Course information contained within the Bulletin is accurate at the time of publication in June 2025 but is subject to change. For the most up-to-date course information, please refer to the Course Catalog.
MATH 5000. Special Topics in Math. 3 Credit Hours.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 5001. Linear Algebra. 3 Credit Hours.
Vector spaces and subspaces over the real and complex numbers; linear independence and bases; linear mappings; dual and quotient spaces; fields and general vector spaces; polynomials, ideals and factorization of polynomials; determinant; Jordan canonical form. Fundamentals of multilinear algebra.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5003. Professional Development Seminar. 1 Credit Hour.
This class advances intentional Professional Development by creating an online professional profile and portfolio that allows employers to determine the strength of a student's candidacy for a specific job. Students develop an online professional profile, attend a Professional Development Workshop and write a White paper which demonstrates analytical and technical writing skills on a topic of interest to the student. The White Paper proposes a change in any STEM area where a lack of efficiency in a process, or gap in knowledge, in an area of research, exists. Finally, students organize a networking event by inviting speakers, hiring managers and graduate students in CST.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Degree Restrictions: Must be enrolled in one of the following Degrees: Prof Science Masters.
College Restrictions: Must be enrolled in one of the following Colleges: Science & Technology.
Repeatability: This course may not be repeated for additional credits.
MATH 5005. Ethics in Computing. 2 Credit Hours.
This course will examine the social, legal, and privacy issues applying to scientific data. Students will be given the opportunity to discuss contemporary case studies, in addition to NIH-sanctioned online training modules (Responsible Conduct in Research). The case-study based approach used in class will expose students to ethics of database management and security, open-access and open-source philosophies, the ethics of collecting, storing, and disseminating data.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Degree Restrictions: Must be enrolled in one of the following Degrees: Prof Science Masters.
College Restrictions: Must be enrolled in one of the following Colleges: Science & Technology.
Repeatability: This course may not be repeated for additional credits.
MATH 5007. Combinatorics. 3 Credit Hours.
Basic theorems and applications of combinatorial analysis, including generating functions, difference equations, Polya's theory of counting, graph theory, matching, and block diagrams.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5011. Algebra and Functions for Teaching. 3 Credit Hours.
This class will broaden and deepen our understanding of algebra and functions and their many applications. We will begin with an examination of the concept of function generally and then look at examples. We will consider the usefulness of functions, both in modeling real phenomena and in solving equations. We will carefully develop more advanced concepts from basic principles and logic as we proceed from polynomial functions to rational, radical, exponential, logarithmic, and trigonometric functions. As we do this, we will explore non-traditional teaching techniques and tools such as the practice of inquiry-based learning and the appropriate use of technology. We will also utilize and discuss the eight Mathematical Practice Standards set forth in the Common Core State Standards.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of C in MATH 1042.
MATH 5012. Introduction to Mathematical Modeling for Teaching. 3 Credit Hours.
In this course, mathematics will be discussed as the language of science. Many students do not have an adequate picture of mathematics: they see it either as a dry formal list of formulas or a dull study of numbers. Mathematics is in fact a network of intriguing and profound ideas that are deeply connected to reality. As a language, mathematics provides penetrating techniques of thought that allow us to analyze physical reality and to look for answers or solutions to some of the most intriguing real-life questions. Students will be asked to read the material in advance and come prepared for class discussion. The in-class activities will be conducted in an inquiry-minded fashion.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of C in MATH 1042.
MATH 5013. Geometry for Teaching. 3 Credit Hours.
This class will provide a deep and complete picture of the underlying concepts needed to teach high school geometry. We'll start by learning about the basic axiomatic method, which is fundamental to all of mathematics. We'll learn about the rigid transformations (reflections, rotations, and translations) and the important role that they can play in defining congruence more generally. We'll finish by looking at some important examples of non-Euclidean geometry, where Euclid's famous parallel postulate does not hold.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of C in MATH 1042.
MATH 5014. Probability Theory and Applications for Teaching. 3 Credit Hours.
Probability is a fundamental topic with applications in nearly every aspect of life. The goal for this course is to equip the student with a large and diverse set of tools with which to tackle a wide variety of problems, both theory and application based. We will focus on the ideas behind the important topics e.g. conditioning, averages, binomial distribution, and explain their origins and applications. This will equip the student to teach their own students this material with an emphasis on the "why", which is essential to maintaining attentiveness.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of C in MATH 1042.
