# MATH - Mathematics

**MATH401 Applications of Linear Algebra (3 Credits)**

Various applications of linear algebra: theory of finite games, linear programming, matrix methods as applied to finite Markov chains, random walk, incidence matrices, graphs and directed graphs, networks and transportation problems.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH461, MATH240, MATH341).

**MATH402 Algebraic Structures (3 Credits)**

For students having only limited experience with rigorous mathematical proofs. Parallels MATH403. Students planning graduate work in mathematics should take MATH403. Groups, rings, integral domains and fields, detailed study of several groups; properties of integers and polynomials. Emphasis is on the origin of the mathematical ideas studied and the logical structure of the subject.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH341, MATH461).

**Restriction:** Must not be in any of the following programs (Mathematics (Master's); Mathematics (Doctoral)).

**MATH403 Introduction to Abstract Algebra (3 Credits)**

Integers; groups, rings, integral domains, fields.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH461, MATH340); and 1 course with a minimum grade of C- from (MATH341, MATH241); and minimum grade of C- in MATH310. Or students who have taken courses with comparable content may contact the department.

**MATH404 Field Theory (3 Credits)**

Algebraic and transcendental elements, Galois theory, constructions with straight-edge and compass, solutions of equations of low degrees, insolubility of the quintic equation, Sylow theorems, fundamental theorem of finite Abelian groups.

**Prerequisite:** Minimum grade of C- in MATH403.

**MATH405 Linear Algebra (3 Credits)**

An abstract treatment of finite dimensional vector spaces. Linear transformations and their invariants.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH461, MATH341); and minimum grade of C- in MATH310.

**MATH406 Introduction to Number Theory (3 Credits)**

Integers, divisibility, prime numbers, unique factorization, congruences, quadratic reciprocity, Diophantine equations and arithmetic functions.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH241, MATH246, MATH340, MATH341, MATH461); or permission of CMNS-Mathematics department.

**MATH410 Advanced Calculus I (3 Credits)**

Subjects covered: sequences and series of numbers, continuity and differentiability of real-valued functions of one variable, the Riemann integral, sequences of functions and power series.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH461, MATH341); and 1 course with a minimum grade of C- from (MATH340, MATH241); and minimum grade of C- in MATH310.

**MATH411 Advanced Calculus II (3 Credits)**

Continuation of MATH410. Topics include: The topology of sets in Rn, the derivative matrix, the general chain rule, inverse and implicit function theorems with applications, smooth curves and surfaces in R3, Lagrange multipliers. Additional topics may include: Metric spaces, the contraction principle, the existence and uniqueness theorem for nonlinear first order differential equations, the Riemann integral of Rn, introduction to integration on curves and surfaces, Green's theorem.

**Prerequisite:** Minimum grade of C- in MATH410; and permission of CMNS-Mathematics department.

**MATH416 Applied Harmonic Analysis: An Introduction to Signal Processing (3 Credits)**

Introduces students to the mathematical concepts arising in signal analysis from the applied harmonic analysis point of view. Topics include applied linear algebra, Fourier series, discrete Fourier transform, Fourier transform, Shannon Sampling Theorem, wavelet bases, multiresolution analysis, and discrete wavelet transform.

**Prerequisite:** Minimum grade of C- in MATH141; and 1 course with a minimum grade of C- from (MATH240, MATH461, MATH341); and familiarity with MATLAB is required.

**MATH420 Mathematical Modeling (3 Credits)**

The course will develop skills in data-driven mathematical modeling through individual and group projects. Emphasis will be placed on both analytical and computational methods, and on effective oral and written presentation of results.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH461, MATH341); and 1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH246, MATH341); and 1 course with a minimum grade of C- from (STAT400, STAT410); and 1 course with a minimum grade C- from (CMSC106, CMSC131). Cross-listed with: AMSC420.

