# Courses

### MATH 100 / ELEMENTS OF FINITE MATHEMATICS AND CALCULUS

Tools for quantitative reasoning and basic college level mathematical concepts for social science students. Mathematics of finance, linear equations and matrices, probability, game theory, derivative, integral, special functions: log, exp, trigonometric and function sketching techniques.Credits: 3

### MATH 101 / FINITE MATHEMATICS

Linear algebra and matrix theory; mathematics of finance; counting and the fundamentals of probability theory; game theory.Credits: 3

### MATH 102 / CALCULUS

Limit of a function; Continuous functions and their properties; Derivative andapplications; Extreme values; Indefinite integral; Riemann integral and fundamental theorem of calculus; Logarithmic and exponential functions; L?Hospital?s rule; Sequence and series of numbers; Power series and their properties;Credits: 3

### MATH 103 / INTRODUCTION TO ABSTRACT MATHEMATICS

Sets; logic and implications; proof techniques with examples; mathematical induction and well-ordering; equivalence relations; functions; cardinality; countable and uncountable sets.Credits: 3

### MATH 104 / DISCRETE MATHEMATICS

Counting problems; combinatorial methods; integers, divisibility and primes; graphs and trees; combinatorics in geometry; introduction to complexity and cryptography.Credits: 3

### MATH 106 / CALCULUS I

Limits and continuity; derivative and properties of differentiable functions; mean value theorems, Taylor's formula, extreme values; indefinite integral and integral rules; Riemann integral and fundamental theorem of calculus; L'Hospital's rule; improper integrals.Credits: 3

### MATH 107 / INTRODUCTION TO LINEAR ALGEBRA

Vectors; matrices and systems of linear equations; vector spaces; linear maps; orthogonality; algebra of complex numbers; eigenvalue problems.Credits: 3

### MATH 200 / MATRIX ALGEBRA AND MULTIVARIABLE CALCULUS

Vector spaces; matrices and determinants; systems of linear equations; eigenvalue problems; diagonalization of matrices. Functions of several variables: partial derivatives, directional derivatives, optimization problems, multiple integrals.Credits: 3

Prerequisite: MATH. 102 or MATH. 106

### MATH 201 / STATISTICS

Descriptive statistics; measures of association, correlation, simple regression; probability theory, conditional probability, independence; random variables and probability distributions; sampling distributions; estimation; inference (confidence intervals and hypothesis testing). Topics are supported by computer applications.Credits: 3

Prerequisite: (MATH.100 or MATH.101 or MATH.106) or consent of the instructor

### MATH 202 / STATISTICS FOR SOCIAL SCIENCES

A course in basic concepts and tools of statistics for students who will study social and Behavioral sciences. Topics to be covered are representation of quantitative information in social sciences, forms of numerical data, creating and interpreting graphical and tabular summaries of data, descriptive statistics, estimation of population parameters, confidence intervals, basic hypothesis testing, t-statistics, chi-squared tests and analysis of variance.Credits: 3

### MATH 203 / MULTIVARIABLE CALCULUS

Functions of several variables; partial differentiation; directional derivatives; exact differentials; multiple integrals and their applications; vector analysis; line and surface integrals; Green?s, Divergence and Stoke?s theorems.Credits: 3

Prerequisite: MATH. 106 or consent of the instructor

### MATH 204 / DIFFERENTIAL EQUATIONS

First order differential equations. Second order linear equations. Series solutions of ODE?s. The Laplace transform and applications. Systems of first order linear equations. Nonlinear equations and systems:existence, uniqueness and stability of solutions. Fourier series and partial differential equations.Credits: 3

Prerequisite: MATH. 107 or consent of the instructor

### MATH 205 / ALGEBRA I

Natural numbers; modular arithmetic; introduction to groups; cyclic and permutation groups; homomorphisms and isomorphisms; normal; factor, simple and free groups; introduction to rings, integral domains, and fields; factor rings and ideals; extension fields; outline of Galois theory.Credits: 3

