MATE 6101. Number Theory II. Three credits. Prerequisites: MATE 3040, 4032. Fundamental non-analytic theory. Theory of congruences. Wilson and Euler theorems and applications. Sums of two squares. Primitive roots. Quadratic reciprocity law.
MATE 6102. Analytic Theory of Numbers. Three credits. Prerequisite: MATE 6101. Introduction to analytic number theory. Selberg and Erdös methods. The Prime Number Theorem.
MATE 6150. Linear Algebra. Three credits. Prerequisite: MATE 4031. Vector spaces over arbitrary fields. Bases. Linear transformations. Matrices. Dual space. Complex vector spaces. Elementary canonical forms. The Rational and Jordan canonical forms. Spectra of transformations. Tensor products. Bilinear forms.
MATE 6180. Introduction to Homological Algebra. Three credits. Prerequisite: MATE 6202. Exact sequences. Projective, injective, and flat modules. Categories. Abelian categories. Functors. Resolutions. Homology. Homological dimension and applications.
MATE 6200. Group Theory. Three credits. Prerequisite: MATE 4033. Fundamental properties of groups. Invariant subgroups. Isomorphism and homomorphism. Free groups. Direct and free products of groups. Permutation groups. Transformation groups.
MATE 6201. Modern Algebra I. Three credits. Prerequisite: MATE 4033. A survey of abstract algebra. Groups, rings, and fields. Introduction to Galois theory.
MATE 6202. Modern Algebra II. Three credits. Prerequisite: MATE 6201. Rings and ideals. Quotient rings. Ring homomorphism. Prime and maximal ideals. Nil radical and Jacobson radical. Modules, submodules, and quotient modules. Module homomorphisms. Finitely generated modules. Exact sequences. Tensor products of modules. Rings and modules of fractions. Primary decomposition. Integral domains.
MATE 6261. Functions of Real Variables I. Three credits. Prerequisite: MATE 5201. Overview of the following topics: set theory, real numbers, completeness of the real numbers, sequences and series, limits and continuity. Metric and topological spaces. Compactness, connectedness, and completeness. Sequences of functions. Uniform continuity and uniform convergence. Differentiation. Riemann-Stieltjes integration. Lebesgue measure and integration on the real line.
MATE 6262. Functions of Real Variables II. Three credits. Prerequisite: MATE 6261. Measure theory on sigma algebras. Inner and outer measure. Measurable functions. Convergence in measure. The Lebesgue integral for real functions of a real variable. The Radon-Nikodym theorem. Multiple integrals. Fubini’s Theorem. Lebesgue’s Theorem. LP spaces. Convergence in the mean.
MATE 6271. Mathematical Analysis I. Three credits. Prerequisite: MATE 5AAA (Advanced Calculus II). Introduction to logic and set theory. Vector spaces. Affine subspaces. Bilinearity. Differential calculus in n dimensional Euclidean space. The implicit function theorem. Taylor’s formula. Compactness and completeness. Metric spaces. Inner product spaces. Orthogonal transformations. Compact transformations. Differential equations.
MATE 6272. Mathematical Analysis II. Three credits. Prerequisite: MATE 6271. Multilinear functions. The exterior algebra. Integration in n dimensional Euclidean space. The change of variable formula. Integral calculus on manifolds. Exterior calculus. Differential forms. Stokes’ theorem.
MATE 6301. Functions of a Complex Variable. Three credits. Prerequisite: MATE 5201. Differentiation and integration of complex functions. Infinite series. Analytic functions. Analytic continuation. Multiple-valued functions. Conformal mappings.
MATE 6400. Series. Three credits. Prerequisite: MATE 5201. The study of series. Tests for convergence and divergence. Operations with series. Bernoulli and Euler numbers. Fourier series.
MATE 6460. Introduction to Functional Analysis. Three credits. Prerequisite: MATE 6540. Fundamental concepts of normed spaces and Banach spaces. Hilbert spaces. Weak convergence and closed transformations. Riesz-Schauder theory. Functions in Banach algebras. Spectral analysis in Hilbert spaces.