MATH 5015. Modern Algebra for Teaching. 3 Credit Hours.
Students will understand the integers and polynomial rings over a field as being specific examples of rings. The idea of quotient spaces will be emphasized with the particular examples of the integers modulo n and factor rings of polynomial rings illustrating and introducing the concept. Moreover, the ability to read and to construct well-written and correct mathematical proofs on these topics is an overarching goal of the course. Written communication skills will also be emphasized.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of C in MATH 2111.
MATH 5016. Mathematical Analysis for Teaching. 3 Credit Hours.
This course will start with a discussion of the basic topology of the real line and the creation of the basic tools using the completeness axiom. From there, the course will proceed to sequences and series, limits, continuity, differentiation, Riemann integration, and Taylor series representations of functions.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of C in MATH 2043.
MATH 5017. Number Theory and Proof for Teaching. 3 Credit Hours.
In this course, we encounter and explore fundamental ideas in number theory. Basic properties of the integers and their two principal operations, addition and multiplication, will form the starting point of our study. Along the way, the course will introduce some basic logic and the rigorous notion of mathematical proof, including mathematical induction, in the context of number theory. Through this elementary foundation, students will experience the richness of mathematics: proof going back as far as Euclid, examples of elementary yet still unproven conjectures, and results that are easy to state and understand but require extremely complicated proofs.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of C in MATH 1042.
MATH 5032. Stochastic Calculus. 3 Credit Hours.
This course is an introduction to stochastic calculus based on Brownian motion and Gaussian processes, covering stochastic differential equations as well as applications to option pricing in finance. Concepts and results are illustrated with examples and numerical projects.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5033. Introduction to Stochastic Processes. 3 Credit Hours.
This course is an introduction to some of the basic models, concepts, results and techniques in the study of stochastic processes and touches upon applications along the way. It covers a large selection of the following topics: finite- and countable-state Markov chains, Poisson and birth-and-death processes, optimal stopping, martingales, renewal processes, Markov chain algorithms, and introduction to Brownian motion.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5034. High-Dimensional Probability. 3 Credit Hours.
This course provides a self-contained introduction to the area of high-dimensional probability and statistics from a non-asymptotic perspective, aimed at students across the mathematical sciences. It will include a focus on core methodology and theory (tail bounds, concentration of measure, random matrices, random graphs and networks) as well as in-depth exploration of various applications (to statistical learning theory, sparse linear and graphical models, community detection, as examples).
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5041. Concepts of Analysis I. 3 Credit Hours.
This is a first semester course in basic real analysis. Topics include the real number system and the completeness property, ordered fields, topology of metric spaces, sequences and series, limits of functions and continuity, and differentiation.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5042. Concepts of Analysis II. 3 Credit Hours.
This is a second semester course in real analysis, representing a continuation of
MATH 5041. Topics include the Riemann integral, infinite series and convergence tests, sequences of functions, uniform convergence, power and Taylor series and operations with them, differentiability of vector valued functions, inverse and implicit function theorems, differential forms, vector analysis, and formula of change of variables in multiple integrals.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 5041.
MATH 5043. Numerical Analysis. 3 Credit Hours.
This course provides a graduate level introduction to classical and modern methods for fundamental problems in computational science and engineering, including approximation and interpolation, numerical integration/quadrature, direct methods for systems of equations, and solution of systems of nonlinear equations. A rigorous mathematical approach to these topics is taken, including floating point arithmetic, error analysis, conditioning and stability, and convergence theorems. The course is accessible to graduate students from all areas of science and engineering interested in a mathematical foundation for the listed computational methods.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5044. Numerical Methods for Ordinary Differential Equations. 3 Credit Hours.
This course introduces numerical methods for ordinary differential equations, including explicit, implicit, and semi-implicit time stepping methods (such as Runge-Kutta, multistep, and Taylor series methods). A rigorous mathematical basis is established, including convergence, stability, and error estimators. Critical challenges commonplace in applications, like stiff problems, and specialized time stepping methods, are also presented.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5045. Ordinary Differential Equations. 3 Credit Hours.
The course covers existence and uniqueness theorems for ordinary differential equations (ODEs), along with continuous and smooth dependence on initial conditions and parameters, assuming familiarity with the material of advanced calculus. Topics include linear differential equations, their qualitative and asymptotic behavior, and the analysis of isolated singularities. The course also introduces nonlinear equations and Sturm-Liouville problems, emphasizing both theoretical aspects and applications. Basic methods for the numerical solution of ODEs are presented, including stability and convergence considerations.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5057. Applied Partial Differential Equations and Optimization. 3 Credit Hours.