Prerequisite: MATH. 103 or consent of the instructor

### MATH 206 / ALGEBRA II

Natural numbers; modular arithmetic; introduction to groups; cyclic and permutation groups; homomorphisms and isomorphisms; normal; factor, simple and free groups; introduction to rings, integral domains, and fields; factor rings and ideals; extension fields; outline of Galois theory.Credits: 3

Prerequisite: MATH. 205 or consent of the instructor

### MATH 207 / COMMUTATIVE ALGEBRA

Review of Unique Factorization Domains and Principal IdeaI Domains, Maximal and prime ideals, Nilradical, Local rings, Modules, Cayley-Hamilton theorem, Nakayama's lemma, Exact and split exact sequences, Noetherian rings, Noetherian modules, Hilbert basis theorem, Integral extensions, Integral closure, Non-singularity, Normal rings, Noether normalization, Hilbert nullstellensatz, Spec(A), Localization, Support of a module and the associated primes, Discrete valuation rings, Trace and separability, Completion, Artin-Rees Lemma, An overview of further topics: Dimension theory, Regular rings, Connections with geometric notions.Credits: 3

### MATH 208 / ADVANCED CALCULUS

Completeness axiom for real numbers; convergent sequences; compactness; continuous functions; differentiation; linear and topological structure of Euclidean spaces; limit, compactness and connectedness in a Euclidean space; continuity and differentiation of functions of several variables; inverse and implicit function theorems.Credits: 3

Prerequisite: MATH. 103 and MATH 107 or consent of the instructor

### MATH 211 / STATISTICS FOR SCIENCES

Descriptive statistics; Probability; Random variables; Special distributions; Estimation; Hypothesis testing; Normal distribution; Two-Sample Inference; Regression.Credits: 3

Prerequisite: MATH. 106 or consent of the instructor

### MATH 301 / REAL ANALYSIS I

Metric spaces and their topology; continuity, compactness and connectedness in a metric space; completion of a metric space; differentiation and Riemann integration; sequences and series of functions; uniform convergence; Ascoli-Arzela theorem; Stone-Weierstrass theorem; Banach fixed-point theorem and its applications.Credits: 3

Prerequisite: MATH. 208 or consent of the instructor

### MATH 302 / ELEMENTS OF FUNCTIONAL ANALYSIS

Normed and Banach spaces; linear operators; duality; inner product and Hilbert spaces; Riesz representation theorem; Hahn-Banach theorem; uniform boundedness principle; open mapping theorem; strong, weak and weak* convergence.Credits: 3

Prerequisite: MATH. 301 and MATH 320 or consent of the instructor

### MATH 303 / APPLIED MATHEMATICS

Review of vector calculus; Fourier series and Fourier transform; Calculus of functions of a complex variable.Credits: 3

Prerequisite: MATH. 204 or consent of the instructor

### MATH 304 / NUMERICAL METHODS

Zeros of functions; solution of linear systems; interpolation; numerical integration and differentiation; numerical solution of differential equations; estimation of error; stability analysis.Credits: 3

Prerequisite: MATH. 203 or consent of the instructor

### MATH 309 / PARTIAL DIFFERENTIAL EQUATIONS

Classification of second order partial equations; well posed problems; method of separation of variables and applications; wave equation: D?Alambert?s solution; Laplace equation: Poisson?s formula, maximum principle, boundary value and eigenvalue problems; heat equation: Cauchy problem, maximum principle.Credits: 3

Prerequisite: MATH. 204 or consent of the instructor

### MATH 310 / NUMBER THEORY

Quadratic Reciprocity, Quadratic Forms, Gauss' Composition Law and Genus Theory, Cubic and Biquadratic Reciprocity, Number Fields, Hilbert Class Field, Orders in Imaginary Quadratic Fields, The Class Number, Class Field Theory and Cebatorev Density Theorem, Norms and Ideles, Elliptic Functions and Theory of Complex Multiplication. Divisibility, Primes, Congruences, Prime Modules and Primitive Roots, Groups, a review of Rings and Fields, Arithmetic Functions, Diophantine Problems, Farey Fractions ad Geometry of Numbers, Continued Fractions, Multiplicative Number Theory and Dirichlet Series.Credits: 3

