**Math 6020/7240 Mathematical Statistics (3)**

*Prerequisites: Math 6070, 6080 and 7150 or permission of the instructor*

Classical parametric families of distributions. Characteristic functions. The distribution of functions of a random variable, the Jacobian method. The Method of Moments. Exponential families and location-and-scale families. Sufficient, minimal, ancillary, complete statistics. Factorization theorem, conditions for minimality or completeness. Basu's theorem. General conditions for existence and uniqueness of the ML estimator. UMVU estimation: risk and loss functions, Information Inequalities. Fisher information, uniqueness of the UMVU estimator. Rao-Blackwellization of estimators. Risk-function-based optimality and sufficiency. Hypothesis testing and the Neyman-Pearson lemma. Optimality of tests: UMP and UMPU tests. Generalized Likelihood Ratio tests. Asymptotics of the likelihood ratio tests. Convergence in law and in probability, the Delta Method. Consistent roots for the likelihood equation. Asymptotic efficiency.

**Math 6030/7030 Stochastic Processes (3)**

*Prerequisite: Math 3070/6070, 3080/6080.*

Markov processes, Poisson processes, queuing models, introduction to Brownian motion.

**Math 6040/7260 Linear Models (3)**

*Prerequisite: Math 3070/6070, 3080/6080.*

Corequisite: Math 3090 or approval of instructor.

Review of linear algebra pertinent to least squares regression. Review of multivariate normal, chi-square, t, F distributions. Classical theory of linear regression and related inference. Regression diagnostics. Extensive practice in data analysis.

**Math 6050/3050 Real Analysis I (3)**

*Prerequisite: Math 2210.*

Introduction to analysis. Real numbers, limits, continuity, uniform continuity, sequences and series, compactness, convergence, Riemann integration. An in-depth treatment of the concepts underlying calculus.

**Math 6070/3070 Introduction to Probability (3)**

*Prerequisite: Math 2210 or equivalent.*

An introduction to probability theory. Counting methods, conditional probability and independence. Discrete and continuous distributions, expected value, joint distributions and limit theorems. Prepares student for future work in probability and statistics.

**Math 6080/3080 Introduction to Statistical Inference (3)**

*Prerequisite: Math 2210, Math 3070.*

Basics of Statistical inference. Sampling distributions, parameter estimation, hypothesis testing, optimal estimates and tests. Maximum likelihood estimates and likelihood ratio tests. Data summary methods, categorical data analysis. Analysis of variance and introduction to linear regression.

**Math 6090/3090 Linear Algebra (4)**

*Prerequisite: Math 2210.*

An introduction to linear algebra emphasizing matrices and their applications. Gaussian elimination, determinants, vector spaces and linear transformations, orthogonality and projections, eigenvector problems, diagonalizability, Spectral Theorem, quadratic forms, applications. MATLAB is used as a computational tool.

**Math 6110/3110 Abstract Algebra (3)**

*Prerequisite: Math 2210.*

An introduction to abstract algebra. Elementary number theory and congruences. Basic group theory: groups, subgroups, normality, quotient groups, permutation groups. Ring theory: polynomial rings, unique factorization domains, elementary ideal theory. Introduction to field theory.

**Math 6210/4210 Differential Geometry (3)**

*Prerequisites: Math 6050 and 6090.*

Theory of plane and space curves including arc length, curvature, torsion, Frenet equations, surfaces in three-dimensional space. First and second fundamental forms, Gaussian and mean curvature, differentiable mappings of surfaces, curves on a surface, special surfaces.

**Math 6240/4240 Ordinary Differential Equations (3)**

*Prerequisites: Math 3090/6090.*

Review of linear algebra, first-order equations (models, existence, uniqueness, Euler method, phase line, stability of equilibria), higher-order linear equations, Laplace transforms and applications, power series of solutions, linear first-order, systems (autonomous systems, phase plane), application of matrix normal forms, linearization and stability of nonlinear systems, bifurcation, Hopf bifurcation, limit cycles, Poincare-Bendixson theorem, partial differential equations (symmetric boundary-value problems on an interval, eigenvalue problems, eigenfunction expansion, initial-value problems in 1D).