MATE 6530. Metric Differential Geometry. Three credits. Prerequisites: MATE 4031, 5201. Elementary theory of curves and surfaces. The fundamental forms. Euler’s theorem. The Codazzi-Mainardi equations. The fundamental theorem of surfaces. Curvature. Geodesics. The Gauss-Bonnet Theorem. Conformal mappings and isometries. Minimal and ruled surfaces.
MATE 6540. Introduction to Topology. Three credits. Prerequisite: MATE 5201. Topology of the line and plane. Abstract topological spaces. Subspaces. Relative topology. Bases and subbases. Continuity. Topological equivalence. Metric spaces. Product topology. Quotient spaces. Separation axioms. Hausdorff spaces. Compact spaces. Connected spaces. Complete metric spaces. Function spaces.
MATE 6545. Advanced Point Set Topology. Three credits. Prerequisite: MATE 6540. The Urysohn metrization theorem. The Tychonoff theorem. Completely regular spaces. The Stone-Čech compactification. Local finiteness. The Nagata-Smirnov metrization theorem. Paracompactness. Complete metric spaces. Compactness in metric spaces. Pointwise and compact convergence. The compact open topology. Ascoli’s theorem. Homotopy of paths. The fundamental group.
MATE 6551. Algebraic Topology I. Three credits. Prerequisite: MATE 6540. Study of topological problems using algebraic methods. Introduction to the theory of categories and functors. Homological algebra and homotopy theory.
MATE 6552. Algebraic Topology II. Three credits. Prerequisite: MATE 6551. Tensor products. Kunneth relations. Cohomological products. Fixed-point theorems. Duality theorems for manifolds.
MATE 6601. Probability and Statistics I. Three credits. Prerequisites: MATE 5001, 5002. Sample spaces. Random variables. Conditional probability and expectation. Moment generating functions. Markov chains. Poisson processes. Queuing theory. Renewal theory. Reliability. Martingales.
MATE 6602. Probability and Statistics II. Three credits. Prerequisite: MATE 6601. Populations and samples. Chi-square, t and F distributions. Estimation. Confidence sets. Simple and compound hypotheses tests. Decision theory. Linear models. Non-parametric methods. Statistics of time series.
MATE 6605. Applied Stochastic Models. Three credits. Prerequisite: MATE 6601. Applications of stochastic processes in areas such as queuing theory, reliability, inventory theory, decision theory, ecology, and population dynamics.
MATE 6606. Applied Stochastic Processes II. Three credits. Prerequisite: MATE 6605. Queueing theory. Single and multiserver queues. Fundamentals of reliability theory. Renewal processes. Semi-Markov processes. Regenerative processes. Applications to reliability theory.
MATE 6610. Sampling Theory. Three credits. Prerequisite: MATE 6602. Sample survey theory and design. Simple random, stratified, systematic, and cluster samples. Probability proportional to size sampling. Estimation of population parameters. Ratio, difference and regression type estimators. Use of auxiliary information. Confidence intervals. Optimum choice of sample size. Strata allocation. Selection probabilities. Double sampling and repetitive surveys. Non-sampling errors, randomized response. Sufficiency principle in sample survey model. Superpopulation models.
MATE 6611. Linear Models I. Three credits. Prerequisite: MATE 5002. Linear statistical models with emphasis on the mathematical foundations of the underlying theory. Linear algebra will be used particularly in the discussion of the main topics: regression and analysis of variance.
MATE 6612. Linear Models II. Three credits. Prerequisite: MATE 6611. Completely randomized block and Latin square designs. Fixed, random, and mixed models. Factorial experiments and confounding. Incomplete block designs. Response surfaces and analysis of covariance. Applications to data analysis.
MATE 6615. Bayesian Statistics and Decision Theory. Three credits. Prerequisite: MATE 6601. Elements of statistical decisions. Frequentist and Bayesian decision theory. Bayesian analysis: estimation, hypothesis testing, model selection, informative and non-informative priors. Approximations and Bayesian computations. Markov Chain Monte Carlo Methods. Introduction to Hierarchical Linear, Dynamic Linear, and Generalized Linear Models.
MATE 6650. Applied Linear Algebra. Three credits. Prerequisite: MATE 4031. Finite dimensional vector spaces. Matrix algebra. Systems of linear equations. Rank. Inverses. Eigenvalues. Linear programming. Canonical forms. Applications.