This course introduces and studies partial differential equations (PDEs) central to applied mathematics and related fields, arising from continuum mechanics and optimization. Material includes continuum mechanics, methods for linear PDEs, calculus of variations, as well as basic constrained optimization and control. Course topics have a wide applicability to the sciences and engineering.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5058. Fundamentals of Mathematical Modeling. 3 Credit Hours.
This course introduces fundamental principles of mathematical modeling. Basics of constructing and analyzing models, including scaling and asymptotics, are studied in large part through dynamical systems and their applications. Examples from many areas across the sciences and engineering disciplines are central to this course.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5061. Fundamentals of Computer Programming for Scientists and Engineers. 4 Credit Hours.
Scientists and engineers use computers for a multitude of purposes. Even with ready-to-use applications, some amount of computer programming is commonly required to adapt to changing technology while attaining the rigorous standards of each specific discipline. This course focuses on fundamental computer programming constructs, introducing the languages Python, C++ and Fortran. Through lectures and intensive exercises students will learn to implement fundamental mathematical constructs and solve basic programming problems relevant to scientific applications. The course briefly reviews also the Linux environment, its software development tools and language interoperability. For each programming language, the course focuses on constructs and syntax designed for performance and numerical accuracy, in connection with methods from applied science, mathematics and engineering. The students taking the course are expected to have sufficient mathematical maturity, as evidenced, for example, by having completed an undergraduate Calculus sequence. The majority of the grade is determined by a mid-term and a final exam, both including a combination of questionnaires and supervised programming assignments.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5062. High Performance Computer Programming for Scientific Modeling. 3 Credit Hours.
This course will provide theory and hands-on experience programming high performance computers for the solution of scientific modeling problems. This includes in particular problems arising from the discretizations of differential equations. Topics covered include domain decomposition and mesh partitioning, quantifying the computation and communication cost, communication avoidance methods, Monte Carlo methods, multithreading, benchmarking and optimization of the parallel computations.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 5061.
MATH 5063. Introduction to High-Performance Computing Technology for Scientists. 4 Credit Hours.
This course is an introduction to the technology used in Linux clusters and supercomputers dedicated to calculations in applied science and engineering. The basic architecture of modern computers (processing units, memory, storage, operating system) is briefly reviewed, emphasizing the role and performance impact of each element in numerical computation. The core of the course focuses on setup and management of computer hardware specialized for scientific computing, and on its impact on commonly used strategies and methods for scientific computation. The material is organized in a combination of lectures and hands-on exercises, using computer hardware hosted at local facilities as well as virtualized resources. The majority of the grade is determined by a mid-term and a final exam, both including a combination of questionnaires and identification of the most efficient solution to common numerical problems.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 5061 (may be taken concurrently)
MATH 5065. Topology. 3 Credit Hours.
This course covers the fundamentals of metric spaces and topological spaces. Core topics include continuity, compactness, connectedness, and convergence. Time permitting, the course will include the classification of compact surfaces and an introduction to algebraic topology, including fundamental groups and covering spaces.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5066. Mathematical Methods for High Performance Computing. 3 Credit Hours.
This course presents mathematical methods for the solution of a variety of discrete and algebraic problems which are at the core of many scientific and engineering applications. The methods covered are especially tailored for high performance computing. Topics include large matrix computations, graphs and networks, fast Fourier transforms, geometric and algebraic multi-grid methods, and constrained optimization.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 5061, MATH 5062, and MATH 5063.
MATH 5067. Introduction to Abstract Algebra I. 3 Credit Hours.
This is the first semester in a year-long introduction to modern algebra. It is a thorough introduction to the theory of groups and rings.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 5068. Introduction to Abstract Algebra II. 3 Credit Hours.
This is the second semester in a year-long introduction to modern algebra. Topics come from the theory of rings, fields, modules and Galois theory.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 5067.
MATH 5071. Mathematical Aspects of Cryptography. 3 Credit Hours.
