## Graduate MATH/MATE Courses

**MATE 6310 Mathematics Teaching and Learning Theories**

This course examines issues, trends and research related to the teaching/learning of secondary school mathematics. Specific topics will vary, but could include: technology in the classroom, mathematical problem solving and the use of applications in the teaching of mathematics.

Prerequisite: Graduate standing in mathematics.

**MATE 6359 Mathematics Education Research Design**

An introduction is given to the methods and tools of mathematics education research and program evaluation in learning and teaching of mathematics. Independent work on assign topics is expected of the student, with presentations on the results in both oral and written form.

Prerequisite: MATE 6310

**MATH 5390 Survey of Topics in Mathematics**

This leveling course will serve as a basic introduction to mathematical analysis and abstract algebra and aims to prepare incoming graduate students for the graduate sequence in both areas. Part of the coursework will bridge the gap between a standard undergraduate course and our graduate classes. After completion of the course, the student will become proficient in explaining and applying the fundamental topics in mathematical analysis and abstract algebra.

Prerequisite: MATH 2415 and MATH 2318

**MATH 6305 History of Mathematics**

This course introduces students to the history of the development of mathematical ideas and techniques from early civilization to the present. The focus will be on both the lives and the works of some of the most important mathematicians.

Prerequisite: Departmental approval.

**MATH 6307 Collegiate Mathematics Teaching**

This course provides opportunities for students to have a practical experience in teaching college-level mathematics courses supervised by faculty.

Prerequisite: Departmental approval.

**MATH 6323 Group Theory**

This course is an introduction to group theory, one of the central areas in modern algebra. Topics will include the theorems of Jordan-Holder, Sylow, and Schur-Zassenhaus, the treatment of the generalized Fitting subgroup, a first approach to solvable as well as simple groups (including theorems of Ph. Hall and Burnside).

Prerequisite: Departmental approval.

**MATH 6325 Contemporary Geometry**

This course contains selected topics in computational, combinatorial and differential geometry as well as combinatorial topology. Topics include the point location problem, triangulations, Voronoi diagrams and Delaunay triangulations, plane curves and curvature, surfaces and polyhedrons and Euler characteristic.

Prerequisite: Departmental approval.

**MATH 6328 Special Topics in Mathematics Teaching**

A critical analysis of issues, trends and historical developments in elementary and/or secondary mathematics teaching with emphasis on the areas of curriculum and methodology. This course may be repeated for credit when topic changes.

Prerequisite: Graduate standing in mathematics.

**MATH 6329 Number Theory**

This course is an introduction to number theory, one of the major branches of modern mathematics. Topics include arithmetic functions, multiplicativity, Moebius inversion, modular arithmetic, Dirichlet characters, Gauss sums, primality testing, distribution of primes, primitive roots, quadratic reciprocity, Diophantine equations, and continued fractions. Applications and further topics include cryptography, partitions, representations by quadratic forms, elliptic curves, modular forms, irrationality, and transcendence.

Prerequisite: Departmental approval.

**MATH 6330 Linear Algebra**

Topics include the proof-based theory of matrices, determinants, vector spaces, linear spaces, linear transformations and their matrix representations, linear systems, linear operators, eigenvalues and eigenvectors, invariant subspaces of operators, spectral decompositions, functions of operators and applications to science, industry and business.

Prerequisite: MATH 2318 Linear Algebra with a grade of C or higher.

**MATH 6330 Linear Algebra**

Topics include the proof-based theory of matrices, determinants, vector spaces, linear spaces, linear transformations and their matrix representations, linear systems, linear operators, eigenvalues and eigenvectors, invariant subspaces of operators, spectral decompositions, functions of operators and applications to science, industry and business.

Prerequisite: MATH 2318 Linear Algebra with a grade of ?C? or higher.

**MATH 6331 Algebra I**

This course is an extension of the undergraduate course in abstract algebra. Topics include polynomial rings over a field and finite field extensions.

Prerequisite: MATH 3363 Modern Algebra I with a grade of ?C? or higher.

**MATH 6331 Algebra I**

This course is an extension of the undergraduate course in abstract algebra. Topics include polynomial rings over a field and finite field extensions.

Prerequisite: MATH 3363 Modern Algebra I with a grade of ?C? or higher.

**MATH 6332 Algebra II**

The purpose of this course is to provide essential background in groups, rings and fields, train the student to recognize algebraic structures in various settings and apply the tools and techniques made available by algebraic structures. Topics include groups, structure of groups, rings, modules, Galois theory, structure of fields, commutative rings and modules.

Prerequisite: MATH 6331.

**MATH 6333 Statistical Learning**

This course introduces the statistical methods for supervised and unsupervised learning, including topics of regression and classification, such as linear regression, multiple regression, logistic regression, K-nearest neighbors, polynomial regression, splines regression, tree regression, random forests, ridge regression and the Lasso, linear and quadratic discriminant analysis, support vector machines, artificial neural networks regularization techniques, and boosting techniques. During the course, we will apply these techniques in several case studies.