MATE 6656. Applied Algebra. Three credits. Prerequisite: MATE 4033. Introduction to concepts of the theory of semigroups, groups, rings, fields, and Boolean algebras. Applications to coding theory. Symmetry groups. Polya’s theory of counting. The theory of machines.
MATE 6680. Computational Analysis. Three credits. Prerequisite: MATE 5201. Introduction to the derivation and analysis of numerical methods. Numerical integration. Approximation of functions. Solution of equations. Eigenvalue problems. Solution of differential equations. Optimization problems.
MATE 6681. Data Structures I. Three credits. Prerequisite: MATE 5050 or MATE 5100. Data structures from a combinatorial viewpoint. Linear, circular, and linked lists. Trees. Binary trees. Applications to computer graphics. Operations on polynomials. Dynamic storage allocation and compilers.
MATE 6682. Data Structures II. Three credits. Prerequisite: MATE 6681. Methods for the design of efficient algorithms. Comparative study of searching and sorting algorithms.
MATE 6685. Computer Applications in Biology. Three credits. Prerequisite: MATE 3026 or 3028. Solution of problems in Bio-Mathematics and Biostatistics through the use of computers. Programming with the most used computer languages to solve problems in modeling and statistics.
MATE 6686. Advanced Data Analysis and Experimental Design. Three credits. Prerequisite: MATE 3026 or equivalent. This is a multidisciplinary course in which the student will be introduced to the most useful methods in Experimental Design and Data Analyses and their application to disciplines such as biology and chemistry, among others. Students will utilize computer-based analyses to increase their understanding and their command of the concepts and techniques learned in the course.
MATE 6690. Computational Analysis II. Three credits. Prerequisites: MATE 6680. Approximation of functions. Numerical differentiation and integration. Numerical methods for ordinary differential equations: initial value problems and boundary value problems. Numerical methods for partial differential equations.
MATE 6700. Projects in Applied Mathematics. Three credits. Prerequisite: Permission of the Director. Introduction to research in applied mathematics. Emphasis on the formulation and solution of problems from the real world in terms of mathematical models and the interpretation of those solutions in the context of the original problems.
MATE 6800. Graduate Seminar. Three credits. Prerequisite: Permission of the Director. Students will present material under the supervision of a faculty member. Choice of topic will be determined by the interests of the participants.
MATE 6881. Linear Programming. Three credits. Prerequisites: MATE 4031, 4100. Theory and algorithms for finite dimensional linear optimization. The Simplex method and its variants. Duality complementary slackness condition. Sensitivity. Decomposition. Dual and primal-dual methods. Optimization in graphs and networks. Combinatorial applications. Algorithmic complexity. Khachian’s algorithm. Karmarkar’s algorithm.
MATE 6882. Non-Linear Programming. Three credits. Prerequisites: MATE 5201, 6881. Theory and applications of non-linear finite dimensional optimization, both constrained and unconstrained. Optimality conditions. Descent method. Conjugate directions and quasi-Newton algorithms. Primal algorithms. Gradient method. Penalty and barrier methods. Duality. Convexification. Augmented Lagrangians. Quadratic programming. Lagrangian methods.
MATE 6896. Thesis Continued. Zero credits. This course will permit masters degree students who have completed all course and research credit requirements to maintain active student status while completing their thesis work.
MATE 6990. Independent Studies. 1-3 credits. Prerequisite: Permission of the Director. Research into some topic of interest by means of informal seminars, lectures, and independent research under the supervision of a member of the Department of Mathematics.
MATE 6996. Master’s Thesis. 1-3 credits. Prerequisite: Permission of the Advisor and Graduate Program Coordinator. Study and research leading to the preparation of a thesis.
MATE 8001. Graph Theory I. Three credits. Prerequisite: MATE 5CCC (Graph Theory). Review of elementary graph theory. Trees. Matrix-tree theorem. Enumerating spanning trees of a graph. Graph decompositions. Connectivity and k-connectivity of graphs. Disjoint paths and connectivity of graphs. Menger’s and Whitney’s theorems. Graph assembling and disassembling of 3-connected and quasi 4-connected graphs. Various Euler problems. Hamiltonian cycles. The k-closure of a graph. Bondy-Chvátal’s theorem. Long cycles in a graph. Dirac’s theorem. Factors. Maximum and perfect matchings. Tutte’s and Edmonds-Gallai’s theorems. Berge-Tutte duality theorem. Perfect graphs. Lovász’ theorem. Edge colorings. Vizing’s theorem. Independent sets and cliques. Vertex colorings. Brook’s theorem. Embeddings of graphs in the plane. Kuratowski’s, Whitney’s, MacLane’s, and Kelmans’ planarity criteria. Five-colour and four-colour theorems on planar graphs. On the hamiltonicity of planar graphs.
