Mathematical and Statistical Sciences
Requirements for Admission
To begin graduate work toward the MS or PhD degrees in Applied Mathematics, a student should have at least the following preparation: 30 semester hours of mathematics with each course grade at B- or better and an overall GPA of 3.0 or better, at least 24 of which are upper-division courses. These courses should include one semester of advanced calculus or introduction to analysis, one semester of linear algebra and one semester of either differential equations, abstract algebra, discrete mathematics or probability.
To begin graduate work toward the MS in Statistics, a student should have at least the following preparation: a baccalaureate (not necessarily in mathematics or statistics) from an accredited college or university with a grade point average (GPA) of 3.0 or above. Students must have taken three semesters of calculus (through multivariate calculus), linear algebra, and a calculus-based statistics course that covers basic probability and statistical distributions. Admitted students are generally expected to have completed several additional upper-division mathematics courses on top of the minimum requirements, though students from non-mathematics backgrounds who meet minimum requirements and have exceptional track records will be considered on a case-by-case basis.
Students who do not have all the prerequisites may be admitted with prerequisite deficiencies with the understanding that those deficiencies must be removed after admission. Students who have a cumulative undergraduate GPA that is less than 3.0 may be eligible for provisional admission to the master’s program (see also the Graduate student admission requirements).
Application Deadlines
Applications to the MS or PhD programs should be submitted by the following target dates to be guaranteed full consideration. International students should submit their applications at least one month prior to these target dates.
| Target Dates for PhD Program | Target Dates for MS Program |
|---|---|
| January 15: fall semester | July 15: fall semester |
| No summer admissions for the PhD program | March 1: summer semester |
| No spring admissions for the PhD program | Nov 1: spring semester |
Applications received after the target dates may still be considered for admission, depending on space availability.
Applied Statistics Graduate Certificate
The minimum admission requirements for students applying for the Graduate Certificate in Applied Statistics are:
- A bachelor's degree (not necessarily in mathematics or statistics) from an accredited college or university
- A grade point average (GPA) of 3.0 or above during their bachelor's degree.
- Students must have taken three semesters of calculus (through multivariate calculus), linear algebra, and a calculus-based statistics course that covers basic probability and statistical distributions.
Subject to approval by the Director of the Statistical Programs and the Graduate Committee, students with prerequisite deficiencies may be admitted with the understanding that those deficiencies must be removed after admission. Courses taken to fulfill admission deficiencies may not be counted toward the certificate.
Faculty
Professors:
Steffen Borgwardt, PhD, Technische Universität München
Troy Butler, PhD, Colorado State University
Stephen Hartke, PhD, Rutgers University
Julien Langou, PhD, Institute National Polytechnique of Toulouse, France
Florian Pfender, PhD, Emory University
Stephanie Santorico, PhD, North Carolina State University
Associate Professors:
Erin Austin, PhD, University of Minnesota
Joshua French, PhD, Colorado State University
Diana White, PhD, University of Nebraska
Assistant Professors:
Yaning Liu, PhD, Florida State University
Farhad Pourkamali Anaraki, PhD, University of Colorado Boulder
Emily Speakman, PhD, University of Michigan
Professor, Teaching Track:
Adam Spiegler, PhD, University of Arizona
Assistant Professor, Teaching Track:
Dmitriy Ostrovskiy, PhD, State University of New York at Stony Brook
Senior Instructors:
Daniel Klie, MS, University of Colorado Denver
Gary Olson, MS, University of Colorado Denver
Robert Rostermundt, PhD, University of Colorado Denver
Pamela Whitten, MA, University of Colorado Boulder
International College of Beijing Faculty:
Thomas Dunn, Instructor, PhD, North Dakota State University
Joseph Quarcoo, Instructor, PhD, University of South Florida
Research Faculty:
Aime Fournier, PhD, Yale University
Emeritus Faculty:
Stephen Billups, Associate Professor Emeritus, PhD, University of Wisconsin-Madison
William Briggs, Professor Emeritus, PhD, Harvard University
William E. Cherowitzo, Professor Emeritus, PhD, Columbia University
Kathryn L. Fraughnaugh, Professor Emeritus, PhD, University of Houston
Michael S. Jacobson, Professor Emeritus, PhD, Emory University
Andrew Knyazev, Professor Emeritus, PhD, Russian Academy of Sciences
Lance Lana, Instructor Emeritus, MS, University of Colorado Denver
Weldon A. Lodwick, Professor Emeritus, PhD, Oregon State University
J. Richard Lundgren, Professor Emeritus, PhD, Ohio State University
Stanley E. Payne, Professor Emeritus, PhD, Florida State University
Jan Mandel, Professor Emeritus, PhD (equivalent), Charles University, Czechoslovakia
Burton Simon, Associate Professor Emeritus,PhD, University of Michigan
Mathematics (MATH) Courses
MATH 5010 - History of Mathematics (3 Credits)
A history of the development of mathematical techniques and ideas from early civilization to the present, including the inter-relationships of mathematics and sciences. Note: this course assumes that students have mathematical knowledge equivalent to MATH 1401. Prereq: Graduate standing. Not open to students who have had MATH 4010. No credit for applied math graduate students. Cross-listed with MATH 4010. Max hours: 3 Credits.