Prerequisite: Consent of instructor

**MATH 6339 Complex Analysis**

This course is an introduction to the fundamentals of complex analysis. Topics include: The Riemann sphere and stereographic projection, elementary functions, analytic functions, the theory of complex integration, power series, the theory of residues, the Cauchy-Riemann equations, conformal and isogonal diffeomorphisms, Weierstrass products, the Mittag-Leffler theorem.

Prerequisite: Departmental approval.

**MATH 6352 Analysis I**

The purpose of this course is to provide the necessary background for all branches of modern mathematics involving analysis and to train the student in the use of axiomatic methods. Topics include metric spaces, sequences, limits, continuity, function spaces, series, differentiation and the Riemann integral.

Prerequisite: MATH 3372 Real Analysis I with a grade of ?C? or higher.

**MATH 6353 Analysis II**

The purpose of this course is to present advanced topics in analysis. Topics may be chosen from (but not restricted to) normed linear spaces, Hilbert spaces, elementary spectral theory, complex analysis, measure and integration theory.

Prerequisite: MATH 6352.

**MATH 6360 Ordinary Differential Equations**

This course examines existence and uniqueness theorems, methods for calculating solutions to systems of ordinary differential equations, the study of algebraic and qualitative properties of solutions, iterative methods for numerical solutions of ordinary differential equations and an introduction to the finite element methods.

Prerequisite: MATH 3341 Differential Equations with a grade of ?C? or higher, or consent of instructor.

**MATH 6361 Partial Differential Equations**

This course considers waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations. Topics include: first-order equations: characteristic ODEs, local existence of smooth solutions, conservation law equations, shocks, rarefaction, integral solutions; second-order partial differentail equations and clssification; Wave equation: fundamental solutions in one, two and three dimensions, Duhamel's principle, energy methods, finite propagation speed; Laplace equation: mean-value property, smoothness, maximum principle, uniqueness of solutions, Hamack inequality, Liouville theorem; Poisson Equation: fundamental solution, Greens functions, energy methods. Heat Equation: fundamental solution, maximum principle, uniqueness of solutions on a bounded domain, energy methods.

Prerequisite: MATH 3341 Differential Equations with a grade of ?C? or higher, or consent of instructor.

**MATH 6362 Fourier Analysis**

The course includes trigonometric series and Fourier Series, Dirichlet Integral, convergence and summability of Fourier Series, uniform convergence and Gibbs phenomena, L2 space, properties of Fourier coefficients, Fourier transform and applications, Laplace transform and applications, distributions, Fourier series of distributions, Fourier transforms of generalized functions and orthogonal systems.

Prerequisite: MATH 6353 or consent of instructor.

**MATH 6363 Integrable Systems**

This course includes solitons and integrable systems. The purpose of the course is to show students how to analyze nonlinear partial differential equations for physical problems and how to solve the equations using traveling wave settings.

Prerequisite: MATH 3349 with a grade of ?C? or higher or consent of instructor.

**MATH 6364 Statistical Methods**

This is a course in the concepts, methods and usage of statistical data analysis. Topics include test of hypotheses and confidence intervals; linear and multiple regression analysis; concepts of experimental design, randomized blocks and factorial analysis; a brief introduction to non-parametric methods; and the use of statistical software.

Prerequisite: Departmental approval.

**MATH 6365 Probability and Statistics**

Topics in this course include set theory and concept of probability, conditional probability, random variables, discrete and continuous probability distributions, distribution and expectations of random variables, moment generating functions, transformation of random variables, order statistics, central limit theorem and limiting distributions.

Prerequisite: MATH 2415 Calculus III with a grade of ?C? or higher, or consent of instructor.

**MATH 6366 Micro-local Analysis**

Topics include: basic concepts and computational technique of distributions (generalized functions, the singular support of distributions, the convolutions of distributions, the structure of distributions, approximations by test functions, Schwartz space; Fourier transforms of test functions and distributions, Paley-Wiener theorem. Schwartz kernel theorem. Sobolev spaces, symbols, pseudo-differential operators (PDOs), the kernel of pseudo-local operators, PDOs and Sobolev spaces, amplitude functions and PDOs, transposed and adjoint to PDO operators. Proper PDOs, Product of PDOs, asymptotic series and expansions, product formula for PDOs, symbols of transposed and adjoint operators, symbol of composition and commutator of PDOs, elliptic operators, wave front set of distributions; Fourier integral operators.

Prerequisite: Departmental approval.

**MATH 6367 Functional Analysis**

This course provides an introduction to methods and applications of functional analysis. Topics include: topological vector spaces; locally convex spaces (Hahn-Banach Theorem, weak topology, dual pairs); normed spaces; theory of distributions (space of test functions, convolution, Fourier transform; Sobolev spaces); Banach spaces (Uniform Boundedness Principle, Open Mapping Theorem, Closed Graph Theorem and applications, Banach-Alaoglus Theorem, Krein-Milman Theorem); C(X) as a Banach space (Stone-Weierstrass Theorem, Riesz Theorem, compact operators); Hilbert spaces; linear operators on Hilbert spaces; eigenvalues and eigenvectors of operators.