MATE 8005. Enumerative Combinatorics I. Three credits. Prerrequites: MATE 6150, MATE 6201, MATE 8001. Review of elementary combinatorics. Outline of the main problems and approaches of enumerative combinatorics. Enumerating trees. Matrix-tree theorem. Coding of trees. Counting Euler cycles in a digraph. Counting and listing of non-isomorphic trees of different types. Generating function method in enumerative combinatorics. Enumerating graphs of different types. Pólyas’s counting theory of non-isomorphic objects. Enumerating non-isomorphic graphs of different types. Principle of inclusion and exclusion. Lattices, their Möbius functions and Möbius algebras. Asymptotic results in enumerative combinatorics.
MATE 8015. Discrete Algorithms. Three credits. Prerequisites: MATE 5CCC (Graph Theory). Efficient algorithms are sought for solving problems in discrete mathematics. Different algorithms and applications to various problems are studied through special reading assignments, lecture presentations, and group discussions. Branch and bound strategy. Dynamic programming principle. Optimal paths in graphs. Optimal spanning trees in graphs. 2-coloring and odd cycles in a graph. Depth-first search in a graph and its applications. Properties of depth-first search tree. Graph decomposition algorithms. Graph assembling algorithms. Graph planarity algorithms. Euler problems. Hamiltonian problems. Some packing and covering problems for graphs. Metric problems on graphs. Set transformation algorithms. Search trees of different types. Various sorting algorithms. The main ideas of the NP-theory (the theory of problem complexity).
MATE 8021. Algebraic Combinatorics I. Three credits. Prerequisites: MATE 6202, MATE 8001. The study of the extremal properties of algebraic structures whose symmetries are of special interest and important for applications in computer and communication sciences. The following topics are studied in depth through special reading assignments, lecture presentations, and group discussions. Linear codes. t-error correcting binary codes and finite fields. Cyclic codes. Perfect codes. Block designs. Latin squares. Pairwise balanced designs. Hadamard matrices. Difference sets. Symmetric designs. Finite geometries. Singer’s theorem. Transversal designs. Group divisible designs. Steiner triple systems. Kirkman triple systems. Strongly regular graphs. Association schemes. Distance-regular graphs. Distance transitivity.
MATE 8031. Combinatorial Optimization I. Three credits. Prerequisites: MATE 6881, MATE 8001. Elements of linear and integer programming: branch and bound methos and its application to combinatorial optimization problems. Network flow theory and its generalizations: statistical maximal flow, feasibility theorems and combinatorial applications, minimal cost flow problems, multi-terminal maximal flows, multi-commodity flows. Matching theory and its generalizations: matchings in bipartite graphs, size and structure of maximum matchings, bipartite graphs with perfect matchings, general graphs with perfect matchings, some graph-theoretical problems related to matchings, matchings and linear programming, matching algorithms, the f-factor problem, vertex packing and covering, some generalizations of matching problems.
MATE 8041. Matroid Theory I. Three credits. Prerequisites: MATE 6150, MATE 8001. Fundamental concepts and axioms of matroid theory. Duality in matroid and matroid operations. Vector representation of matroids. The matroid of a graph and graph planarity. Greedy algorithms on matroids. The union of matroids and its rank function. Efficient algorithms for some combinatorial optimization problems (packing, covering, intersection, etc.) on matroids with applications to a variety of combinatorial objects (e.g. graphs, matrices, algebraic dependencies, transversals).