Grading Basis: Letter Grade
Restriction: Restricted to Graduate and Graduate Non-Degree Majors
MATH 5017 - Topics in Mathematics for Teachers (0.3-50 Credits)
Topics vary from semester to semester. Designed for professional mathematics teachers. Note: This course will not count toward a degree in applied mathematics. Prereq: permission of instructor. Repeatable. Max Hours: 50 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 50.
MATH 5027 - Topics in Applied Mathematics (3 Credits)
Selected topics in mathematical problems arising from various applied fields such as mechanics, electromagnetic theory, economics and biological sciences. Prereq: Graduate stadning in Applied Mathematics, Statistics, or Dual MATH MA/ECON MA, or permission of the instructor. Repeatable. Term offered: fall, spring. Max hours: 18 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 18.
Restriction: Graduate standing in Applied Mathematics, Statistics, or Dual MATH MA/ECON MA.
Typically Offered: Fall, Spring.
MATH 5070 - Applied Analysis (3 Credits)
Metric spaces, uniform convergence, elements of Banach spaces, elements of functions of complex variable. Problem solving and independent proof writing. Review of selected advanced topics in analysis for the PhD preliminary examination. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of two semesters of undergraduate real analysis (e.g., MATH 4310 and MATH 4320). Term offered: spring. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Spring.
MATH 5110 - Theory of Numbers (3 Credits)
Every other year. Topics include divisibility, prime numbers, congruences, number theoretic functions, quadratic reciprocity, and special diophantine equations, with applications in engineering. Prereq: Graduate Standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of an undergraduate-level course in mathematical proof (e.g. MATH 3000). Cross-listed with MATH 4110. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 5135 - Functions of a Complex Variable (3 Credits)
Infrequent. The complex plane, infinite series and products, elementary special functions, Cauchy-Riemann equations, conformal mapping, complex integration, Cauchy integral theory, and residue theory. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have mathematical knowledge equivalent to two semesters of undergraduate-level real analysis (e.g. MATH 4310, MATH 4320) or to a semester of graduate-level real analysis (e.g., MATH 5070). Max Hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 5310 - Probability (3 Credits)
The course covers axioms of probability, combinatorial probability, conditional probability, random variables (discrete, continuous, and multivariate), expected value (mean, moments, variance, covariance, etc.), limit theorems (laws of large numbers, Central Limit Theorem), Poisson processes and Markov chains. Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS. Note: This course assumes that students have the equivalent of differential and integral calculus (e.g., MATH 2411). Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 5320 - Statistical Inference (3 Credits)
Methods for constructing sampling distributions; sufficient, minimal sufficient, ancillary and complete statistics; methods for constructing and evaluating point estimators; estimator optimality; methods for constructing and evaluating hypothesis tests; methods for constructing and evaluating confidence interval estimators; asymptotic properties of estimators. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 5337 - Intro to Statistical and Machine Learning (3 Credits)
This is an applied, hands-on course in statistical and machine learning. This course will introduce students to the general framework, best practices, model training, and assessment for machine learning methods from the viewpoint of statistics. Both supervised and unsupervised methods are covered including penalized regression, knearest neighbors, clustering, and neural networks. Additional machine learning topics such as random forests and support vector machines are included as time permits. Ultimately, students will learn how and why to use a particular method, how to validate and explain the results, and apply the methods to real data. Note: It is recommended that students are comfortable learning a statistical computing language such as R or Python as these will be taught alongside the course material. Students with minimal programming experience should expect to spend more time learning the programming language throughout the course. Prereq: MATH 4387 or MATH 5387 or MATH 4830 or MATH 5830 or BIOL 3763 with a C- or higher. Students who have completed a different statistics course that contains regression and computing may seek instructor permission to enroll. Cross-listed with MATH 4337. Max hours: 3 Credits.