**MATH423 Linear Optimization (3 Credits)**

Mathematical formulation of linear programming, graphical solutions, simplex method, duality, transportation problems, assignment problems, and game theory.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH401, MATH461, or MATH341).

**Credit Only Granted for:** MATH423 or MATH498T.

**Formerly:** MATH498T.

**MATH424 Introduction to the Mathematics of Finance (3 Credits)**

Introduction to the mathematical models used in finance and economics with emphasis on pricing derivative instruments. Designed for students in mathematics, computer science, engineering, finance and physics. Financial markets and instruments; elements from basic probability theory; interest rates and present value analysis; normal distribution of stock returns; option pricing; arbitrage pricing theory; the multiperiod binomial model; the Black-Scholes option pricing formula; proof of the Black-Scholes option pricing formula and applications; trading and hedging of options; Delta hedging; utility functions and portfolio theory; elementary stochastic calculus; Ito's Lemma; the Black-Scholes equation and its conversion to the heat equation.

**Prerequisite:** Minimum grade of C- in MATH141; and 1 course with a minimum grade of C- from (STAT400, STAT410); and permission of CMNS-Mathematics department.

**Recommended:** MATH246, MATH240, MATH241, MATH340, or MATH341.

**MATH430 Euclidean and Non-Euclidean Geometries (3 Credits)**

Hilbert's axioms for Euclidean geometry. Neutral geometry: the consistency of the hyperbolic parallel postulate and the inconsistency of the elliptic parallel postulate with neutral geometry. Models of hyperbolic geometry. Existence and properties of isometries.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH341, MATH461).

**MATH431 Geometry for Computer Applications (3 Credits)**

Topics from projective geometry and transformation geometry, emphasizing the two-dimensional representation of three-dimensional objects and objects moving about in the plane and space. The emphasis will be on formulas and algorithms of immediate use in computer graphics.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH461, MATH240, MATH341).

**MATH432 Introduction to Topology (3 Credits)**

Metric spaces, topological spaces, connectedness, compactness (including Heine-Borel and Bolzano-Weierstrass theorems), Cantor sets, continuous maps and homeomorphisms, fundamental group (homotopy, covering spaces, the fundamental theorem of algebra, Brouwer fixed point theorem), surfaces (e.g., Euler characteristic, the index of a vector field, hairy sphere theorem), elements of combinatorial topology (graphs and trees, planarity, coloring problems).

**Prerequisite:** Minimum grade of C- in MATH410.

**MATH436 Differential Geometry of Curves and Surfaces I (3 Credits)**

Curves in the plane and Euclidean space, moving frames, surfaces in Euclidean space, orientability of surfaces; Gaussian and mean curvatures; surfaces of revolution, ruled surfaces, minimal surfaces, special curves on surfaces, "Theorema Egregium"; the intrinsic geometry of surfaces.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH461, MATH240, MATH341); and must have completed two 400-level MATH courses with a minimum grade of C- (not including MATH461, and 480's).

**MATH437 Differential Forms (3 Credits)**

Introduction to differential forms and their applications, and unites the fundamental theorems of multivariable calculus in a general Stokes Theorem that is valid in great generality. It develops this theory and technique to perform calculations in analysis and geometry. Topics include an introduction to topological spaces, the Gauss-Bonnet Theorem, Gauss's formula for the linking number, and the Cauchy Integral Theorem. Applications include Maxwell's equations of electromagnetism, connections and gauge theory, and symplectic geometry and Hamiltonian dynamics.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH240, MATH341, MATH461).

**Recommended:** MATH405, MATH403, MATH436, MATH410, or MATH432.

**MATH445 Elementary Mathematical Logic (3 Credits)**

Elementary development of propositional and predicate logic, including semantics and deductive systems and with a discussion of completeness, incompleteness and the decision problem.