**Math 6250/4250 Mathematical Foundations of Computer Security (3)**

*Prerequisites: Calculus, Math 2170 and Math 3110 or the permission of the instructor.*

This course studies the mathematics underlying computer security, including both public key and symmetric key cryptography, crypto-protocols and information flow. The course includes a study of the RSA encryption scheme, stream and clock ciphers, digital signatures and authentication. It also considers semantic security and an analysis of secure information flow.

**Math 6280/3280 Information Theory (3, 3)**

*Prerequisites: Math 3050 or 3090 and familiarity with discrete probability.*

This introduction to information theory will address fundamental concepts, such as information, entropy, relative entropy, and mutual information. In addition to giving precise definitions of these concepts, the course will include a probabilistic approach based on equipartitions. Many of the applications of information will be discussed, including Shannon's basic theorems on channel capacity and related coding theorems. In addition to channels and channel capacity, the course will discuss applications of information theory to mathematics, statistics ,and computer science.

**Math 6310/3310 Scientific Computing (3)**

*Prerequisites: Math 2210, 2240, and Computer Science 1010 or equivalent.*

Errors. Curve fitting and function approximation, least squares approximation, orthogonal polynomials, trigonometric polynomial approximation. Direct methods for linear equations. Iterative methods for nonlinear equations and systems of nonlinear equations. Interpolation by polynomials and piecewise polynomials. Numerical integration. Single-step and multi-step methods for initial-value problems for ordinary differential equations, variable step size. Current algorithms and software.

**Math 6350 Numerical Optimization (3)**

*Prerequisite: Linear Algebra (MATH 3090 or equivalent) and the material in an elementary graduate numerical analysis course.*

The course will cover the derivation, analysis, and implementation of numerical methods for constrained and unconstrained optimization of continuous multivariate functions. The focus will be derivative and derivative-free algorithms using both direct-search and model-based approaches. The course will provide an efficient introduction to the fundamental ideas for the practicing engineer or scientist interested in optimizing functions that are expensive to evaluate. The students will be working in a MATLAB programming environment.

*Additional Information:*

The first term will focus on optimizing functions with fewer than 10 variables. The second term will focus on optimizing functions with many (100s to 1000s) of variables.

**Math 6370/7370: Time Series Analysis**

*Prerequisities: One course from MATH 6020/7240, MATH 6040/7260 or MATH 7360; one course from MATH 7150, MATH 7550, MATH 6050/3050 or MATH 6710/7210.*

This course provides an introduction to time series analysis at the graduate level. The course is about modeling based on three main families of techniques: (i) the classical decomposition into trend, seasonal and noise components; (ii) ARIMA processes and the Box and Jenkins methodology; (iii) Fourier analysis. If time permits, other possible topics include state space modeling and fractional processes. The course is focused on the theory, but some key examples and applications are also covered and implemented in the software package R.

**Math 6470/4470 Analytical Methods of Applied Mathematics (3)**

*Prerequisites: Math 2210 and 2240.*

Derivations of transport, heat/reaction-diffusion, wave. Poisson's equations; well-posedness; characteristics for first order PDE's; D'Alembert formula and conservation of energy for wave equations; propagation of waves; Fourier transforms; heat kernel, smoothing effect; maximum principles; Fourier series and Sturm-Liouville eigenexpansions; method of separation of variables; frequencies of wave equations, stable and unstable modes, long-time behavior of heat equations; delta function; fundamental solution of Laplace equation, Newton potential; Green's function and Poisson formula; Dirichlet Principle.

**Math 6550/7510, 6560/7520 Differential Geometry I, II (3, 3)**

Differential manifolds. Vector fields and flows. Tangent bundles. Frobenius theorem. Tensor fields. Differential forms, Lie derivatives. Integration and deRham's theorem. Riemannian metrics, connections, curvature, parallel translation, geodesics, and submanifolds, including surfaces. First and second variation formulas, Jacobi fields, Lie groups. The Maurer-Cartan equation. Isometries, principal bundles, symmetric spaces, Kähler geometry.