This course on the mathematical aspects of cryptography proceeds in four parts. In Part 1, we go over groups, rings, and finite fields. Part 2 is devoted to symmetric key cryptosystems, including both classical cryptosystems (affine cipher, substitution cipher, Hill cipher) and modern cryptosystems, e.g., the Advanced Encryption Standard (AES, Rijndael). Part 3 is devoted to aspects of elementary number theory related to cryptography. Part 4 is devoted to public key cryptosystems. In this part, we learn the discrete logarithm problem, the Diffie-Hellman key exchange, the Pohlig-Hellman algorithm and the collision algorithm, the ElGamal public key cryptosystem, the Rivest-Shamir-Adleman (RSA) public key cryptosystem, and attacks on RSA. The course involves programming problems and a study project.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8001. Candidates Seminar. 1 to 3 Credit Hour.
Challenging problems from many different areas of mathematics are posed and discussed.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 8002. Candidates Seminar. 1 to 3 Credit Hour.
Challenging problems from many different areas of mathematics are posed and discussed.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 8011. Abstract Algebra I. 3 Credit Hours.
This is the first semester of a year-long sequence in Abstract Algebra. Topics in group theory include group actions, the Sylow theorems, nilpotent and solvable groups, and free groups. Topics in ring theory include structure results on factorization and polynomial rings. The structure theorem for finitely generated modules over principal ideal domains is proved. Throughout, the theme of universal properties and basic category theory is developed.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8012. Abstract Algebra II. 3 Credit Hours.
This is the second semester of a year-long sequence in Abstract Algebra. The course covers fields and Galois theory, module theory including canonical forms for matrices, and Noetherian rings and modules.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8011.
MATH 8013. Numerical Linear Algebra. 3 Credit Hours.
This course presents the fundamental numerical techniques for Linear Algebra, aimed at graduate students in mathematics and related areas who seek a mathematical introduction. Topics include orthogonalization and QR factorization in theory and practice, iterative methods for solving linear systems, Krylov subspace methods, variational formulations for symmetric and nonsymmetric problems, convergence estimates, relations to orthogonal polynomials, preconditioning techniques, incomplete factorizations, multigrid methods, domain decomposition, and basic methods for eigenvalue problems.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8014. Numerical Linear Algebra II. 3 Credit Hours.
The syllabus includes iterative methods, classical methods, nonnegative matrices. Semi-iterative methods. Multigrid methods. Conjugate gradient methods. Preconditioning. Domain decomposition. Direct Methods. Sparse Matrix techniques. Graph theory. Eigenvalue Problems.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8013.
MATH 8023. Numerical Methods for Partial Differential Equations. 3 Credit Hours.
This course is designed for graduate students of all disciplines and areas who are interested in numerical methods for partial differential equations, focusing on a rigorous mathematical derivation and analysis. Topics include finite difference, finite element, Fourier, and spectral methods for boundary value problems, elliptic equations, transport, diffusion/heat equation, and wave propagation problems. Fundamental concepts like stability, convergence, and error analysis are a key focus.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8024. Numerical Methods for Nonlinear Partial Differential Equations. 3 Credit Hours.
This course is designed for graduate students of all disciplines and areas who are interested in numerical methods for partial differential equations, focusing on a rigorous mathematical derivation and analysis. Topics include finite difference, finite element, Fourier, and spectral methods for boundary value problems, elliptic equations, transport, diffusion/heat equation, and wave propagation problems. Fundamental concepts like stability, convergence, and error analysis are a key focus.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8031. Probability Theory. 3 Credit Hours.
This course provides a rigorous treatment of probability theory, covering the following: the axioms of probability, random variables and their distributions, expectation and variance, conditional probability and independence, Markov chains, random walks, laws of large numbers, weak convergence of measures, characteristic functions, the central limit theorem, and additional topics if time permits.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8032. Stochastic Processes. 3 Credit Hours.
This course provides a rigorous treatment of stochastic processes, focusing on the following: existence of stochastic processes, Poisson process, conditional expectation and martingales, continuous-time Markov processes, stationary processes, ergodicity and mixing, Brownian motion and its properties.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8031.
MATH 8041. Real Analysis I. 3 Credit Hours.
This course focuses on the development of the Lebesgue measure and integration theory. It covers core areas in analysis, including functions of bounded variation, Lebesgue measure and outer measure, measurable functions, Lebesgue integral, modes of convergence, and repeated integration. This course presumes knowledge of undergraduate Real Analysis.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8042. Real Analysis II. 3 Credit Hours.
This course is a continuation of
MATH 8041. It covers core areas in analysis including Lebesgue differentiation theory, maximal functions, absolute continuity, approximation of identity, L^p spaces, Hilbert spaces, abstract measures and integration.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8041.
MATH 8051. Functions of a Complex Variable. 3 Credit Hours.