**Prerequisite:** Minimum grade of C- in MATH141.

**MATH446 Axiomatic Set Theory (3 Credits)**

Development of a system of axiomatic set theory, choice principles, induction principles, ordinal arithmetic including discussion of cancellation laws, divisibility, canonical expansions, cardinal arithmetic including connections with the axiom of choice, Hartog's theorem, Konig's theorem, properties of regular, singular and inaccessible cardinals.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH403, MATH410).

**MATH452 Introduction to Dynamics and Chaos (3 Credits)**

An introduction to mathematical dynamics and chaos. Orbits, bifurcations, Cantor sets and horseshoes, symbolic dynamics, fractal dimension, notions of stability, flows and chaos. Includes motivation and historical perspectives, as well as examples of fundamental maps studied in dynamics and applications of dynamics.

**Prerequisite:** MATH341; or MATH246 and one of (MATH240 or MATH461). Cross-listed with: AMSC452.

**MATH456 Cryptography (3 Credits)**

The theory, application, and implementation of mathematical techniques used to secure modern communications. Topics include symmetric and public-key encryption, message integrity, hash functions, block-cipher design and analysis, number theory, and digital signatures.

**Prerequisite:** (CMSC106, CMSC131, or ENEE150; or equivalent programming experience); and (2 courses from (CMSC330, CMSC351, ENEE324, or ENEE380); or any one of these courses and a 400-level MATH course, or two 400-level MATH courses); and Permission of CMNS-Mathematics department or permission of instructor. Cross-listed with: CMSC456, ENEE456.

**MATH461 Linear Algebra for Scientists and Engineers (3 Credits)**

Basic concepts of linear algebra. This course is similar to MATH240, but with more extensive coverage of the topics needed in applied linear algebra: change of basis, complex eigenvalues, diagonalization, the Jordan canonical form.

**Prerequisite:** Minimum grade of C- in MATH141; and must have completed a MATH or STAT course with a prerequisite of MATH141.

**Credit Only Granted for:** MATH240, MATH341, or MATH461.

**Additional Information:** This course may not be used towards the upper level math requirements for MATH/STAT majors.

**MATH462 Partial Differential Equations (3 Credits)**

Linear spaces and operators, orthogonality, Sturm-Liouville problems and eigenfunction expansions for ordinary differential equations. Introduction to partial differential equations, including the heat equation, wave equation and Laplace's equation. Boundary value problems, initial value problems and initial-boundary value problems.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH246, MATH341).

**MATH463 Complex Variables (3 Credits)**

The algebra of complex numbers, analytic functions, mapping properties of the elementary functions. Cauchy integral formula. Theory of residues and application to evaluation of integrals. Conformal mapping.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH241, MATH340).

**MATH464 Transform Methods (3 Credits)**

Fourier transform, Fourier series, discrete fast Fourier transform (DFT and FFT). Laplace transform. Poisson summations, and sampling. Optional Topics: Distributions and operational calculus, PDEs, Wavelet transform, Radon transform and applications such as Imaging, Speech Processing, PDEs of Mathematical Physics, Communications, Inverse Problems.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH246, MATH341).

**MATH470 Mathematics for Secondary Education (3 Credits)**

An advanced perspective on some of the core mathematics underlying high school mathematics courses. Topics include number systems, functions of one variable, equations, inequalities, trigonometric functions, curve fitting, and polynomials. The course includes an analysis of alternate approaches to mathematical ideas and problems, and makes connections between ideas that may have been studied separately in different high school and college courses.

**Prerequisite:** MATH141 and MATH140; and must have completed one 400-level MATH course (not to include MATH461, 478, and 480's).

**Restriction:** Must be in the Secondary Math Education major.

**MATH475 Combinatorics and Graph Theory (3 Credits)**

General enumeration methods, difference equations, generating functions. Elements of graph theory, matrix representations of graphs, applications of graph theory to transport networks, matching theory and graphical algorithms.

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH341, MATH461); and 1 course with a minimum grade of C- from (MATH241, MATH340); and permission of CMNS-Mathematics department. Cross-listed with CMSC475.