MATE 8051. Convex Polytopes. Three credits. Prerequisite: MATE 6150. The basic concepts of linear and affine geometry. Convex sets and their support properties. Supporting hyperplanes. The theorems of Radon, Helly, and Caratheodory. Convex polytopes and their faces. Polarity and duality in convex polytopes. Cell-complexes and Schlegel diagrams. Shelling the boundary complexes. The cubical complexes. The graph of a d-polytope and its properties. 3-polytopes and Steinitz’ theorem. Affine and projective transformations. The fundamental theorem of projective geometry. Simplicial and simple polytopes. Euler’s theorem and the Dehn-Sommerville equations. Lower bound and upper bound theorems for convex polytopes.
MATE 8309. Complex Analysis II. Three credits. Prerequisite: MATE 6301. Analytic continuation. Algebraic functions. Elliptic functions. Entire and meromorphic functions. Normal families. Conformal mappings.
MATE 8465. Spectral Theory and Differential Equations. Three credits. Prerequisites: MATE 6460, MATE 8469. Unbounded operators. Graphs and symmetric operators. Self-adjointness. Cayley transforms. Extensions of symmetric operators. Resolutions of the identity and the spectral theorem. Stone’s theorem. Introduction to distributions. Sobolev spaces. Variational methods. The Laplace operator. The heat equation.
MATE 8469. Functional Analysis II. Three credits. Prerequisite: MATE 6460. Banach algebras. Commutative Banach algebras. Bounded operators on a Hilbert space. Unbounded operators. Topological vector spaces. Duality. Compact operators. Distributions and their applications to the theory of partial differential equations.
MATE 8605. Simulation and the Monte Carlo Method. Three credits. Prerequisite: MATE 6606. Random number generation. Random variate generation. Monte Carlo integration. Variance reduction techniques. Simulation of stochastic processes. Regenerative methods for simulation analysis. Simulation of queueing systems. Monte Carlo optimization. Simulation languages. Statistical evaluation of simulation results. Markov Chain Monte Carlo Methods (MCMC). Gibbs Sampler. Metropolis-Hastings Algorithm. Applications.
MATE 8680. Iterative Solution of Systems of Nonlinear Equations. Three credits. Prerequisites: MATE 6680, MATE 6690. Gâteaux and Fréchet derivatives. Mean value theorems and second derivatives. Contractions. Inverse and implicit function theorems. Sard’s theorem. Monotone operators. Degree theory. Properties of the degree and basic existence theorems. General iterative methods. Newton and secant methods and generalizations. Continuation methods. Predictor-corrector methods. Numerical bifurcation theory.
MATE 8685. Parallel Algorithms-Design and Analysis. Three credits. Prerequisite: MATE 6682. Parallel architecture classifications. Parallel computational models. Network models such as arrays, trees, and hypercubes. Efficiency and scalability measures for parallel algorithms. Techniques of parallel algorithm design. Primitives for parallel algorithm design. Efficient parallel algorithms for integer and matrix computations, fast Fourier transform, sorting and graph problems. Algorithm mappings for network models. Data routing algorithms.
MATE 8800. Doctoral Seminar. Three credits. Prerequisite: Permission of the Graduate Program Coordinator. Advanced seminar in research areas related to the doctoral program.
MATE 8899. Doctoral Dissertation Continued. Zero credits. Prerequisite: Permission of the Graduate Program Coordinator. This course will permit doctoral degree students who have completed all course and research credit requirements to maintain active student status while completing their dissertation work.
MATE 8980. Topics in Pure Mathematics. 1-3 credits. Prerequisite: Permission of the professor. Topics will be chosen according to the interests and availability of faculty and students.
MATE 8985. Topics in Pure Mathematics. 1-3 credits. Prerequisite: Permission of the professor. Topics will be chosen according to the interests and availability of faculty and students.
MATE 8986. Topics in Discrete Mathematics. 1-3 credits. Prerequisite: Permission of the professor. Topics will be chosen according to the interests and availability of faculty and students.
MATE 8990. Topics in Applied Mathematics. 1-3 credits. Prerequisite: Permission of the professor. Topics will be chosen according to the interests and availability of faculty and students.
MATE 8991. Doctoral Dissertation. 1-3 credits. Prerequisite: Permission of the Graduate Program Coordinator. Study and research leading to the preparation of the doctoral dissertation.
MATE 8995. Topics in Computational Mathematics. 1-3 credits. Prerequisite: Permission of the professor. Topics will be chosen according to the interests and availability of faculty and students.