Prerequisite: MATH. 205 or consent of the instructor

### MATH 312 / MATHEMATICAL STATISTICS

Decision theory; estimation; confidence intervals; hypotheses testing; large-sample theory; efficiency of alternative statistical procedures.Credits: 3

Prerequisite: MATH. 203 or consent of the instructor

### MATH 313 / PROBABILITY THEORY

Review of elementary probability; multivariate random variables and their functions; conditional distribution and expectation; generating functions and transforms; order statistics; multivariate normal distribution; types of convergence; laws of large numbers; central limit theorem.Credits: 3

Prerequisite: Math 203 and Math 211 or consent of the instructor

### MATH 320 / LINEAR ALGEBRA

Finite-dimensional real and complex vector spaces, bases of a vector space, linear maps, dual spaces, quadratic forms, self-adjoint and unitary transformations, eigenvalue problem, canonical form of a linear transformation, tensors, and applications.Credits: 3

Prerequisite: MATH. 107 or consent of the instructor

### MATH 350 / SELECTED TOPICS IN MATHEMATICS I

Detailed examination of current topics in Mathematics.Credits: 3

Prerequisite: MATH. 203 or consent of the instructor

### MATH 351 / SELECTED TOPICS IN MATHEMATICS II

Detailed examination of current topics in Mathematics.Credits: 3

Prerequisite: MATH. 203 or consent of the instructor

### MATH 390 / INDEPENDENT STUDY I

Investigation of one or more topics of interest with the guidance of an instructor. Presentation of a research proposal at the end of the term.Credits: 3

### MATH 395 / INDEPENDENT STUDY

Investigation of one or more topics of interest with the guidance of an instructor. Presentation of a research proposal at the end of the term.Credits: 1.5

### MATH 401 / COMPLEX ANALYSIS

Complex numbers and functions; exponential and trigonometric functions; infinite series and products; limits, continuity and derivatives of complex functions; Cauchys theorem; Taylor and Laurent series; conformal mapping.Credits: 3

Prerequisite: MATH. 301 or consent of the instructor

### MATH 402 / TOPOLOGY

Topological spaces, subspaces, continuous functions, base for a topology, separation axioms, compactness, locally compact spaces, connectedness, path connectedness, finite product spaces, set theory and Zorn?s lemma, infinite product spaces, quotient spaces, homotopic paths, the fundamental group,induced homomorphisms, covering spaces, applications of the index, homotopic maps, maps into the punctured plane, vector fields, the Jordan curve theorem.Credits: 3

Prerequisite: MATH. 301 or consent of the instructor

### MATH 403 / FUNCTIONAL ANALYSIS

Basic principles of normed spaces. Normed and Banach spaces; Hilbert spaces; linear operators; dual spaces. Basic principles of functional analysis: Hahn-Banach theorem; open mapping theorem; uniform boundedness theorem, Krein-Milman theorem. Applications.Credits: 3

Prerequisite: MATH. 301 or consent of the instructor

### MATH 404 / GRAPH THEORY

Fundamental concepts in graph theory; trees; matchings in graphs; connectivity and planarity; the colorings of graphs and diagraphs; Hamilton cycles; matroids.Credits: 3

### MATH 405 / DIFFERENTIAL GEOMETRY

Differential geometry of curves and surfaces in three-dimensional space; intrinsic geometry; geodesics; curvature; Gauss-Bonnett theorem.Credits: 3