This course is an introduction to the theory of functions of one complex variable. Topics include holomorphic functions, derivatives and path integrals, power series, exponential and trigonometric functions, the Cauchy integral theorem and consequences (the maximum principle, Liouville's theorem, Morera's theorem, Schwarz' reflection principle, and Goursat's theorem), residue calculus, convergence in the space of holomorphic functions, Hurwitz' theorem, Montel's theorem, conformal maps and the Riemann mapping theorem. This course presumes knowledge of undergraduate Real Analysis.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8052. Intermediate Complex Variables. 3 Credit Hours.
The topics and the textbook choice are at the instructor's discretion. Topics of focus include: Phragmen-Lindelof method, Runge's theorem, Weierstrass factorization theorem, Mittag-Leffler theorem, theory of entire functions, elements of theory of elliptic functions, several complex variables, theory of Riemann surfaces.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8051.
MATH 8061. Smooth Manifolds. 3 Credit Hours.
This is a standard introductory graduate course in differential topology. Topics include: smooth structures on manifolds; smooth functions and maps between manifolds; tangent and cotangent bundles; vector bundles; differential forms; tensors; integration and Stokes' theorem.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8062. Algebraic Topology. 3 Credit Hours.
This is a standard introductory graduate course in algebraic topology. Topics include: the fundamental group; van Kampen's theorem; covering space theory; singular homology and cohomology; exact sequences; de Rham cohomology and de Rham's theorem; Poincare duality.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8107. Mathematical Modeling for Science, Engineering, and Industry. 3 Credit Hours.
In this course, students form research groups for projects provided by partners in industry, engineering, or in other disciplines of science. Throughout the semester, the teams are advised by the course instructors and the external partners. The problems themselves are formulated in non-mathematical language, so the students are required to translate these into mathematical language and develop solutions using methods of mathematical modeling. This means that the mathematical and computational methods must be selected or created by the students themselves. At the end, research findings and final results need to be formulated in a language accessible to the external partner. Students disseminate their progress and achievements in weekly presentations, a mid-term and a final project report, as well as a final presentation. Group work with and without the instructors' involvement is a crucial component in this course. Graduate students from disciplines other than mathematics, interested in collaborating in teams on mathematical modeling, and with a background in principles of modeling and computational methods, are welcome to this course.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8141. Partial Differential Equations. 3 Credit Hours.
This course introduces fundamental concepts and results in the theory of elliptic, parabolic, and hyperbolic differential equations, preparing students for problem-solving in PDEs and their applications. Topics covered: Laplace's equation (maximum principles, Dirichlet problem, Green's function and Poisson kernel), Heat equation (initial boundary value problems, fundamental solution, mean value formula, maximum principle, uniqueness theorems, examples of non-uniqueness, backward heat equation, and energy methods), Wave equation (D'Alembert's formula, plane waves, solution by spherical means, Huygens' Principle, Energy methods: uniqueness, domain of dependence). Solution of the Cauchy problem for nonlinear first order PDEs. This course presupposes knowledge of undergraduate Real Analysis.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8142. Intermediate Partial Differential Equations. 3 Credit Hours.
This course builds on the foundations of graduate-level PDE theory. The choice of content is at the instructor's discretion. Topics to be covered include nonlinear PDEs, Fourier transform methods, dispersive equations, Sobolev spaces, energy methods, and the Fredholm alternative.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8141.
MATH 8161. Topology. 3 Credit Hours.
Point set topology through the Urysohn Metrization Theorem; fundamental group and covering spaces. Differential forms; the DeRham groups.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 5041.
MATH 8200. Topics in Applied Mathematics. 3 Credit Hours.
Variable topics, such as control theory and transform theory, will be treated.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 8210. Topics in Applied Mathematics II. 3 Credit Hours.
Variable topics, such as control theory and transform theory, will be treated.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 8700. Topics Computer Program. 3 Credit Hours.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 8710. Topics Computer Program. 3 Credit Hours.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 8981. Graduate Development Seminar. 1 Credit Hour.