**MATH478 Selected Topics For Teachers of Mathematics (1-3 Credits)**

**Prerequisite:** Permission of CMNS-Mathematics department.

**Additional Information:** Math majors may not use this course to fulfill the upper-level math requirement.

**MATH480 Algebra for Middle School Teachers (3 Credits)**

Prepares teachers with elementary certification to teach Algebra 1 in middle school. Focuses on basic algebra concepts and related theoretical ideas.

**Prerequisite:** MATH214.

**Restriction:** Must be a middle school teacher; and permission of CMNS-Mathematics department.

**Credit Only Granted for:** MATH480 or MATH483.

**Additional Information:** Not applicable to MATH/STAT major or minor requirements.

**MATH481 Statistics and Data Analysis for Middle School Teachers (3 Credits)**

Prepares teachers with elementary certification to teach simple data analysis and probability in middle school. Focuses on understanding basic statistics, data analysis, and related theoretical ideas.

**Prerequisite:** MATH214.

**Restriction:** Must be a middle school teacher; and permission of CMNS-Mathematics department.

**Credit Only Granted for:** MATH481 or MATH485.

**Additional Information:** Not applicable to MATH/STAT major or minor requirements.

**MATH484 Geometry for High School Teachers (3 Credits)**

Focuses on concepts related to geometry, including several geometry axiom schemes, transformations, and similarity. Includes constructions with Geometer's Sketchpad.

**Prerequisite:** MATH141; or students who have taken courses with comparable content may contact the department.

**Restriction:** Senior standing.

**Credit Only Granted for:** MATH482, MATH484, or MATH498E.

**Formerly:** MATH498E.

**MATH487 Number for Middle Grades Teachers (3 Credits)**

The rational number and proportional reasoning concepts developed in the middle grades and the larger mathematical context for these. Multiple representations of relationships, including verbal descriptions, diagrams, tables, graphs, and equations. Common misconceptions.

**Prerequisite:** Must have admission to M.A. or M.Ed. with concentration in Mathematics Education; or permission of CMNS-Mathematics department.

**Restriction:** This course may not be used towards the upper level math requirements for the MATH/STAT major.

**Credit Only Granted for:** MATH487 or MATH498K.

**Formerly:** MATH498K.

**MATH489 Research Interactions in Mathematics (1-3 Credits)**

Students participate in a vertically integrated (undergraduate, graduate and/or postdoctoral, faculty) mathematics research group. Format varies. Students and supervising faculty will agree to a contract which must be approved by the department. Up to three credits of MATH489 may be applied to the mathematics degree requirements. See the department's MATH489 online syllabus for further information.

**Prerequisite:** Permission of CMNS-Mathematics department.

**Repeatable to: ** 10 credits if content differs.

**MATH498 Selected Topics in Mathematics (1-9 Credits)**

Topics of special interest to advanced undergraduate students will be offered occasionally under the general guidance of the departmental committee on undergraduate studies.

**Repeatable to: ** 9 credits if content differs.

**MATH600 Abstract Algebra I (3 Credits)**

Groups with operators, homomorphism and isomorphism theorems, normal series, Sylow theorems, free groups, Abelian groups, rings, integral domains, fields, modules. Topics may include HOM (A,B), Tensor products, exterior algebra.

**Prerequisite:** MATH405 and MATH403; or students who have taken courses with comparable content may contact the department.

**MATH601 Abstract Algebra II (3 Credits)**

Field theory, Galois theory, multilinear algebra. Further topics from: Dedekind domains, Noetherian domains, rings with minimum condition, homological algebra.

**Prerequisite:** MATH600.

**MATH602 Homological Algebra (3 Credits)**

Projective and injective modules, homological dimensions, derived functors, spectral sequence of a composite functor. Applications.

**Prerequisite:** MATH600.

**MATH603 Commutative Algebra (3 Credits)**

Ideal theory of Noetherian rings, valuations, localizations, complete local rings, Dedekind domains.