Prerequisite: MATH. 203

### MATH 406 / ACTUARIAL MATHEMATICS

Contingency mathematics in the areas of life and health insurance, annuities, and pensions from both the probabilistic and deterministic approaches. Survival distribution and life tables; life insurance; life annuities; net premiums; net premium reserves; multiple life functions; multiple decrement models; valuation theory for pension plans; the expense factor; and non-forfeiture benefits and dividends.Credits: 3

### MATH 407 / COMBINATORIAL ANALYSIS

Problems of enumeration, structure, and optimization in such finite or discrete systems as graphs, matroids, partially ordered sets, lattices, partitions, codes and block designs.Credits: 3

### MATH 408 / GAME THEORY

Games in extensive form; pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations and solution concepts; related topics and applications.Credits: 3

### MATH 409 / OPTIMIZATION

Convergence of sequences in Rn, multivariate Taylor's theorem. Optimality conditions for unconstrained optimization. Newton's and quasi-Newton methods for unconstrained optimization. Equality-constrained optimization, Karush-Kuhn-Tucker theorem for constrained optimization. Inequality-constrained optimization. Interior point methods for constrained optimization. Linear and quadratic programs, their numerical solution.Credits: 3

### MATH 411 / STOCHASTIC CALCULUS AND STOCHASTIC SYSTEMS I

Modeling of stochastic systems. Introduction to Markov chains, renewal processes, queuing theory, reliability and time series models; Ito Calculus, Fokker-Planck and Kolmogorov differential equations; applications to the problems of environmental as well as physical systems such as allocation of resources, inventory control, transportation and finance.Credits: 3

### MATH 412 / STOCHASTIC CALCULUS AND STOCHASTIC SYSTEMS II

Modeling of stochastic systems. Introduction to Markov chains, renewal processes, queuing theory, reliability and time series models; Ito Calculus, Fokker-Planck and Kolmogorov differential equations; applications to the problems of environmental as well as physical systems such as allocation of resources, inventory control, transportation and finance.Credits: 3

### MATH 414 / ALGEBRAIC GEOMETRY

Basic notions of commutative algebra and homological algebra: category of modules over a ring, flatness, Ext and Tor. General properties of schemes: affine schemes. projective schemes, dimension, projective and proper morphisms. Normal and regular schemes. Flat and smooth morphisms. Zariski's main theorem and applications. Coherent sheaves and Cech Cohomology.Credits: 3

Prerequisite: MATH. 206 or consent of the instructor

### MATH 450 / SELECTED TOPICS IN MATHEMATICS I

Detailed examination of current topics in Mathematics.Credits: 3

### MATH 451 / SELECTED TOPICS IN MATHEMATICS II

Detailed examination of current topics in Mathematics.Credits: 3

Prerequisite: MATH. 302 or consent of the instructor

### MATH 490 / INDEPENDENT STUDY II

Work on the research proposal resulting from MATH 390 with the guidance of an instructor, culminating in a research paper suitable for presentation or publication.Credits: 3

### MATH 491 / HONORS PROJECT

Available to students with a GPA equal to or greater than 3.00 and with consent of the instructor.Credits: 3

### MATH 495 / INDEPENDENT STUDY

Credits: 1.5

### MATH 503 / APPLIED MATHEMATICS I

Linear algebra: Vector and inner product spaces, linear operators, eigenvalue problems; Vector calculus: Review of differential and integral calculus, divergence and Stokes' theorems. Ordinary differential equations: Linear equations, Sturm-Liouville theory and orthogonal functions, system of linear equations; Methods of mathematics for science and engineering students.Credits: 3

### MATH 504 / NUMERICAL METHODS I

A graduate level introduction to matrix-based computing. Stable and efficient algorithms for linear equations, least squares and eigenvalue problems. Both direct and iterative methods are considered and MATLAB is used as a computing environment.Credits: 3

### MATH 505 / APPLIED MATHEMATICS II

Calculus of variations; Partial differential equations: First order linear equations and the method of characteristics; Solution of Laplace, wave, and diffusion equations; Special functions; Integral equations.Credits: 3