Prerequisite: MATH 3372 Real Analysis I with a grade of ?C? or higher or consent of instructor.

**MATH 6369 Mathematical Methods**

Special functions, perturbation methods, asymptotic expansion, partial differential equation models, existence and uniqueness, integral transforms, Green functions for ODEs and PDEs, calculus of variations, methods of least squares, Ritz-Rayleigh and other approximate methods, integral equations, generalized functions.

Prerequisite: Graduate standing, and MATH 2415 Calculus III with a grade of "C" or higher.

**MATH 6370 Topology**

This course is a foundation for the study of analysis, geometry and algebraic topology. Topics include set theory and logic, topological spaces and continuous functions, connectedness, compactness, countability and separation axioms.

Prerequisite: MATH 4355 Topology with a grade of "C" or higher, or consent of instructor.

**MATH 6371 Differential Geometry**

The course will introduce students to the study of smooth manifolds, fiber bundles, differential forms, and Lie groups. Thereafter, Euclidean geometries and their common generalizations Klein and Riemannian geometries will be discussed with a focus on examples. If time allows the unifying notion of a Cartan geometry will also be introduced.

Prerequisite: MATH 6352 or consent of instructor.

**MATH 6373 Algebraic Geometry**

The course will begin with an introduction to polynomials and ideals, Grobner bases, and affine varieties. This includes the Hilbert Basis Theorem, the Nullstellensatz, and the ideal-variety correspondence. Thereafter, the course will focus on examples and computations. Topics include solving systems by elimination, resultants, computations in local rings, modules and syzygies, and polytopes and toric varieties.

Prerequisite: MATH 6331 or consent of instructor.

**MATH 6375 Numerical Analysis**

This course provides a fundamental introduction to numerical techniques used in mathematics, computer science, physical sciences and engineering. The course covers basic theory on classical fundamental topics in numerical analysis such as: computer arithmetic, approximation theory, numerical differentiation and integrations, solution of linear and nonlinear algebraic systems, numerical solution of ordinary differential equations and error analysis of the abovementioned topics. Connections are made to contemporary research in mathematics and its applications to the real world.

Prerequisite: MATH 2318 Linear Algebra and 2415 Calculus III with grades of ?C? or better; and computer programming; or consent of instructor.

**MATH 6376 Numerical Methods for Partial Differential Equations**

This course provides a fundamental introduction to numerical techniques used in mathematics, computer science, physical sciences and engineering. The course covers basic theory and applications in the numerical solutions of elliptic, parabolic and hyperbolic partial differential equations.

Prerequisite: MATH 2318 Linear Algebra, 2415 Calculus III and MATH 3349 Numerical Methods with ?C? or better or graduate-level Numerical Analysis with a ?B? or better, some familiarity with ordinary and partial differential equations and computer programming or consent

**MATH 6377 Mathematical Fluid Mechanics**

This course provides an introduction to fundamental aspects of mathematical fluid mechanics. Topics include classification of fluids, flow characteristics, dimensional analysis, derivations of Euler, Bernoulli and Navier-Stokes equations, complex analysis for two-dimensional potential flows, exact solutions for simple cases of flow such as plane Poiseuille flow, and Couette flow.

Prerequisite: Departmental approval.

**MATH 6378 Inverse Problem and Image Reconstruction**

Topics include: inverse problem of linear PDEs, Maxwell equation and Fourier integral operator, Back-projection operator and applications in radar image reconstruction, including synthetic aperture radar image and inverse synthetic aperture radar images arranged by antenna.

Prerequisite: Consent of instructor

**MATH 6385 Cryptology and Codes**

Topics include: elementary ciphers, error-control codes, public key ciphers, random number generator- error codes, and Data Encryption Standard. Supporting topics from number theory, linear algebra, group theory, and ring theory will also be studied.

Prerequisite: MATH 3363 Modern Algebra I with a grade of ?C? or better.

**MATH 6388 Discrete Mathematics**

This course is an introduction to modern finite mathematics. Topics include methods of enumeration, analytic methods, generating functions, the theory of partitions, graphs, partially ordered sets, and an introduction to Polya's theory of enumeration.

Prerequisite: MATH 3363 Modern Algebra I or consent of instructor.

**MATH 6391 Master's Project**

Individual work or research on advanced mathematical problems conducted under the direct supervision of a faculty member.

Prerequisite: Consent of instructor.

**MATH 6399 Special Topics in Mathematics**

This course covers special topics in graduate level mathematics that are not taught elsewhere in the department. May be repeated for credit when topic is different.

Prerequisite: Consent of instructor

**MATH 7100 Thesis Extension**

This course is to be taken by those requiring an extension to complete a Masters Thesis after taking the two course thesis sequence.