Grading Basis: Letter Grade
MATH 5387 - Applied Regression Analysis (3 Credits)
Topics include simple and multiple linear regression, model diagnostics and remediation, and model selection. Emphasis is on practical aspects and applications of linear models to the analysis of data in business, engineering and behavioral, biological and physical sciences. Prereq: Graduate standing in Applied Mathematics or Statistics or instructor permission. AMEN-MS/PHD/STAT-MS. Note: This course assumes that students have the equivalent of an undergraduate-level course in statistics (e.g., MATH 4820). No co-credit with MATH 4830/5830. Cross-listed with MATH 4387. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 5388 - Machine Learning Methods (3 Credits)
Regression, neural networks, clustering, support vector machines, random forests, and other prediction/classification techniques will be used to solve supervised and unsupervised learning problems. This course will connect each topic with the underlying mathematical foundation such as optimization methods and statistical inference. A key focus is deriving the methods and their properties to guide proper application. Students will learn how to apply methods using standard libraries from Python, R, or Matlab. Prereq: Graduate standing in Applied Mathematics or Statistics or in one of the 4+1 BS-MS programs. Cross-listed with MATH 4388. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics or in one of the 4+1 BS-MS programs. (AMEN-MS, AMEN-PHD, STAT-MS, MATH BS-BMA).
MATH 5390 - Game Theory (3 Credits)
Zero-sum and non-zero-sum games; Nash equilibrium and the principle of indifference; Shapley value and other concepts of fair division; Evolutionary game theory, ESS, and evolutionary population dynamics. Applications in economics, business, and biology. Note: this course assumes that students have the equivalent of MATH 2421, 3191 and 3800 or 4810. Prereq: Graduate standing in Applied Mathematics. Cross-listed with MATH 4390. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 5490 - Network Flows (3 Credits)
Every other year. Begins with the classical min-cost flow problem, defined on an ordinary network. Other problems, such as shortest path, are also shown in this class. Both theory and algorithms are presented. Extensions include generalized networks, nonlinear costs, fixed charges, multi-commodity flows and additional applications, such as in communications networks. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Term offered: spring of even years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Spring.
MATH 5593 - Linear Programming (3 Credits)
A linear program is an optimization problem that seeks to minimize or maximize a linear function subject to a system of linear in equalities and equations. This course begins with examples of linear programs and variations in their representations. Basic theoretical foundations covered include polyhedra, convexity, linear inequalities and duality. Two classes of solution algorithms are given: simplex methods and interior point methods. The primary emphasis of this course is on mathematical foundations, and applications are used to illustrate the main results. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of a course in linear algebra (e.g., MATH 3191). Term offered: fall. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Fall.
MATH 5660 - Numerical Analysis I (3 Credits)
A first semester course in numerical methods and analysis fundamental to many algorithms encountered in scientific computing, data science, machine learning, and computational models in science and engineering. Rounding errors and numerical stability of algorithms; solution of linear and nonlinear equations; data modeling with interpolation and least-squares; and optimization methods. This course assumes that students have the equivalent of differential and integral calculus (e.g., MATH 2411), linear algebra (e.g., MATH 3191 or 3195), and computer programming (e.g., MATH 1376 or CSCI 1410). Cross-listed with CSCI 4650, 5660, and MATH 4650. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 5661 - Numerical Analysis II (3 Credits)
A second semester course in numerical methods and analysis fundamental to many algorithms encountered in scientific computing, data science, machine learning, and computational models in science and engineering. Numerical differentiation and integration; random numbers and stochastic modeling; Fast Fourier Transform; data compression; eigenvalues and singular value decompositions with application to regression and dimension reduction. This course assumes that students have the equivalent of differential and integral calculus (e.g., MATH 2411), linear algebra (e.g., MATH 3191 or 3195), and computer programming (e.g., MATH 1376 or CS 1410). Cross-listed with MATH 4660, CSCI 4660 and 5661. Term offered: spring. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Spring.