This course aims to familiarize first-year PhD students with the structure of a PhD in Mathematics. A significant focus of the course is professional development, wherein students learn about important milestones in the program and are trained in the related responsibilities. Students enrolled in this course must attend at least one seminar or colloquium per week, in order to be exposed to research-level mathematics and best practices for communicating mathematics. The seminar itself features a weekly discussion on a topic of interest, led by the Director of Graduate Studies and/or a senior TA. Topics covered in the seminar should include: Basics of departmental structure; effective study techniques for graduate courses and qualifying exams; best practices for professional conduct; creating a professional webpage; written and oral communication of research-level mathematics; research topics studied by faculty in the department; the process of finding a PhD advisor, e.g. through independent study courses; organizing PhD studies with perspective of post-PhD career goals; finding and applying for summer internships in industry and education; and applying for post-PhD employment, in and out of academia.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 8985. Teaching in Higher Education. 1 to 3 Credit Hour.
This course is required for any student seeking Temple's Teaching in Higher Education Certificate. The course focuses on the research on learning theory and the best teaching practices, with the aim of preparing students for effective higher education teaching. All educational topics will be considered through the lens of teaching mathematics and quantitative thinking.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 9000. Topics in Number Theory I. 3 Credit Hours.
Analytic and algebraic number theory. Classical results and methods and special topics such as partition theory, asymptotic, Zeta functions, transcendence, modular functions.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9003. Modular Forms. 3 Credit Hours.
This course introduces students to modular forms. Topics include Eisenstein series, finiteness of spaces of modular forms, Hecke operators, modular curves, L-functions of modular forms, and moduli descriptions of modular curves and modular forms. Additional possible topics include congruences between modular forms and an introduction to modularity.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8051.
MATH 9005. Combinatorial Mathematics. 3 Credit Hours.
Topics include: Enumeration, Trees, Graphs, Codes, Matchings, Designs, Chromatic Polynomials, Coloring, Networks.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
MATH 9011. Homological Algebra. 3 Credit Hours.
The course is devoted to fundamental notions of homological algebra: chain complexes, abelian categories, derived functors, and spectral sequences. A portion of this course is also devoted to rudiments of category theory. Students will learn how to apply constructions of homological algebra and category theory to questions from abstract algebra, topology and deformation theory.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8012.
MATH 9012. Representation Theory. 3 Credit Hours.
This course primarily focuses on representations of finite groups over a field of characteristic zero. The bulk of the course covers the basics of subrepresentations, irreducibility, semisimplicity, character theory and orthogonality relations, induced representations, and Mackey theory. Additional possible topics include Schur indices, pseudorepresentations, representations of Lie algebras, and modular representation theory.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8012.
MATH 9014. Commutative Algebra. 3 Credit Hours.
This course focuses on the fundamental concepts of commutative algebra. Topics of the course include ideals of commutative rings, modules, Noetherian and Artinian rings, Noether normalization, Hilbert's Nullstellensatz, rings of fractions, primary decomposition, discrete valuation rings and the rudiments of dimension theory.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8012.
MATH 9015. Algebraic Geometry. 3 Credit Hours.
This course introduces students to the vast subject of algebraic geometry. Topics of the course include affine and projective varieties, morphisms of algebraic varieties, birational equivalence, and basic intersection theory. Students will also learn about schemes, morphisms of schemes, coherent sheaves, and divisors.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8012.
MATH 9021. Riemannian Geometry. 3 Credit Hours.
The goal of this course is to provide a solid introduction to the two central concepts of Riemannian geometry, namely, geodesics and curvature. Topics include Riemannian metrics, Riemannian connections, geodesics, curvature (sectional, Ricci, and scalar), the Jacobi equation, the second fundamental form, and global results such as the Gauss-Bonnet theorem, Hopf-Rinow theorem, and Hadamard's theorem.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8061 and MATH 8062 (may be taken concurrently)
MATH 9023. Knot Theory. 3 Credit Hours.
This course covers the theory of knots and links, with a focus on different flavors of invariants. The course covers a large selection of the following topics: polynomial invariants, connections to braid groups, Dehn surgery, geometric invariants, quantum invariants of 3-manifolds.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8062.
MATH 9024. Low-Dimensional Topology. 3 Credit Hours.
This course is a broad introduction to low-dimensional topology with an emphasis on 3-manifolds, especially their construction, classification results, and invariants. Potential topics include: gluing constructions, Dehn surgery, foliations and flows, geometric structures, and connections to group theory.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8062.
MATH 9041. Functional Analysis. 3 Credit Hours.
