribution approach to financial risk management and estimating risk measures like value-at-risk and expected shortfall. Using extreme value theory to model tail behavior. Multivariate modeling, dimension reduction, and the use of copulas. Introduction to statistical inference for continuous time asset pricing models.

**ACMS 70780. Categorical Discrete Data**

(3-0-3)

Methods and applications of statistical techniques for analyzing categorical data in two and higher dimensions. Specific topics include tests of association for multiway contingency tables, logistic regression, loglinear models, and regression models for ordinal data. The theory of generalized linear models will be emphasized, as well as estimation methods such as the Newton-Raphson and Fisher scoring. Concepts will be illustrated using examples with real datasets from the social and biological sciences. Calculations will be performed in R. Knowledge of statistical inference and linear regression needed.

**ACMS 70860. Stochastic Analysis**

(3-0-3)

This course is a sequel to ACMS 60850 (Applied Probability). It gives an introduction to stochastic modeling and stochastic differential equations, with application to models from biology and finance. Some topics covered will be: stochastic versus deterministic models; Brownian motion and related processes, e.g., the Ornstein-Uhlenbeck Process; diffusion processes and stochastic differential equations; discrete and continuous Markov chain models with applications; the long run behavior of Markov chains; the Poisson processes with applications; and numerical methods for stochastic processes.

**ACMS 80770. Topics in Applied Mathematics**

(3-0-3)

The subject matter of this course will be an advanced topic in applied mathematics.

**ACMS 80870. Topics in Statistics**

(3-0-3)

The subject matter of this course will be an advanced topic in statistics.