Prerequisite: Graduate standing and consent of thesis advisor.

**MATH 7300 Thesis I**

First part of two course sequence.

Prerequisite: Graduate standing and consent of thesis advisor.

**MATH 7301 Thesis II**

Second part of two course sequence.

Prerequisite: Graduate standing and consent of thesis advisor.

**MATH 8323 Representation Theory**

This course will cover the key concepts in the representation theory of finite groups, Lie groups, and Lie algebras. First representations and characters of finite groups will be introduced, with emphasis on the symmetric group, covering results such as the Frobenius formula. Then, Lie groups and Lie algebras will be introduced, and the theory of their representations will be built to cover the Weyl Character formula and Cartan's classification of simple complex Lie algebras. There will be an emphasis on examples and interdisciplinary applications.

Prerequisite: Consent of Instructor.

**MATH 8329 Analytic Number Theory**

This course is an introduction to algebraic and analytic number theory, as well as their applications. Topics include algebraic number fields, ideal class groups, Dedekind domains, elliptic curves, Dirichlet series, the prime number theorem, and cryptography.

Prerequisite: Consent of Instructor

**MATH 8330 Advanced Linear Algebra**

Topics include the proof-based theory of matrices, determinants, vector spaces, linear maps, dual spaces, linear systems, linear operators, multilinear maps, eigenvalues and eigenvectors, invariant subspaces of operators, spectral decompositions, functions of operators, and applications to science, industry, and business.

Prerequisite: Consent of Instructor.

**MATH 8331 Abstract Algebra**

This is a first graduate course in abstract algebra that emphasizes a rigorous approach to concepts and proofs. Topics will include a review of groups, and will then focus on rings, fields, and modules, including Euclidean rings, principal ideal domains, unique factorization domains, as well as polynomial rings over a field and finite field extensions.

Prerequisite: Consent of Instructor

**MATH 8332 Commutative Algebra**

Commutative Algebra studies commutative rings, their ideals, and modules over such rings. This field relates to several other areas of mathematics, such as algebraic geometry and algebraic number theory. Topics covered will include ideals for Noetherian and Artinian rings, the Hilbert Basis Theorem and the Nullstellensatz, an introduction to affine algebraic geometry, Dedekind domains, and dimension theory.

Prerequisite: Consent of Instructor.

**MATH 8333 Advanced Statistical Learning**

This course is a survey of topics on fundamental theory and applications of statistical methods for supervised and unsupervised learning, including multiple linear and logistic regression models, discriminant analysis, regression splines, generalized additive models, model selection and regularization methods (ridge and lasso), tree-based methods, random forests, bagging and boosting, support vector machines, artificial neural networks techniques, principal component analysis, factor analysis, k-means, and hierarchical clustering.

Prerequisite: Consent of Instructor.

**MATH 8334 Machine Learning**

This course is an introduction to fundamental concepts in Machine Learning, including clustering, regression, classification, association rules mining, and time series analysis.

Prerequisite: Consent of instructor.

**MATH 8335 Deep Learning**

This course aims to present the mathematical, statistical and computational challenges of building stable representations for high-dimensional data, such as images, text and data. Topics include recent models from both supervised and unsupervised learning, convolutional architectures, invariance learning, and non-convex optimization.

Prerequisite: Consent of instructor.

**MATH 8336 Introduction to Data Science**

This course provides an overview of Data Science, covering a broad selection of key challenges in and methodologies for working with big data. Topics include data collection, integration, management, modeling, analysis, visualization, prediction and informed decision making, as well as data security and data privacy. This introductory course is integrative across the core disciplines of Data Science, including databases, data warehousing, statistics, data mining, data visualization, high performance computing, cloud computing, and business intelligence.

Prerequisite: Consent of Instructor

**MATH 8337 Information Theory**

This course covers conditional Shannon entropy, mutual information, entropy diagrams, data compression, channel coding, Gaussian channel and its capacity, Information theory and statistics, information-theoretic security, and encryption.

Prerequisite: Consent of instructor.

**MATH 8338 Mathematical Foundations of Statistical and Quantum Mechanics**

This course will introduce students to key concepts in the mathematics of statistical and quantum mechanics. Topics covered will include Hilbert spaces, linear operators, Poisson algebras, canonical quantization, and measure spaces. Applications will include the quantum harmonic oscillator, statistical ensembles, partition functions, Bose-Einstein statistics, entropy, and ergodic theorems.

Prerequisite: Consent of instructor.

**MATH 8339 Advanced Complex Analysis**

The focus of this course is on the study of holomorphic functions and their most important basic properties. Topics include: Complex numbers and functions; complex limits and differentiability; analytic functions; complex line integrals; Cauchy's theorem and the Cauchy integral formula; Taylor's theorem; zeros of holomorphic functions; Rouche's Theorem; the Open Mapping theorem and Inverse Function theorem; Schwarz' Lemma; automorphisms of the ball, the plane and the Riemann sphere; isolated singularities and their classification; Laurent series; the Residue Theorem; calculation of definite integrals and evaluation of infinite series using residues. An overview of theory of harmonic functions will be covered, including the Laplacian; relation to analytic functions; conjugate harmonic functions; Dirichlet problem; and applications. Additional topics such as the Gamma and Zeta functions and the prime number theorem may be included. Application of methods of complex analysis in the course include propagation of acoustic waves relevant for the design of jet engines, problems arising in solid and fluid mechanics, as well as conformal geometry in imaging, shape analysis and computer vision.