**Prerequisite:** MATH600.

**MATH606 Algebraic Geometry I (3 Credits)**

Prime and primary ideals in Noetherian rings, Hilbert Nullstellensatz, places and valuations, prevarieties (in the sense of Serre), dimension, morphisms, singularities, varieties, schemes, rationality.

**MATH607 Algebraic Geometry II (3 Credits)**

Topics in contemporary algebraic geometry chosen from among: theory of algebraic curves and surfaces, elliptic curves, Abelian varieties, theory of schemes, theory of zeta functions, formal cohomology, algebraic groups, reduction theory.

**Prerequisite:** MATH606.

**MATH620 Algebraic Number Theory I (3 Credits)**

Algebraic numbers and algebraic integers, algebraic number fields of finite degree, ideals and units, fundamental theorem of algebraic number theory, theory of residue classes, Minkowski's theorem on linear forms, class numbers, Dirichlet's theorem on units, relative algebraic number fields, decomposition group, inertia group and ramification group of prime ideals with respect to a relatively Galois extension.

**Prerequisite:** MATH601.

**MATH621 Algebraic Number Theory II (3 Credits)**

Valuation of a field, algebraic function fields, completion of a valuation field, ramification exponent and residue class degree, ramification theory, elements, differents, discriminants, product formula and characterization of fields by the formula, Gauss sum, class number formula of cyclotomic fields.

**Prerequisite:** MATH600. And MATH620; or students who have taken courses with comparable content may contact the department.

**MATH630 Real Analysis I (3 Credits)**

Lebesgue measure and the Lebesgue integral on R, differentiation of functions of bounded variation, absolute continuity and fundamental theorem of calculus, Lp spaces on R, Riesz-Fischer theorem, bounded linear functionals on Lp, measure and outer measure, Fubini's theorem.

**Prerequisite:** MATH411; or students who have taken courses with comparable content may contact the department.

**MATH631 Real Analysis II (3 Credits)**

Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, Radon-Nikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, Banach-Steinhaus theorem, topological spaces, Arzela-Ascoli and Stone-Weierstrass theorems, compact sets and Tychonoff's theorem.

**Prerequisite:** MATH630.

**MATH632 Functional Analysis (3 Credits)**

Introduction to functional analysis and operator theory: normed linear spaces, basic principles of functional analysis, bounded linear operators on Hilbert spaces, spectral theory of selfadjoint operators, applications to differential and integral equations, additional topics as time permits.

**Prerequisite:** MATH631.

**MATH634 Harmonic Analysis (3 Credits)**

L1 theory: Fejer theorem, inversion theorem, ideal structure, Tauberian theorem. L2 theory: Plancherel-Parseval theorems, Paley-Wiener theorem. Lp theory: Hausdorff-Young theorem. Distribution theory: Bochner's theorem, Wiener continuous measures theorem, Malliavin theorem, Schwartz theory, almost periodic functions.

**Prerequisite:** MATH630.

**MATH636 Representation Theory (3 Credits)**

Introduction to representation theory of Lie groups and Lie algebras; initiation into non-abelian harmonic analysis through a detailed study of the most basic examples, such as unitary and orthogonal groups, the Heisenberg group, Euclidean motion groups, the special linear group. Additional topics from the theory of nilpotent Lie groups, semisimple Lie groups, p-adic groups or C-algebras.

**Prerequisite:** MATH631.

**MATH642 Dynamical Systems I (3 Credits)**

Foundations of topological dynamics, homeomorphisms, flows, periodic and recurrent points, transitivity and minimality, symbolic dynamics. Elements of ergodic theory, invariant measures and sets, ergodicity, ergodic theorems, mixing, spectral theory, flows and sections. Applications of dynamical systems to number theory, the Weyl theorem, the distribution of values of polynomials, Vander Waerden's theorem on arithmetic progressions.