Functional Analysis provides the foundation for high-dimensional geometry, analysis, and probability. This course introduces the fundamental concepts and methods of functional analysis, with an emphasis on concrete examples. Topics include the Baire Category Theorem and its applications, particularly to Banach space theory principles such as the Uniform Boundedness Principle, the Open Mapping Theorem, and the Closed Graph Theorem. The course also explores different notions of convergence, weak topology, duality, and convexity. In addition, it introduces linear operator theory, including spectral theory for compact operators, self-adjoint operators, Fredholm theory, and their applications.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in (MATH 5042 or MATH 8042)
MATH 9042. Fourier Analysis and Distribution Theory. 3 Credit Hours.
This course is a basic introduction to Fourier Analysis and Distribution Theory. Topics include Fourier series, the Fourier transform, Schwartz functions, tempered distributions, convolutions, and applications to partial differential equations. Knowledge of
MATH 9041, Functional Analysis, is recommended but not required.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8042.
MATH 9043. Calculus of Variations. 3 Credit Hours.
The course covers the first variation and Euler-Lagrange equations, including null Lagrangians and Caratheodory's "Royal Road". It introduces geodesic coverings, the eikonal equation, and the Hamilton-Jacobi equation. The second variation is explored through Jacobi's theory of conjugate points and Weierstrass's E-function. The Hamiltonian formalism, convex duality, and Hilbert's invariant integral are also discussed, highlighting connections between variational principles, geometry, and physics.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8042.
MATH 9044. Harmonic Analysis. 3 Credit Hours.
This course presents a rapid introduction to modern tools of Harmonic Analysis. These tools are relevant to problems in a very dynamic area of mathematics at the interface of several branches such as Fourier Analysis, Functional Analysis, Partial Differential Equations, Index Theory, and Geometric Measure Theory. Topics of focus include Hardy spaces, singular integrals, maximal operators, functions of bounded and vanishing mean oscillation, square functions, applications to elliptic partial differential equations in the upper half space.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8042.
MATH 9061. Lie Groups. 3 Credit Hours.
This course develops Lie theory from the ground up. Starting with basic definitions of Lie groups and Lie algebras, the course develops structure theory, analytic and algebraic aspects, structure of nilpotent and solvable Lie groups, and the classification of semisimple Lie groups.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8062.
MATH 9063. Riemann Surfaces. 3 Credit Hours.
This course covers the analytic and geometric theory of Riemann surfaces. Topics include: uniformization, algebraic perspectives, calculus on Riemann surfaces, moduli space. Moduli and Teichmuller spaces for compact Riemann surfaces; introduction to modular forms; embedding of compact Riemann surfaces in complex projective spaces. Branched coverings and maps onto the Riemann sphere.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8051.
MATH 9071. Hyperbolic Geometry. 3 Credit Hours.
This is an introduction to hyperbolic manifolds, their structure, and their deformations. Potential topics include: thick-thin decomposition, rigidity results, hyperbolic Dehn filling, arithmetic manifolds, and the general theory of Kleinian groups.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8061 and MATH 8062 (may be taken concurrently)
MATH 9073. Geometric Group Theory. 3 Credit Hours.
This course surveys the rapidly expanding field of geometric group theory, focusing on the role played by negative curvature. The course begins with classical combinatorial techniques used to construct and study infinite discrete groups. After introducing basic concepts in coarse geometry, it turns attention to Gromov's notion of hyperbolic groups. In addition to studying geometric, algebraic, and algorithmic properties of these groups, the course keeps an eye towards several generalizations of hyperbolicity that have recently played a large role in understanding many geometrically significant groups. As examples, it also touches on the study of mapping class groups, outer automorphism groups of free groups, and cubical groups.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may not be repeated for additional credits.
Pre-requisites: Minimum grade of B- in MATH 8062.
MATH 9082. Independent Study. 1 to 3 Credit Hour.
Independent research supervised by a Mathematics faculty member.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9083. Independent Study. 1 to 3 Credit Hour.
Independent research supervised by a Mathematics faculty member.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9100. Topics in Algebra. 3 Credit Hours.
This course covers variable topics in theory of commutative and non-commutative rings, groups, algebraic number theory, and algebraic geometry.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9200. Topics in Numerical Analysis. 3 Credit Hours.
This course covers a variable set of timely topics in Numerical Analysis, including, but not limited to, numerical optimization, computational fluid dynamics, numerical methods for general flow problems, or advanced finite element methods.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9210. Topics in Applied Mathematics. 3 Credit Hours.
This course covers a variable set of timely topics in Applied Mathematics, including, but not limited to, mathematical biology, multiscale modeling and methods, material science, or control theory.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9300. Topics in Probability. 3 Credit Hours.