Prerequisite: Consent of Instructor.

**MATH 8343 Linear Models**

This course introduces linear models with and without random effects as well as linear mixed models. Full and non-full rank models will be addressed in detail. Topics include simple and multiple regression, analysis of variance for factorial and block designs, generalized linear models, and generalized linear mixed models. The necessary statistical analyses will be done using statistical software such as SAS, R, etc.

Prerequisite: Consent of instructor.

**MATH 8344 Function Space Methods in System Theory**

This is a mathematics course on the theory and methods of functional analysis for the modeling and analysis of systems. Topics include vector spaces, normed linear and Hilbert spaces, linear operators, optimization of functionals, and duality.

Prerequisite: Consent of the instructor.

**MATH 8346 Hydrodynamic Stability**

This course covers basic concepts in hydrodynamic Stability such as stability system, Kelvin-Helmholtz instability, derivation of linear stability system, Rayleigh-Taylor instability, shear instability, break up of a liquid jet, parallel shear flows, inviscid theory, Squire's transformations, Rayleigh stability system, Rayleigh's inflexion-point theorem, viscous instability, the Orr-Sommerfeld equation, critical Reynolds number for simple flows, temporal and spatial instabilities, thermal instability, flow patterns, bifurcation theory, limit cycle and attractors, the Landau nonlinear equation and solution, secondary instability, transition to turbulence, and experimental and technological applications.

Prerequisite: Consent of instructor.

**MATH 8347 Turbulence**

Laminar & turbulence flows, flow instabilities and transition to turbulence, mathematical description of turbulence, methods of Taking averages, concept of ergodicity, random fields of fluid dynamic variables, statistical specification of turbulence, mean values of velocity products, velocity correlation and spectrum tensors, vorticity correlation and spectrum tensors, dynamics of decay, equilibrium theory, energy-containing eddies, probability distribution of velocity vector, experimental aspects, semi-empirical theories, technological applications.

Prerequisite: Consent of the instructor

**MATH 8348 Survival Analysis**

Applied statisticians in many fields must frequently analyze time to event data. The focus of this course is on applications of the techniques to biology and medicine. However, the statistical tools discussed in this course are more broadly applicable to data from medicine, biology, public health, epidemiology, engineering, economics, and demography. This course presents models and statistical methods for the analysis of recurrent event data. The analysis is complicated by issues of censoring, where an individual's life length is known to occur only in a certain period or truncation, where individuals enter the study only if they survive a sufficient length of time or if the event has occurred by a given date. Both parametric and nonparametric models are included with procedures for estimation, testing, and model checking.

Prerequisite: Consent of instructor.

**MATH 8349 Loss Models**

The goal of this course to gain a conceptual understanding and practical facility with the statistical/probabilistic techniques that form the syllabus for the SOA exam, Construction and Evaluation of Actuarial Models. Topics include Frequency models, aggregate models, Severity models, construction of empirical models, construction and selection of parametric models, Bayesian estimation, estimating failure time and loss, determining the acceptability of a fitted model, credibility, and simulation.

Prerequisite: Consent of the instructor

**MATH 8350 Actuarial Risk Theory**

This course introduces the main topics in actuarial risk theory such as utility theory and insurance, individual risk models, collective risk models, ruin theory, premium principles, risk measures, credibility theory, and R in actuarial risk theory.

Prerequisite: Consent of the instructor

**MATH 8351 Nonlinear hyperbolic PDEs**

The course will introduce students to the fundamental solutions for hyperbolic partial differential equations with variable coefficients. Topics include applications to the problem of global in time existence of solutions of the Cauchy problem for nonlinear equations.

Prerequisite: Consent of instructor.

**MATH 8352 Advanced Analysis I**

The purpose of this course is to provide the necessary background for all branches of modern mathematics involving analysis and to train the students in the use of axiomatic methods. The course includes notions from set theory, the real number system, metric spaces, continuous functions, differentiation, Riemann integration, interchange of limit operations, the method of successive approximations, partial differentiation, and multiple integrals.

Prerequisite: Consent of Instructor

**MATH 8353 Measure Theory**

The purpose of this course is to present Lebesgue's theory of integration and its applications. The concept of measure spaces will be introduced, and topics including Carathéodory's extension theorem, Borel and Lebesgue measures will be covered. The Lebesgue integral and measurable functions will be defined, together with various properties, such as the convergence theorems, Lusin's theorem and Egorov's theorem. Further topics will include Fubini's Theorem, the Radon-Nikodym Theorem, Lebesgue Differentiation Theorem, and L^p spaces. Applications will include Fourier Transforms and their properties.