**Prerequisite:** MATH432. And MATH630; or students who have taken courses with comparable content may contact the department.

**MATH643 Dynamical Systems II (3 Credits)**

Entropy theory, variational principle for the entropy, expansiveness, measures with maximal entropy. Smooth systems on manifolds, diffeomorphisms and flows, periodic points, stable and unstable manifolds, homoclinic points, transversality, the Krupka-Smale theorem, Morse-Smale systems. Hyperbolicity, Anosov systems, distributions and foliations, strange attractors, Bowen's measure.

**Prerequisite:** MATH642; or students who have taken courses with comparable content may contact the department.

**MATH660 Complex Analysis I (3 Credits)**

Linear transformations, analytic functions, conformal mappings, Cauchy's theorem and applications, power series, partial fractions and factorization, elementary Riemann surfaces, Riemann's mapping theorem.

**Prerequisite:** MATH410 and MATH463; or students who have taken courses with comparable content may contact the department.

**MATH661 Complex Analysis II (3 Credits)**

Introduction to techniques of several complex variables, with focus on geometric topics: Hartogs phenomena, Cousin problems, Dolbeault cohomology, L2 techniques, embedding of Stein manifolds, and the theory of coherent analytic sheaves.

**MATH669 Selected Topics in Riemann Surfaces (1-3 Credits)**

Construction of Riemann surfaces, hyperbolic geometry, Fuchsian and Kleinian groups, potential theory, uniformisation spaces of meromorphic functions, line bundles, Picard variety, Riemann-Roch, Teichmueller theory.

**Restriction:** Permission of instructor.

**Repeatable to: ** 99 credits if content differs.

**MATH670 Ordinary Differential Equations I (3 Credits)**

Existence and uniqueness, linear systems usually with Floquet theory for periodic systems, linearization and stability, planar systems usually with Poincare-Bendixson theorem.

**MATH671 Ordinary Differential Equations II (3 Credits)**

The content of this course varies with the interests of the instructor and the class. Stability theory, control, time delay systems, Hamiltonian systems, bifurcation theory, and boundary value problems.

**MATH673 Partial Differential Equations I (3 Credits)**

Analysis of boundary value problems for Laplace's equation, initial value problems for the heat and wave equations. Fundamental solutions, maximum principles, energy methods. First order nonlinear PDE, conservation laws. Characteristics, shock formation, weak solutions. Distributions, Fourier transform.

**Prerequisite:** MATH411; or students who have taken courses with comparable content may contact the department. Cross-listed with: AMSC673.

**MATH674 Partial Differential Equations II (3 Credits)**

Boundary value problems for elliptic partial differential equations via operator-theoretic methods. Hilbert spaces of functions. Duality, weak convergence. Sobolev spaces. Spectral theory of compact operators. Eigenfunction expansions.

**Prerequisite:** MATH673 or AMSC673; or permission of instructor. Cross-listed with: AMSC674.

**MATH675 Analysis and PDEs (3 Credits)**

Introduction to tools of modern analysis which have been used in recent years in the study of partial differential equations: Fourier transform, Calderon-Zygmund theory, interpolation, Lebesgue spaces, Lorentz spaces, Sobolev spaces, Besov spaces, Littlewood-Paley theory, multipliers, Bernstein inequalities, the fractional Leibniz rule, Strichartz estimates, velocity averaging lemma. Applications to some of the following PDEs: the Navier-Stokes equations, Euler equations, nonlinear Schrodinger equations, nonlinear wave equations, the Patlak Keller Segel model.

**Prerequisite:** MATH630 or MATH674; or permission of instructor.

**MATH689 Research Interactions in Mathematics (1-3 Credits)**

The students participate in a vertically integrated (undergraduate, graduate and/or postdoctoral, faculty) research group. Format varies, but includes regular meetings, readings and presentations of material. See graduate program's online syllabus or contact the graduate program director for more information.