### MATH 506 / NUMERICAL METHODS II

Development and analysis of numerical methods for ODEs, an introduction to numerical optimization methods, and an introduction to random numbers and Monte Carlo simulations. The course starts with a short survey of numerical methods for ODEs. The related topics include stability, consistency, convergence and the issue of stiffness. Then it moves to computational techniques for optimization problems arising in science and engineering. Finally, it discusses random numbers and Monte Carlo simulations. The course combines the theory and applications (such as programming in MATLAB) with the emphasis on algorithms and their mathematical analysis.Credits: 3

### MATH 509 / OPTIMIZATION

Convergence of sequences in Rn, multivariate Taylor's theorem. Optimality conditions for unconstrained optimization. Newton's and quasi-Newton methods for unconstrained optimization. Equality-constrained optimization, Karush-Kuhn-Tucker theorem for constrained optimization. Inequality-constrained optimization. Interior point methods for constrained optimization. Linear and quadratic programs, their numerical solution.Credits: 3

### MATH 510 / ADVANCED ORDINARY DIFFERENTIAL EQUATIONS

Existence and uniqueness theorems; continuation of solutions; continuous dependence and stability, Lyapunovs direct method; differential inequalities and their applications; boundary-value problems and Sturm-Liouville theory.Credits: 4

### MATH 511 / PARTIAL DIFFERENTIAL EQUATIONS I

First order equations, method of characteristics; the Cauchy-Kovalevskaya theorem; Laplace's equation: potential theory and Greens?s function, properties of harmonic functions, the Dirichlet problem on a ball; heat equation: the Cauchy problem, initial boundary-value problem, the maximum principle; wave equation: the Cauchy problem, the domain of dependence, initial boundary-value problem.Credits: 4

### MATH 512 / PARTIAL DIFFERENTIAL EQUATIONS II

Review of functional spaces and embedding theorems; existence and regularity of solutions of boundary-value problems for second-order elliptic equations; maximum principles for elliptic and parabolic equations; comparison theorems; existence, uniqueness and regularity theorems for solutions of initial boundary-value problems for second-order parabolic and hyperbolic equations.Credits: 4

Prerequisite: MATH. 511 or consent of the instructor

### MATH 514 / ALGEBRAIC GEOMETRY

Basic notions of commutative algebra and homological algebra: category of modules over a ring, flatness, Ext and Tor. General properties of schemes: affine schemes. projective schemes, dimension, projective and proper morphisms. Normal and regular schemes. Flat and smooth morphisms. Zariski's main theorem and applications. Coherent sheaves and Cech Cohomology.Credits: 3

Prerequisite: MATH. 206 or consent of the instructor

### MATH 521 / ALGEBRA I

Free groups, group actions, group with operators, Sylow theorems, Jordan-Hölder theorem, nilpotent and solvable groups. Polynomial and power series rings, Gauss?s lemma, PID and UFD, localization and local rings, chain conditions, Jacobson radical.Credits: 4

### MATH 522 / ALGEBRA II

Galois theory, solvability of equations by radicals, separable extensions, normal basis theorem, norm and trace, cyclic and cyclotomic extensions, Kummer extensions. Modules, direct sums, free modules, sums and products, exact sequences, morphisms, Hom and tensor functors, duality, projective, injective and flat modules, simplicity and semisimplicity, density theorem, Wedderburn-Artin theorem, finitely generated modules over a principal ideal domain, basis theorem for finite abelian groups.Credits: 4

### MATH 525 / ALGEBRATIC NUMBER THEORY

Valuations of a field, local fields, ramification index and degree, places of global fields, theory of divisors, ideal theory, adeles and ideles, Minkowski's theory, extensions of global fields, the Artin symbol.Credits: 4

### MATH 527 / NUMBER THEORY

Method of descent, unique factorization, basic algebraic number theory, diophantine equations, elliptic equations, p-adic numbers, Riemann zeta function, elliptic curves, modular forms, zeta and L-functions, ABC-conjecture, heights, class numbers for quadratic fields, a sketch of Wiles? proof.Credits: 4