MATH 5718 - Applied Linear Algebra (3 Credits)
Topics include: Vector spaces, practical solution of systems of equations, projections, eigenvalues and eigenvectors, unitary transformations, Schur QR, singular value decompositions, similarity transformations, Jordan forms, and positive definite matrices. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of an undergraduate-level course in linear algebra (e.g., MATH 3191). Term offered: fall. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Fall.
MATH 5733 - Partial Differential Equations (3 Credits)
Initial/Boundary value problems for first-order, wave, heat and Laplace equations; maximum principles; Fourier series and applications. Prereq: Graduate standing in Applied Mathematics. Cross-listed with MATH 4733. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 5779 - Math Clinic (3 Credits)
The clinic is intended to illustrate the applicability and utility of mathematical concepts. Research problems investigated originate from a variety of sources--industry, government agencies, educational institutions, or nonprofit organizations. Prereq: Graduate standing in Applied Mathematics or Statistics or permission of instructor. Cross-listed with MATH 4779. Repeatable. Max Hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 5792 - Probabilistic Modeling (3 Credits)
Every other year. Markov chains; Poisson processes, continuous time Markov chains, elementary topics in queuing theory, and some mathematical aspects of Monte Carlo simulation, including random variate generation, variance reduction, and output analysis. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of an undergraduate-level course in probability (e.g., MATH 4810) and some programming experience. Cross-listed with MATH 4792. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 5830 - Applied Statistics (3 Credits)
Review of estimation, confidence intervals and hypothesis testing; ANOVA; categorical data analysis; non-parametric tests; linear and logistic regression. Restriction: Restricted to Graduate and Graduate Non-Degree majors. Note: This course assumes that students have the equivalent of an introductory course in statistics (e.g., MATH 2830). No co-credit with MATH 4387 or 5387 and doesn't count for Math degrees. Cross-listed with MATH 4830. Max hours: 3 Credits.
Grading Basis: Letter Grade
Restriction: Restricted to Graduate and Graduate Non-Degree Majors
MATH 5840 - Independent Study (1-3 Credits)
Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Prereq: Graduate standing in Applied Mathematics or Statistics and instructor permission. Repeatable. Max Hours: 9 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 9.
Prereq: Graduate standing in Applied Mathematics
MATH 5880 - Directed Research (1-6 Credits)
Students will engage in original research projects supervised and mentored by faculty. Students must work with faculty prior to registration to develop a proposal for their project and receive permission to take this course. Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Repeatable. Max Hours: 6 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 6.
MATH 5939 - Internship (1-6 Credits)
Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Repeatable. Max Hours: 9 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 9.
Prereq: Graduate standing in Applied Mathematics
MATH 5950 - Master's Thesis (1-8 Credits)
Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Prereq: Graduate standing in Applied Mathematics or Statistics and instructor permission. No co-credit with MATH 5960 or MATH 6960. Repeatable. Max hours: 8 Credits.
Grading Basis: Letter Grade with IP
Repeatable. Max Credits: 8.
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
Additional Information: Report as Full Time.
MATH 5960 - Master's Project (1-8 Credits)
Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Prereq: Graduate standing in Applied Mathematics or Statistics and instructor permission. No cocredit with MATH 5950 or MATH 6960.
Grading Basis: Letter Grade with IP
Repeatable. Max Credits: 8.
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
Additional Information: Report as Full Time.