Prerequisite: Consent of Instructor.

**MATH 8360 Advanced Ordinary Differential Equations**

The topics in this course include existence and uniqueness theorems; methods for calculating solutions to systems of nonlinear ordinary differential equations (ODEs) and dynamical systems (DSs) relevant to applications in various areas; iterative methods for numerical solutions of ODES and DSs; and finite element methods combined with machine learning and deep learning.

Prerequisite: Consent of Instructor

**MATH 8361 Advanced Partial Differential Equations**

This course considers waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations. Topics include: first-order equations: characteristic ODEs, existence of smooth solutions, conservation law equations, shocks, rarefaction, integral solutions; second-order partial differential equations and classification; Wave equation: fundamental solutions in one, two and three dimensions, Duhamel's principle, energy methods, finite propagation speed; the Laplace equation: mean-value property, smoothness, maximum principle, uniqueness of solutions, Harnack's inequalities, Liouville's theorem; fundamental solution to the Poisson Equation, Green's functions, energy methods. the fundamental solution to the heat equation, the maximum principle, uniqueness of solutions on a bounded domain, and energy methods. In addition, the theory of second order linear PDEs will be covered, including the existence of weak solutions, regularity, maximum principles. The courses will include fixed point methods, and method of subsolutions and supersolutions. The PDE models considered in the course appear in physical models and have numerous applications in physics and engineering.

Prerequisite: Consent of instructor.

**MATH 8362 Advanced Fourier Analysis**

Topics in this course include orthonormal systems, Fourier series, continuous and discrete Fourier transform theory, generalized functions, and an introduction to spectral analysis. Applications are focused on physical problems, linear systems theory, signal processing, and some recent developments in Fourier analysis.

Prerequisite: Consent of instructor.

**MATH 8363 Solitons and Integrable Models**

The purpose of this course is to demonstrate how to construct soliton and multi-soliton solutions to nonlinear integrable models appearing in physics and other nonlinear sciences and how to test integrability using the Lax pair approach. Research topics include solitary wave solutions, multi-soliton solutions, peakon and cuspon solutions, Lax pairs, Poisson brackets, symplectic structures, and canonical Hamiltonian structures.

Prerequisite: Consent of instructor.

**MATH 8364 Advanced Statistical Methods**

This is a course in the concepts, methods and usage of statistical data analysis. Topics include test of hypotheses and confidence intervals; statistical inference illustrated with general and generalized linear models, diagnostics for regression models; concepts of longitudinal and clustered data analysis; concepts of experimental design, randomized blocks and factorial analysis; a brief introduction to non-parametric methods; and the use of statistical software.

Prerequisite: Consent of Instructor.

**MATH 8365 Advanced Probability & Statistics**

Topics in this course include conditional probability, random variables, discrete and continuous probability distributions, distribution and expectations of random variables, moment generating functions, the transformation of random variables, order statistics, central limit theorem, and limiting distributions. The course also covers measure theory and foundations of probability theory, zero-one laws, probability inequalities, weak and strong laws of large numbers, the central limit theorems, use of characteristic functions, introduction to martingales and random walks, modes of convergence, and the asymptotic properties of estimators and tests.

Prerequisite: Consent of Instructor.

**MATH 8366 Advanced Microlocal Analysis**

This course provides an introduction to methods and applications of Microlocal Analysis. Microlocal Analysis is an instrument that was developed to apply Fourier transform to solve partial differential equations and carry out a qualitative analysis of the solutions. Topics include: basic concepts and computational technique of distributions, generalized functions, the local theory of distributions, the singular support of distributions, the convolutions of distributions, the structure of distributions, approximations by test functions, Schwartz space; Fourier transforms of test functions and distributions, Paley-Wiener theorem, Schwartz kernel theorem, Sobolev spaces, symbols, pseudo-differential operators (PDOs), the kernel of pseudo-local operators, PDOs and Sobolev spaces, amplitude functions and PDOs, transpose and adjoint of PDO operators, proper PDOs, product of PDOs, asymptotic series and expansions, product formula for PDOs, symbols of transposed and adjoint operators, symbol of composition and commutator of PDOs, elliptic operators, wave front set of distributions, propagation of singularities, Fourier integral operators, applications to elliptic and parabolic partial differential equations on manifolds, and propagation of singularities for hyperbolic partial differential equations.

Prerequisite: Consent of instructor.

**MATH 8367 Advanced Functional Analysis**

This course provides an introduction to methods and applications of functional analysis. Topics include: topological vector spaces, locally convex spaces, Hahn-Banach Theorem, weak topology, dual pairs, normed spaces, theory of distributions, space of test functions, convolution, Fourier transform, Sobolev spaces, Banach spaces, Uniform Boundedness Principle, Open Mapping Theorem, Closed Graph Theorem and applications, Banach-Alaoglus Theorem, Krein-Milman Theorem, Stone-Weierstrass Theorem, Riesz Theorem, compact operators, Hilbert spaces, linear operators on Hilbert spaces, eigenvalues and eigenvectors of operators, spectral theorem and functional calculus for compact normal operators, unbounded self-adjoint and symmetric operators and their spectral decomposition, Cayley transform, unbounded normal operators and the spectral theorem, and Fredholm Theory.