**Restriction:** Permission of instructor.

**Repeatable to: ** 6 credits if content differs.

**MATH695 Teaching Seminar (1 Credit)**

A course intended for first year teaching assistants. Topics include: everyday mechanics of teaching; teaching methods and styles; technology; course enrichment, diversity in the classroom; sexual harassment; teacher-student interactions; presentations by students.

**Restriction:** Must be in one of the following programs (Mathematics (Doctoral); Mathematics (Master's); Mathematical Statistics (Doctoral); Applied Mathematics and Scientific Computation (Master's); Mathematical Statistics (Master's); Applied Mathematics and Scientific Computation (Doctoral)).

**MATH712 Mathematical Logic I (3 Credits)**

Sentential logic, first-order languages, models and formal deductions. Basic model theory including completeness and compactness theorems, other methods of constructing models, and applications such as non-standard analysis.

**MATH713 Mathematical Logic II (3 Credits)**

Incompleteness and undecidability results of Godel, Church, Tarski and others. Recursive function. Basic proof theory and axiomatic set theory.

**Prerequisite:** MATH712.

**MATH730 Fundamental Concepts of Topology (3 Credits)**

Survey of basic point set topology, fundamental group, covering spaces, Van Kampen's theorem, simplicial complexes, simplicial homology, Euler characteristics and classification of surfaces.

**Prerequisite:** MATH403, MATH410, and MATH411; or students who have taken courses with comparable content may contact the department.

**MATH734 Algebraic Topology (3 Credits)**

Singular homology and cohomology, cup products, Poincare duality, Eilenberg-Steenrod axioms, Whitehead and Hurewicz theorems, universal coefficient theorem, cellular homology.

**Prerequisite:** MATH600; and MATH432 or MATH730; or students who have taken courses with comparable content may contact the department.

**MATH740 Fundamental Concepts of Differential Geometry (3 Credits)**

Manifolds, tangent vectors and differential forms, Riemannian metrics, connections, curvature, structure equations, geodesics, calculus of variations.

**Prerequisite:** MATH405, MATH411, and MATH730; or students who have taken courses with comparable content may contact the department.

**MATH742 Geometric Analysis (3 Credits)**

Calculus of Variations, Bochner technique, Morse theory, weak solutions and elliptic regularity, maximum principle for elliptic and parabolic equations, Green's function of the Laplacian, isoperimetric and Sobolev inequalities, continuity method, curvature and comparison results, harmonic maps, curvature prescription problems.

**MATH744 Lie Groups I (3 Credits)**

An introduction to the fundamentals of Lie groups, including some material on groups of matrices and Lie algebras.

**Prerequisite:** MATH405, MATH403, MATH411, and MATH432; or students who have taken courses with comparable content may contact the department.

**MATH799 Master's Thesis Research (1-6 Credits)**

**MATH808 Selected Topics in Algebra (1-3 Credits)**

Advanced topics of current interest.

**Restriction:** Permission of instructor.

**Repeatable to: ** 18 credits.

**MATH818 Selected Topics in Logic (1-3 Credits)**

Advanced topics of current interest.

**Restriction:** Permission of instructor.

**Repeatable to: ** 18 credits.

**MATH848 Selected Topics in Geometry and Topology (1-3 Credits)**

Advanced topics of current interest.

**Restriction:** Permission of instructor.

**Repeatable to: ** 18 credits.

**MATH858 Selected Topics in Analysis (1-3 Credits)**

Advanced topics of current interest.

**Restriction:** Permission of instructor.

**Repeatable to: ** 18 credits.

**MATH868 SelectedTopics in Complex Analysis (1-3 Credits)**

Advanced topics of current interest.

**Restriction:** Permission of instructor.

**Repeatable to: ** 18 credits.

**MATH898 Pre-Candidacy Research (1-8 Credits)**

**MATH899 Doctoral Dissertation Research (1-8 Credits)**