MATH 6023 - Topics in Discrete Math (3 Credits)
Topics may include graph theory, combinatorics, matroid theory, combinatorial matrix theory, finite geometry, design theory, and discrete algorithms. Note: Since topic varies by semester, students may register for this course more than once. Note: students should obtain permission from the instructor prior to enrolling in this course. Prereq: Graduate standing in Applied Mathematics. Repeatable. Max Hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
Prereq: Graduate standing in Applied Mathematics
MATH 6101 - Uncertainty Quantification (3 Credits)
The field of uncertainty quantification is evolving rapidly due to increasing emphasis on models of physical and biological systems that have quantified uncertainties for large-scale applications, novel algorithm development, and new computational architectures that facilitate implementation of these algorithms. In this course, we develop the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties for a variety of simulation models. The topics will include concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, surrogate model construction, and local and global sensitivity analysis. Note: A basic knowledge of probability, linear algebra, ordinary and partial differential equations, and introductory numerical analysis techniques is assumed. Coursework will typically consist of projects. Prereq: Graduate standing in Applied Mathematics or Statistics. AMENMS/PHD/STAT-MS. Recommended preparation MATH 5070, MATH 5718, MATH 5660, MATH 5733. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 6131 - Real Analysis (3 Credits)
Every other year. Lebesque measure and integration, general measure and integration theory, Radon-Nikodyn Theorem, Fubini Theorem. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of a two semester course in undergraduate analysis or advanced calculus (e.g. MATH 4310 and 4320) or introductory graduate-level coursework in analysis (e.g. MATH 5070). Term offered: fall. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Fall.
MATH 6330 - Workshop in Statistical Consulting (3 Credits)
Students participate as consultants in a drop-in consulting service operated by the department. Seminars provide students with supervised experience in short term statistical consulting. Note: Since problems vary each semester, students may register for this course more than once. Prereq: Graduate standing in Applied Mathematics or Statistics or instructor permission. AMEN-MS/PHD/STAT-MS. Note: This course assumes that students have the equivalent of graduate-level coursework in regression analysis (e.g. MATH 5387). Term offered: fall. Repeatable. Max Hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
Typically Offered: Fall.
MATH 6380 - Stochastic Processes (3 Credits)
Every other year. Markov processes in discrete and continuous time, renewal theory, martingales, Brownian motion, branching processes, and stationary processes. Applications include queuing theory, performance evaluation of computer and communication systems and finance. Prereq: Graduate standing in Applied Mathematics or Statistics or instructor permission. AMEN-MS/PHD/STAT-MS. Note: This course assumes that students have the equivalent of undergraduate-level coursework in linear algebra (e.g. MATH 3191) and ordinary differential equations (e.g. MATH 3200), along with undergraduate-level coursework in probability (e.g. MATH 4810). Term offered: fall of odd years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 6384 - Spatial Data Analysis (3 Credits)
This course will cover various statistical methods for spatial data. This will include assessing cluster identification for point process and regional data, as well as quantifying spatial dependence and making predictions for regional and geostatistical spatial data. Prereq: Graduate standing in Applied Mathematics or Statistics or instructor permission. AMEN-MS/PHD/STAT-MS. Term offered: fall of odd years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
Typically Offered: Fall.
MATH 6388 - Statistical and Machine Learning (3 Credits)
This course covers a variety of statistical and machine learning methods. Both supervised and unsupervised methods are covered with an emphasis on model training and error estimation. Topics include penalized regression, principal components, k-nearest neighbors, clustering, and neural networks. Additional higher-level topics such as random forests, support vector machines, and boosting are also covered as time permits. Students will gain exposure to high performance computing by working on a Linux cluster. Prereq: Graduate standing in Statistics or Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in regression analysis (e.g. MATH 5387). Term offered: fall of odd years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
Typically Offered: Fall.
MATH 6404 - Applied Graph Theory (3 Credits)
Every other year. Emphasis on graph theory. Topics will include trees, diagraphs and networks, intersection graphs, coloring, clique coverings, distance, paths and cycles. Topics are motivated by applications. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Term offered: fall. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Fall.