This course presents research topics related to probability theory, depending on the interests of the students and the instructor. Potential topics include interacting particle systems, random matrix theory, probability models in mathematical physics, statistical mechanics, homogenization, stochastic optimal control, differential and mean-field games.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9400. Topics in Analysis. 3 Credit Hours.
This course focuses on advanced topics in Analysis and the theory of Partial Differential Equations with the goal of introducing students to open problems in the field. The focus changes with the instructor. Potential topics covered include non-linear PDEs, Monge-Ampere equations, elliptic PDE in irregular domains, spectral theory, singular integrals, optimal transport, regularity theory of solutions to elliptic PDEs, mathematical analysis of fluid equations, and analysis on quasi-metric spaces.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9420. Topics in Differential Equations II. 3 Credit Hours.
This is a year-long sequence. Topics covered may include the theory of elliptic partial differential equations in divergence form and non-divergence form, and nonlinear PDEs. These courses may also focus on pseudodifferential operators and Fourier integral operators.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9500. Topics in Geometry and Topology. 3 Credit Hours.
Variable topics in geometric topology and related areas. Possibilities include: Morse theory, surfaces and their diffeomorphisms, mapping class groups, Teichmuller theory, braids, dynamics of group actions, and homogeneous dynamics.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
Pre-requisites: Minimum grade of B- in MATH 8062.
MATH 9510. Topics in Differential Geometry and Topology II. 3 Credit Hours.
Variable topics in geometric topology and related areas. Topics include: surfaces and their diffeomorphisms, mapping class groups, braids, 3-dimensional manifolds, geometric structures on manifolds, and group actions on geometric objects.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
Pre-requisites: Minimum grade of B- in MATH 8061 and MATH 8062.
MATH 9991. Master's Research Projects. 1 to 6 Credit Hour.
Short-term, limited research project or laboratory project in the field. This course is not the capstone project course, nor can it be used for thesis based research. The course is for master's students only, including PSM, MA or MS. This class will not confer full-time program status unless nine credits are taken.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Degree Restrictions: Must be enrolled in one of the following Degrees: Master of Arts, Master of Science, Prof Science Masters.
Repeatability: This course may be repeated for additional credit.
MATH 9994. Preliminary Examination Preparation. 1 to 6 Credit Hour.
This course is required for students who are preparing for the preliminary or candidacy examination. Students should enroll after coursework is completed or when preparing for the candidacy exam until the time that the preliminary or candidacy examination is completed. This course will confer full-time status at the minimum credit hour registration limit of one credit. All students must complete a minimum of one credit of this course. Students must complete a total of 6 credit hours of 9994, 9998 and 9999.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9995. Capstone Project. 1 to 6 Credit Hour.
Capstone project for master's students including students in PSM, MA or MS. This class will provide full-time status. Students in PSM programs need to register for at least one credit of this course to fulfill program requirements. Additional credits may be required for specific programs. This course will confer full-time status at the minimum credit hour registration limit of one credit.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Degree Restrictions: Must be enrolled in one of the following Degrees: Master of Arts, Master of Science, Prof Science Masters.
Repeatability: This course may be repeated for additional credit.
MATH 9996. Master's Thesis Research. 1 to 6 Credit Hour.
Course for master's thesis research. Only intended for students in thesis bearing master's programs. A minimum of one credit is required. This course will confer full-time status at the minimum credit hour registration limit of one credit.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9998. Pre-Dissertation Research / Elevation to Candidacy. 1 to 6 Credit Hour.
This course is intended for students who are performing research prior to candidacy. Students can register for this course after required courses are completed. This course will confer full-time status at the minimum credit hour registration limit of one credit. Students must be registered for this course during the semester that they are to be elevated to candidacy examination. Students must complete a total of 6 credit hours of 9994, 9998 and 9999.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Repeatability: This course may be repeated for additional credit.
MATH 9999. Dissertation Research. 1 to 6 Credit Hour.
The course is for Ph.D. students who have been elevated to candidacy. During the course of their candidacy students must complete a minimum of two credits of dissertation research. This course will confer full-time status at the minimum credit hour registration limit of one credit. Students must complete a total of 6 credit hours of 9994, 9998 and 9999.
Level Registration Restrictions: Must be enrolled in one of the following Levels: Graduate.
Student Attribute Restrictions: Must be enrolled in one of the following Student Attributes: Dissertation Writing Student.
Repeatability: This course may be repeated for additional credit.