### MATH 528 / ANALYTIC NUMBER THEORY

Primes in arithmetic progressions, Gauss' sum, primitive characters, class number formula, distribution of primes, properties of the Riemann zeta function and Dirichlet L-functions, the prime number theorem, Polya- Vinogradov inequality, the large sieve, average results on the distribution of primes.Credits: 4

Prerequisite: MATH. 533 or consent of the instructor

### MATH 531 / REAL ANALYSIS I

Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, Radon-Nikodym theorem, extension of measures, Fubini's theorem.Credits: 4

### MATH 532 / REAL ANALYSIS II

Normed and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, Baire category and uniform boundedness theorems, strong, weak and weak*-convergence, open mapping theorem, closed graph theorem.Credits: 4

Prerequisite: MATH. 531 or consent of the instructor

### MATH 533 / COMPLEX ANALYSIS I

Review of the complex number system and the topology of C, elementary properties and examples of analytic functions, complex integration, singularities, maximum modulus theorem, compactness and convergence in the space of analytic functions.Credits: 4

### MATH 534 / COMPLEX ANALYSIS II

Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function.Credits: 4

Prerequisite: MATH. 533 or consent of the instructor

### MATH 535 / FUNCTIONAL ANALYSIS

Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, Krein-Milman theorem and applications, Schauder fixed point theorem, Krein-Shmulian theorem, Eberlein-Shmulian theorem, linear operators on Banach spaces.Credits: 4

Prerequisite: MATH. 532 or consent of the instructor

### MATH 536 / APPLIED FUNCTIONAL ANALYSIS I

Review of linear operators in Banach spaces and Hilbert spaces; Riesz ·Schauder theory; fixed point theprems of Banach and Schauder; semigroups of linear operators; Sobolev spaces and basic embedding theorems; boundary - value problems for elliptic equations; eigenvalues and eigenvectors of second order elliptic operators; initial boundary-value problems for parabolic and hyperbolic equations.Credits: 4

### MATH 537 / APPLIED FUNCTIONAL ANALYSIS II

Existence and uniqueness of solutions of abstract evolutionary equations. Global non-existence and blow up theorems. Applications to the study of the solvability and asymptotic behavior of solutions of initial boundary-value problems for reaction diffusion equations, Navier-Stokes equations, nonlinear Klein-Gordon equations and nonlinear Schrödinger equations.Credits: 4

### MATH 538 / DIFFERENTIAL GEOMETRY

Differentiable manifolds; differentiable forms; integration on manifolds; de Rhamm cohomology; connections and curvatureCredits: 4

### MATH 541 / PROBABILITY THEORY

An introduction to measure theory, Kolmogorov axioms, independence, random variables, product measures and joint probability, distribution laws, expectation, modes of convergence for sequences of random variables, moments of a random variable, generating functions, characteristic functions, distribution laws, conditional expectations, strong and weak law of large numbers, convergence theorems for probability measures, central limit theorems.Credits: 4

### MATH 544 / STOCHASTIC PROCESS AND MARTINGALES

Stochastic processes, stopping times, Doob-Meyer decomposition, Doob's martingale convergence theorem, characterization of square integrable martingales, Radon-Nikodym theorem, Brownian motion, reflection principle, law of iterated logarithms.Credits: 4

Prerequisite: MATH. 541 or consent of the instructor

### MATH 545 / MATHEMATICS OF FINANCE

From random walk to Brownian motion, quadratic variation and volatility, stochastic integrals, martingale property, Ito formula, geometric Brownian motion, solution of Black-Scholes equation, stochastic differentialequations, Feynman-Kac theorem, Cox-Ingersoll-Ross and Vasicek term structure models, Girsanov's theorem and risk neutral measures, Heath-Jarrow-Morton term structure model, exchange-rate instruments.Credits: 4