MATH 6595 - Nonlinear Programming (3 Credits)
Every other year. Introduces fundamental algorithms and theory for nonlinear optimization problems. Topics include Newton, quasi-Newton and conjugate direction methods; line search and trust-region methods; active set, penalty and barrier methods for constrained optimization; convergence analysis and duality theory. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of a two semester course in undergraduate analysis (e.g. MATH 4310 and 4320) and graduate-level coursework in linear algebra (e.g. MATH 5718). Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 6653 - Introduction to Finite Element Methods (3 Credits)
Every other year. The Finite Element Method (FEM) is introduced as a generic tool for the approximation of partial differential equations that model engineering and physics problems of interest. Elliptic, hyperbolic, and parabolic equations are solved with FEM. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in numerical analysis (e.g. MATH 5660). Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 6840 - Independent Study (1-3 Credits)
Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Prereq: Graduate standing in Applied Mathematics or Statistics and instructor permission. Repeatable. Max Hours: 3 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 3.
Prereq: Graduate standing in Applied Mathematics
MATH 6960 - Research Methods in Mathematics and Statistics (3 Credits)
The goal of the course is to guide students through the process of performing rigorous mathematical and statistical research. Topics include performing a proper literature review, developing technical writing skills, and learning best practices regarding oral presentations. Students will discuss their ongoing research projects in groups and individually with the instructor, write a research paper, and present their results in a classroom seminar. Master's students should be entering their final year of coursework. Prereq: Graduate standing in Applied Mathematics or Statistics and instructor permissions. No co-credit with MATH 5950, MATH 5960, or ECON 6073. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 7132 - Functional Analysis (3 Credits)
Every other year. Linear metric and topological spaces, duality, weak topology, spaces of functions, linear operators, compact operators, elements of spectral theory, and operator calculus. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate level coursework in real analysis (e.g. MATH 6131). Term offered: spring of odd years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Spring.
MATH 7384 - Mathematical Probability (3 Credits)
Every other year. Measurable spaces, probability measures, random variables, conditional expectations and martingales. Convergence in probability, almost sure convergence, convergence in distribution, limit theorems (law of large numbers, central limit theorem, laws of iterated logarithm). Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of undergraduate-level coursework in probability (e.g. MATH 4810) and graduate-level coursework in analysis (e.g. MATH 5070 or 6131). Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 7385 - Stochastic Differential Equations (3 Credits)
Brownian motion, Ito integral, Ito formula, Dynkin's formula, stochastic optimal control, boundary value problems, Girsanov theorem, mathematical finance, optimal stopping. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in mathematical probability (e.g. MATH 7384). Max Hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 7386 - Monte Carlo Methods (3 Credits)
This course introduces Monte Carlo integration, variance reduction methods, quasi-Monte Carlo, Markov chain Monte Carlo, Metropolis-Hastings algorithm, Gibbs sampler, simulated annealing, expectation-maximization algorithm, sequential Monte Carlo methods. Prereq: Graduate standing in Applied Mathematics or Statistics. AMENMS/PHD/STAT-MS. Recommended preparation: MATH 5310
and MATH 5320. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 7393 - Bayesian Statistics (3 Credits)
Prior and posterior distributions, conjugate models, single and multiparameter models, hierarchical models, numerical methods for evaluating posteriors, Monte Carlo methods, and Markov chain Monte Carlo. Prereq: Graduate standing in Applied Mathematics or Statistics or instructor permission. AMEN-MS/PHD/STAT-MS. Programming experience is strongly recommended. Term offered: spring of odd years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
Typically Offered: Spring.
MATH 7397 - Nonparametric Statistics (3 Credits)
Every three years. Statistical inference without strong model assumptions. Hypothesis testing and estimation using permutations and ranks, analysis of variance, and nonparametric model fitting. Prereq: Graduate standing in Applied Mathematics or Statistics or instructor permission. AMEN-MS/PHD/STAT-MS. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 7405 - Advanced Graph Theory (3 Credits)
Continuation of MATH 6404. Topics to be covered include: trees and optimization, encoding and embedding of graphs, generalized colorings and applications, perfect graphs, extremal problems, substructures, connectedness' and cycles. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in graph theory (e.g. MATH 6404). Term offered: spring of even years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Spring.