Prerequisite: Consent of instructor.

**MATH 8369 Mathematical Methods in Applied Sciences**

This course covers topics from classical and modern mathematical theories and methods for applied sciences.

Prerequisite: Consent of instructor.

**MATH 8371 Differential Geometry**

The course will introduce the study of smooth manifolds, fiber bundles, connections on bundles, differential forms, Lie groups, and Riemannian manifolds. This will be followed by integration on manifolds and Stoke' theorem, as well as de Rham cohomology. Finally, characteristic classes and aspects of Chern-Weil theory will be introduced. Applications will include topics in gauge theory.

Prerequisite: Consent of Instructor.

**MATH 8374 Applications of Differential Geometry**

This course develops Riemannian and Pseudo-Riemannian geometry, the concept of curvature of a manifold, and mathematical aspects of General Relativity. Topics will include linearized gravity, introduction to gravitational waves, black hole solutions, and cosmological solutions.

Prerequisite: Consent of instructor.

**MATH 8375 Advanced Numerical Analysis**

This course introduces numerical techniques used in mathematics, computer science, physical sciences, and engineering. The course covers basic theory on classical fundamental topics in numerical analysis such as computer arithmetic, conditioning, and stability, direct methods for solving systems of linear equations including LU, Cholesky, QR, and SVD factorization, matrix eigenvalue problems, solutions to nonlinear equations, interpolation of polynomials, numerical differentiation and integration, numerical solutions of ordinary differential equations, Monte Carlo methods and error analysis of the above-mentioned topics. Connections are made to contemporary research in mathematics and its applications. Familiarity with computer programming is required.

Prerequisite: Consent of Instructor.

**MATH 8376 Numerical Methods for Differential Equations**

This course provides a fundamental introduction to numerical techniques used in mathematics, computer science, physical sciences, and engineering. The course covers basic theory and applications in the numerical solutions of elliptic, parabolic and hyperbolic partial differential equations. Computer programming assignments form an essential part of the course. The course introduces students to numerical methods for (1) ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, convergence and stability; (2) finite difference, finite element, and integral equation methods for elliptic partial differential equations; (3) spectral methods and the fast Fourier transform, exponential temporal integrators, and multigrid iterative solvers; and (4) finite difference and finite volume parabolic and hyperbolic partial differential equations.

Prerequisite: Consent of Instructor.

**MATH 8377 Advanced Fluid Mechanics**

This course provides theoretical aspects of mathematical fluid mechanics and applications. Topics include classification of fluids, flow characteristics, dimensional analysis, derivations of Euler, Bernoulli, and Navier-Stokes equations, complex analysis for two-dimensional potential flows, exact solutions for simple cases of flow such as plane Poiseuille flow, and Couette flow. The course also introduces mathematical methods applied to problems in fluid dynamics. Particular attention is given to the power of dimensional analysis and scaling arguments. Topics also include particle motion, flow kinematics, conservation laws and vorticity, boundary layers and asymptotic models, and water waves. Course material may be supplemented by classroom and laboratory demonstrations.

Prerequisite: Consent of Instructor.

**MATH 8378 Advanced Inverse Problems and Image Reconstruction**

This course focuses on inverse problems and image reconstruction. Topics include the inverse problem of linear PDEs, the Maxwell equation and the Fourier integral operator, Back-projection operator, and applications in radar image reconstruction, including synthetic aperture radar (SAR) image and inverse synthetic aperture radar images arranged by antenna. SAR systems, algorithms, and simulations will be covered.

Prerequisite: Consent of Instructor.

**MATH 8379 Advanced Stochastic Processes**

This is an advanced course in stochastic processes. The course includes discrete-time Markov chains, continuous-time Markov processes, Poisson processes, branching stochastic processes, elements of queuing theory, renewal and regenerative processes, diffusion processes, Brownian motion, Ito integration, and some recent developments in the stochastic areas.

Prerequisite: Consent of Instructor.

**MATH 8381 Advanced Mathematical Statistics**

This course includes topics in advanced mathematical statistics such as sampling distributions, sufficient statistics, theory of estimation, point estimation, minimum variance estimation, Rao-Cramer inequality, maximum likelihood estimation, interval estimation, Bayes estimators, credible intervals, Neyman-Pearson theory of hypothesis testing, most powerful tests, likelihood ratio tests, chi-square tests, Bayesian hypothesis testing, simple and multiple regression analysis, and some recent developments in this area.

Prerequisite: Consent of Instructor.