### MATH 550 / SELECTED TOPICS IN PROBABILITY AND STATISTICS

Credits: 3

### MATH 551 / SELECTED TOPICS IN ANALYSIS I

Credits: 3

### MATH 552 / SELECTED TOPICS IN ANALYSIS II

Credits: 3

### MATH 553 / SELECTED TOPICS IN FOUNDATIONS OF MATHEMATICS

Credits: 3

### MATH 554 / SELECTED TOPICS IN ALGEBRA AND TOPOLOGY

Credits: 3

### MATH 555 / SELECTED TOPICS IN PROBABILITY AND STATISTICS

Credits: 3

### MATH 556 / SELECTED TOPICS IN DIFFERENTIAL GEOMETRY

Credits: 3

### MATH 557 / SELECTED TOPICS IN DIFFERENTIAL EQUATIONS

Credits: 3

### MATH 558 / SELECTED TOPICS IN APPLIED MATHEMATICS

Credits: 3

### MATH 559 / SELECTED TOPICS IN COMBINATORICS

Credits: 3

### MATH 563 / ALGEBRIC CODING THEORY

Error correcting coding theory. Hamming, Golay, cyclic, 2-error correcting BCH codes, Reed-Solomon, Convolutional, Reed-Muller and Preparata codes. Interaction of codes and combinatorial designs.Credits: 4

### MATH 564 / COMBINATORIAL DESIGN THEORY

Balanced incomplete block designs, group divisible designs and pairwise balanced designs. Resolvable designs, symmetric designs and designs having cyclic automorphisms. Pairwise orthogonal latin squares. Affine and projective geometries. Embeddings and nestings of designs.Credits: 4

### MATH 565 / GRAPH THEORY

Matchings, edge colorings and vertex colorings of graphs. Connectivity, spanning trees, and disjoint paths in graphs. Cycles in graphs, embeddings. Planar graphs, directed graphs. Ramsey Theory, matroids, random graphs.Credits: 4

### MATH 566 / TOPICS IN MODULE AND RING THEORY

Generalities on modules, categories, and functors. The socle and the Jacobson radical of a module. Semisimple modules. Chain conditions on modules. The Hopkins-Levitzki Theorem. The Wedderburn-Artin Theorem and its applications toM linear representations of finite groups. The ?Hom? functors and exactness. Injective modules. Essential monomorphisms, injective hulls. Projective modules. Superfluous epimorphisms, projective covers. Indecomposable direct sum decompositions of modules. The Krull-Remak-Schmidt-Azumaya Theorem. Krull dimension and Goldie dimension of modules and lattices.Credits: 3

### MATH 571 / TOPOLOGY

Topological spaces, subspaces, continuous functions, base for a topology, separation axioms, compactness, locally compact spaces, connectedness, path connectedness, finite product spaces, set theory and Zorn?s lemma, infinite product spaces, quotient spaces, homotopic paths, the fundamental group,induced homomorphisms, covering spaces, applications of the index, homotopic maps, maps into the punctured plane, vector fields, the Jordan curve theorem.Credits: 4

### MATH 572 / ALGEBRAIC TOPOLOGY

Fundamental group, Seifert-van Kampen theorem, CW complexes, covering spaces and deck transformations; simplicial and singular homology, homotopy invariance, exact sequences and excision, cellular homology, Mayer-Vietoris sequences; cohomology, universal coefficient theorem, cup product, Kunneth formula, orientation, Poincare duality.Credits: 4

### MATH 579 / READINGS IN MATHEMATICS

Literature survey and presentation on a subject determined by the instructor.Credits: 1

### MATH 584 / SELECTED TOPICS IN ANALYSIS&ALGEBRATIC NUBERS

Credits: 3

### MATH 590 / GRADUATE SEMINAR

Credits: 0

### MATH 591 / THESIS

Credits: 0

### MATH 592 / TERM PROJECT

Individual term project accompanied with the advisor.Credits: 0

### MATH 695 / PHD THESIS

Credits: 0