MATH 7409 - Applied Combinatorics (3 Credits)
Every other year. Emphasis is on enumerative combinatorics. Topics include multinomial coefficients, generating functions, SDRs, Polya's enumeration theory, pigeon-hole principle, inclusion/exclusion and Moebius inversion of finite posets. Topics may also include introduction to designs and finite geometry. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 7410 - Combinatorial Structures (3 Credits)
Every other year. Finite combinatorial structures; existence, construction and applications. Topics include Latin squares, Hadamard matrices, block designs, finite geometries and extremal and non-constructive combinatorics. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in combinatorics (e.g. MATH 7409). Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 7593 - Advanced Linear Programming (3 Credits)
Every three years. A Ph.D. level course that goes deeper into linear programming, starting from where a graduate-level course (5593) ends. Topics include advanced sensitivity analysis, sparse matrix techniques, and special structures. Additional topics, which vary, include deeper analysis of algorithms, principles of model formulation and solution analysis. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in linear programming (e.g. MATH 5593). Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 7594 - Integer Programming (3 Credits)
Every three years. A Ph.D. level course that uses linear programming (5593), especially polyhedral theory, to introduce concepts of valid inequalities and superadditivity. Early group-theoretic methods by Gomory and Chvatal's rounding function are put into modern context, including their role in algorithm design and analysis. Duality theory and relaxation methods are presented for general foundation and analyzed for particular problem classes. Among the special problems considered are knapsack, covering, partitioning, packing, fix-charge, traveling salesman, generalized assignment matchings. Matroids are introduced and some greedy algorithms are analyzed. Additional topics, which vary, include representability theory, heuristic search and complexity analysis. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in linear programming (e.g. MATH 5593). Term offered: spring of odd years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Spring.
MATH 7595 - Advanced Nonlinear Programming (3 Credits)
Every three years. Focuses primarily on the fundamental theory of nonlinear programming. Topics include convex analysis, optimality criteria, Lagrangian and conjugate duality, stability and sensitivity analysis. Other topics vary depending on the research interests of the instructor. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework addressing computational methods in nonlinear programming (e.g. MATH 6595). Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 7663 - Finite Difference Methods for Partial Differential Equations (3 Credits)
Every other year. Consistency, stability, and convergence for difference schemes. Derivations based on Taylor series and finite series. Methods for parabolic and hyperbolic initial value problems and initial-boundary-value problems, elliptic boundary-value problems, some nonlinear problems. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in numerical analysis (e.g. MATH 5070) and partial differential equations (e.g. MATH 5733). Max Hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 7665 - Numerical Linear Algebra (3 Credits)
Every other year. Solution of linear equations, eigenvector and eigenvalue calculation, matrix error analysis, orthogonal transformation, iterative methods. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in numerical analysis (e.g. MATH 5660) and linear algebra (e.g. MATH 5718). Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 7822 - Topics in Linear Algebra (3 Credits)
Infrequent. Topics may include canonical forms, bilinear and quadratic forms, and combinatorial matrix theory. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in linear algebra (e.g. MATH 5718). Repeatable. Max Hours: 48 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 48.
Prereq: Graduate standing in Applied Mathematics
MATH 7823 - Topics in Discrete Math (3 Credits)
Infrequent. Advanced topics in discrete mathematics; will change from semester to semester. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: Students should contact the course instructor to determine the course focus, and to determine if any prior undergraduate- or graduate-level coursework is assumed. Repeatable. Max Hours: 48 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 48.
Prereq: Graduate standing in Applied Mathematics
MATH 7824 - Topics in Computational Mathematics (3 Credits)
Infrequent. Topics include methods for differential equations, numerical optimization, approximation theory, inverse problems, and Fourier analysis. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: Students should contact the course instructor to determine the course focus, and to determine if any prior undergraduate- or graduate-level coursework is assumed. Repeatable. Max Hours: 48 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 48.
Prereq: Graduate standing in Applied Mathematics
MATH 7825 - Topics in Optimization (3 Credits)
Infrequent. Some topics are extensions of those introduced in MATH 6595, while other topics are new. Examples of topics are: duality, stability, sensitivity, consistency, redundancy, principles of optimality, control theory, calculus of various global (non-convex) optimization and model reformulation. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: Students should contact the course instructor to determine the course focus, and to determine if any prior undergraduate- or graduate-level coursework is assumed. Repeatable. Max hours: 48 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 48.