**Math 7010/6510, 7020/6520 Topology I and II (3, 3)**

*Prerequisites: Math 3050 and 4060.*

Point set topology. Connectedness, product and quotient spaces, separation properties, metric spaces. Classification of compact connected surfaces. Homotopy. Fundamental group and covering spaces. Singular and simplicial homology. Eilenberg-Steenrod axioms. Computational techniques, including long exact sequences. Mayer-Vietoris sequences, excision, and cellular chain complexes. Introduction to singular cohomology.

**Math 7110/6610, 7120 /6620 Algebra I and II (3, 3)**

*Prerequisites: Math 3090 and 3110.*

Vector Spaces: matrices, eigenvalues, Jordan canonical form.

Elementary Number Theory: primes, congruences, function, linear Diophantine equations, Pythagorean triples.

Group Theory: cosets, normal subgroups, homomorphisms, permutation groups, theorems of Lagrange, Cayley, Jordan-Hölder, Sylow, Finite abelian groups, free groups, presentations.

Ring Theory: prime and maximal ideals, fields of quotients, matrix and Noetherian rings.

Fields: algebraic and transcendental extensions, survey of Galois theory.

Modules and Algebras: exact sequences, projective and injective and free modules, hom and tensor products, group algebras, finite dimensional algebras.

Categories: axioms, subobjects, kernels, limits and colimits, functors and adjoint functors.

**Math 7150 Probability Theory I**

*Prerequisites: Math 3070/6070, Math 3080/6080 and Math 6050/3050*

This course provides a measure-theoretic and mathematically complete treatment of fundamental topics in Probability Theory at the graduate level. This includes probability measures, distributions, random variables and vectors, moments and conditional expectations, laws of large numbers, the different notions of convergence, characteristic functions, central limit theorems and conditional probability. One important difference between this course and Probability Theory II (Math 7550) is the fact that Measure Theory is not a prerequisite; instead, the necessary measure-theoretic concepts will be introduced in class. The course provides the adequate probabilistic background for students who plan on taking Mathematical Statistics (Math 6020/7240).

**Math 7210/6710, 7220/6720 Analysis I and II (3, 3)**

*Prerequisites: Math 3050, 3090, and 4060.*

Lebesgue measure on R. Measurable functions (including Lusin’s and Egoroff’s theorems). The Lebesgue integral. Monotone and dominated convergence theorems. Radon-Nikodym Theorem. Differentiation: bounded variation, absolute continuity, and the fundamental theorem of calculus. Measure spaces and the general Lebesgue integral (including summation and topics in R ⁿ such as the Lebesgue differentiation theorem). Lp spaces and Banach spaces. Hahn-Banach, open mapping, and uniform boundedness theorems. Hilbert space. Representation of linear functionals. Completeness and compactness. Compact operators, integral equations, applications to differential equations, self-adjoint operators, unbounded operators.

**Math 7260/6040 Linear Models (3)**

*Prerequisite: Math 6070, 6080.*

Corequisite: Math 6090 or approval of instructor.

Review of linear algebra pertinent to least squares regression. Review of multivariate normal, chi-square, t, F distributions. Classical theory of linear regression and related inference. Regression diagnostics. Extensive practice in data analysis.

**Math 7291 Algebraic Geometry I (3)**

This is the first semester of a second year course for graduate students with research interest in Algebraic Geometry and related areas. The course will give students a necessary background preparation for research in Algebraic Geometry or to read and understand papers in this area. Topics in this course include: affine and projective varieties, morphisms of varieties, nonsingular varieties, and category theory..

**Math 7292 Algebraic Geometry II (3)**

Courses on special topics list of subject titles include: Algebra, Analysis, Applied Math, Computation, Differential Equations, Geometry, Probability and Statistics, Theoretical Computer Science and Topology offered every year. Each course is designed to cover advanced material not included in one of the regular courses listed above.

**Math 7310, 7320 Applied Mathematics I, II (3, 3)**

This is a first year graduate course in Applied Mathematics. A solid working knowledge of linear algebra and advanced calculus is the necessary background for this class. The topics covered include a mix of analytical and numerical methods that are used to understand models described by differential equations. We will emphasize applications from science and engineering, as they are the driving force behind each of the topics addressed.