**MATH 8382 Advanced Statistical Computing**

This is a course in modern computationally intensive statistical methods including simulation, optimization methods, Monte Carlo integration, maximum likelihood /EM parameter estimation, Monte Carlo simulations, Markov chain Monte Carlo methods, resampling methods, non-parametric density estimation, writing functions using R, and some recent developments in this area. Topics include power and sample size determinations.

Prerequisite: Consent of Instructor.

**MATH 8384 Advanced Biostatistics**

This course is a survey of crucial topics in biostatistics; application of regression in biostatistics; analysis of correlated data; logistic and Poisson regression for binary or count data; design and analysis of clinical trials; sample size calculation by simulation; bootstrap techniques for assessing statistical significance; introduction to high-throughput data analysis, design of high-throughput experiments, and some recent developments in biostatistics.

Prerequisite: Consent of Instructor.

**MATH 8385 Advanced Cryptology & Codes**

This course covers secure communications and related topics. Topics include classic cryptography, public-key ciphers and RSA, linear codes, error-correcting codes, decoding algorithms, introduction to elliptic-curve ciphers and codes. Supporting topics from number theory, group theory, linear algebra, and discrete geometry will also be studied.

Prerequisite: Consent of Instructor.

**MATH 8387 Advanced Mathematical Modeling**

This course presents the theory and application of mathematical modeling. Topics include dynamic models, stable and unstable motion, stability of linear and nonlinear systems, Lyapunov functions, feedback, growth and decay, the logistic model, population models, cycles, bifurcation, catastrophe, biological and biomedical models, chaos, strange attractors, deterministic and random behavior.

Prerequisite: Consent of Instructor.

**MATH 8388 Advanced Discrete Mathematics and Combinatorics**

This course is an introduction to modern discrete mathematics and its applications. Topics include methods of enumeration, analytic methods, generating functions, the theory of partitions, graphs and trees, partially ordered sets, introduction to probabilistic, algebraic and topological methods in discrete mathematics and combinatorics.

Prerequisite: Consent of Instructor.

**MATH 8398 Interdisciplinary Course**

This course will allow students to gain interdisciplinary background in a field of study taught by faculty outside of the School of Mathematical and Statistical Sciences. The coursework will include contributions from academic disciplines such as physics, engineering, computer science, biology, biomedical science, environmental science, or economics, that build knowledge requisite for research in multiple areas.

Prerequisite: Graduate standing and consent of the instructor or program director

**MATH 8399 Advanced Topics in Mathematics and Statistics**

This course covers special topics in the program not taught elsewhere in the school. The topics could be chosen from any area out of the four concentrations, including Applied Mathematics, Computational Mathematics, Mathematical Physics, Data Science, Applied Statistics, Nonlinear Mechanics, Mathematical Biology, Computer Science, Computer Engineering. The course emphasis is at the discretion of the student and advisor and supports specialized work on the student's Ph.D. dissertation. May be repeated for credit when the topic varies.

Prerequisite: Consent of Instructor

**MATH 9101 Graduate Research Seminar**

The MSIA Student Seminar is a sequence of courses taken by all MSIA Ph.D. students (MSIA M.S. students will join the course for the first two semesters of study). This course could be repeated upon the student's needs before taking MSIA Dissertation course. It is specifically designed to assist with many of the unique challenges confronting MSIA graduate students. For Ph.D. students, one of these challenges is the choice of a dissertation committee that includes two different co-advisors, one from mathematics and one from another partner discipline. Another challenge common to the interests of both M.S. and Ph.D. students is the development of a sound understanding of the way that mathematics plays a role in diverse application areas. MSIA 9101 is a course only for MSIA graduate students. It is coordinated with the SMSS Colloquium Research Seminar, and students are required to participate in the research seminar on a weekly basis as a course requirement. Students will therefore meet once each week during the regular course time and again during the time of the research seminar.

Prerequisite: Consent of the instructor or program director

**MATH 9901 Dissertation I**

The dissertation course is the first part of the capstone of the MSIA Ph.D. program with a rewarding process of carrying out the interdisciplinary research in collaboration with the two co- advisors, one is in mathematics or statistics and the other one in interdisciplinary sciences. When both co-advisors are satisfied with the quantity and quality of the completed research, the student must write a dissertation. The dissertation is a document that describes the entire research project, including scholarly background, methodology of research, and original results. The dissertation must be prepared in accordance with the regulations in the Dissertation Manual published by the UTRGV Graduate College

Prerequisite: MSIA 9101 Graduate Seminar or consent of the instructor

**MATH 9902 Dissertation II**

The Dissertation II course is a continuation of the Dissertation I and the second part of the capstone of the MSIA Ph.D. program. The student must closely work with both co-advisors about the research topics toward completing a dissertation. The dissertation must be prepared in accordance with the regulations in the Dissertation Manual published by the UTRGV Graduate College. The dissertation must be defended before the dissertation committee consisting of at least five faculty members from mathematics/statistics and interdisciplinary disciplines.

Prerequisite: MSIA 9901 Dissertation I