Prereq: Graduate standing in Applied Mathematics
MATH 7826 - Topics in Probability and Statistics (3 Credits)
Infrequent. Topics may include generalized linear models, information theory, robust methods, spatial statistics, sequential analysis, Monte Carlo methods, queuing theory. Note: Since topics vary each semester, students may register for this course more than once. Prereq: Graduate standing in Applied Mathematics or Statistics or instructor permission. AMEN-MS/PHD/STAT-MS. Note: Students should contact the course instructor to determine the course focus, and to determine if any prior undergraduate- or graduate-level coursework is assumed. Repeatable. Max hours: 48 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 48.
Prereq: Graduate standing in Applied Mathematics or Statistics. AMEN-MS/PHD/STAT-MS
MATH 7827 - Topics in Applied Mathematics (3 Credits)
Infrequent. Topics include problems in differential equations, optimization, mathematical modeling, Fourier analysis and approximation theory. Note: Since topics vary each semester, students may register for this course more than once. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Repeatable. Max Hours: 48 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 48.
Prereq: Graduate standing in Applied Mathematics
MATH 7840 - Independent Study (1-3 Credits)
Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Prereq: Graduate standing in Applied Mathematics or Statistics and instructor permission. Repeatable. Max Hours: 3 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 3.
Prereq: Graduate standing in Applied Mathematics
MATH 7921 - Readings in Mathematics (1 Credit)
Annual. Seven readings courses are offered regularly primarily for Ph.D. students at the research level in the designated fields. The seminar format requires significant student participation. Prereq: permission of instructor. Department consent required. Repeatable. Max hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
MATH 7922 - Rdgs:Math Fndts-Cmptr Sc (1 Credit)
Department consent required. Repeatable. Max hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
MATH 7923 - Readings: Discrete Mathematics (1 Credit)
Department consent required. Repeatable. Max hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
MATH 7924 - Rdgs:Comp Mathematics (1 Credit)
Department consent required. Repeatable. Max Hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
MATH 7925 - Readings: Optimization (1 Credit)
Department consent required. Repeatable. Max hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
MATH 7926 - Rdgs:Applied Prob/Stats (1 Credit)
Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Prereq: Graduate standing in Applied Mathematics or Statistics and instructor permission. Repeatable. Max Hours: 99 Credits.
Grading Basis: Letter Grade
Repeatable. Max Credits: 99.
Prereq: Graduate standing in Applied Mathematics
MATH 7927 - Rdgs:Comp/Math Biology (1 Credit)
Department consent required. Max hours: 1 Credits.
Grading Basis: Letter Grade
MATH 8660 - Mathematical Foundations of Finite Element Methods (3 Credits)
Every other year. Theoretical foundations of finite element methods for elliptic boundary value problems, Sobolev spaces, interpolations of Sobolev spaces, variational formulation of elliptic boundary-value problems, basic error, estimates, applications to elasticity, practical aspects of finite element methods. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in finite element methods (e.g. MATH 6653) or equivalent programming experience, and graduate-level coursework in analysis or functional analysis (e.g. MATH 6131 or MATH 7132). Term offered: spring of odd years. Max hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
Typically Offered: Spring.
MATH 8664 - Iterative Methods in Numerical Linear Algebra (3 Credits)
Every other year. Preconditioned iterative methods for linear systems and eigen problems, conjugate gradients, multigrid and domain decomposition. Prereq: Graduate standing in Applied Mathematics or permission of the instructor. Note: This course assumes that students have the equivalent of graduate-level coursework in numerical analysis (e.g. MATH 5660) and numerical linear algebra (e.g. MATH 7665). Max Hours: 3 Credits.
Grading Basis: Letter Grade
Prereq: Graduate standing in Applied Mathematics
MATH 8990 - Doctoral Dissertation (1-10 Credits)
Only for students working on their Ph.D. research. Department consent required. Note: Students must submit a special processing form completely filled out and signed by the student and faculty member, describing the course expectations, assignments and outcomes, to the CLAS Graduate Academic Services Coordinator for approval. Repeatable. Max hours: 50 Credits.
Grading Basis: Letter Grade with IP
Repeatable. Max Credits: 50.
Additional Information: Report as Full Time.