**Math 7350 Scientific Computing I (3)**

*Prerequisites: MATH 3310 or MATH 7310-7320.*

Introduction to numerical analysis: well-posedness and condition number, stability and convergence of numerical methods, a priori and a-posteriori analysis, sources of error in computational models, machine representation of numbers. Linear operators on normed spaces. Root finding for nonlinear equations. Polynomial interpolation. Numerical integration. Orthogonal polynomials in approximation theory. Numerical solution of ordinary differential equations.

**Math 7360 Data Analysis (3)**

*Prerequisites: Math 6070, 6080 and 6040/7260 (or equivalent background in mathematical statistics and linear models).*

This course covers the statistical analysis of data sets using the R software package. The R environment, an Open Source system based on the S language, is one of the most versatile and powerful tools available for statistical data analysis, and is widely used in both academic and industrial research. Key topics include graphical methods, generalized linear models, clustering, classification, time series analysis, and spatial statistics. No prior knowledge of R is required.

**Math 7510/6550, 7520/6560 Differential Geometry I, II (3, 3)**

Differentiable manifolds. Vector fields and flows. Tangent bundles. Frobenius theorem. Tensor fields. Differentiable forms, Lie derivatives. Integration and deRham’s theorem. Riemannian metrics, connections, curvature, parallel translation, geodesics, and submanifolds, including surfaces. First and second variation formulas, Jacobi fields, Lie groups. The Maurer-Cartan equation. Isometries, principal bundles, symmetric spaces, Kähler geometry.

**Math 7530, 7540 Partial Differential Equations I and II (3, 3)**

*Prerequisites: MATH 3050, 4060, 4470/6470/7310, 7210 and 7220 or by instructor's approval.*

Classical weak and strong maximum principles for 2nd order elliptic and parabolic equations, Hopf boundary point lemma, and their applications. Sobolev spaces, weak derivatives, approximation, density theorem, Sobolev inequalities, Kondrachov compact imbedding. L² theory for second order elliptic equations, existence via Lax-Milgram Theorem, Fredholm alternative, a brief introduction to L² estimates, Harnack inequality, eigenexpansion. L² theory for second order parabolic and hyperbolic equations, existence via Galerkin method, uniqueness and regularity via energy method. Semigroup theory applied to second order parabolic and hyperbolic equations. A brief introduction to elliptic and parabolic regularity theory, the Lp and Schauder estimates. Nonlinear elliptic equations, variational methods, method of upper and lower solutions, fixed point method, bifurcation method. Nonlinear parabolic equations, global existence, stability of steady states, traveling wave solutions. Conservation laws, Rankine-Hugoniot jump condition, uniqueness issue, entropy condition, Riemann problem for Burger's equation, p-systems.

**Math 7550 Probability Theory II**

*Prerequisites: MATH 6710/7210 and MATH 7150*

This course covers multiple topics in Probability theory at the PhD level. Measure theory is a requirement. The topics include the construction of stochastic processes and Brownian motion, martingale theory and stochastic calculus. The prerequisites are Math 6710/7210 and Math 7150.

**Math 7560 Stochastic Processes II**

*Prerequisites: MATH 6710/7210, MATH 7150 and MATH 6030/7030*

This course provides covers multiple topics in the theory of continuous time stochastic processes and stochastic analysis at the PhD level. This includes probability theory on metric spaces, Prohorov's theorem, weak convergence and tightness in the spaces C and D, stochastic differential equations and Markov processes.

**Math 7570 Scientific Computing II (3)**

*Prerequisites: MATH 7350.*

Floating point arithmetic (limitations and pitfalls). Numerical linear algebra, solving linear systems by direct and iterative models, eigenvalue problems, singular value decompositions, numerical integration, interpolation, iterative solution of nonlinear equations, unconstrained optimization.

**Math 7580 Scientific Computing III (3)**

*Prerequisites: MATH 7350 and 7570.*

Solution of ODE, both initial and boundary value problems. Numerical PDE. Introduction to fluid dynamics and other areas of application. Detailed Syllabus »

**Math 7710 - 7790 Special Topics (3)**

*Prerequisites: defined by the instructor.*

Courses on special topics list of subject titles include: Algebra, Analysis, Applied Math, Computation, Differential Equations, Geometry, Probability and Statistics, Theoretical Computer Science and Topology offered every year. Each course is designed to cover advanced material not included in one of the regular courses listed above.